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01 Matter and Energy
Chemistry is all about matter, its properties and transformations and the associated energy changes. The properties
of a sample of matter are dictated by its composition and structure. For chemists, the smallest functional building blocks of
matter are atoms. For a defined grouping of atoms (a system), the identity of atoms involved and their spatial (3dimensional) arrangements determine, among other properties, the energy of that assembly. Changes in those
arrangements are accompanied by changes in energy. For transformations where atom identities are preserved, the
mass-conservation and energy-conservation laws govern the changes. For nuclear reactions, where atoms may change
their identity, mass-energy is preserved (E = mc2).
01-1 Atomic structure
An atom consists of a nucleus, occupied by neutrons and positively-charged protons, and a negatively-charged electron
cloud that surrounds the nucleus and is attracted to the protons within it by electrostatic force. The chemical identity of an atom (its
assigned element symbol) is determined by the number of protons in the nucleus (i.e. its atomic number). The elements often exist
as a mixture of isotopes—variants of atoms with the same atomic number but a different number of neutrons.
01-2 Avogadro’s number
On a macroscopic scale, chemists “count” atoms by weighing. Since the atomic mass unit (amu) is defined such that 1 g of a
substance contains Avogadro’s number of such units, the mass in g that is numerically equal to the atomic mass in amu (for
atoms) or molecular mass in amu (for molecules) contains Avogadro’s number of atoms or molecules. Such a number of atoms or
molecules is defined as a mole.
01-3 Ions and molecules
Atoms rarely exist in unchanged atomic form. Their electronic structure is modified when they become ions by loss or gain of
electrons, or when they combine to form molecules or extended networks by rearranging their electron clouds. The details of the
3-dimensional arrangements of atoms in such compounds are responsible for overall energy and other properties.
01-4 Introduction to energy
Chemical systems have two dominant energy components: the potential energy of the electrostatic interactions between
charged particles, and the kinetic energy of particles in motion. Energy can be exchanged with the surroundings through the
transfer of heat and work. Adding heat or performing work on the system increases the energy of the system (ΔE > 0), while
removing heat or having the system perform work decreases its energy (ΔE < 0).
01-5 Enthalpy
For processes taking place under constant pressure, the heat exchanged between the system and the surroundings is called
enthalpy. Enthalpy is an extensive function (dependent on the amount of matter) and is a function of state. It depends on the state
(P, V, T), but not on the path used to reach that state. Changes in enthalpy are a convenient measure of changes in the internal
energy of a system that is undergoing a chemical or physical transformation. Processes with ΔH < 0 are called exothermic, and
those with ΔH > 0 are endothermic.
01-6 Light energy
Electromagnetic radiation, the purest form of energy, sometimes exhibits wave-like properties and is typically characterized
by its wavelength (λ) or frequency (ν). The electromagnetic spectrum covers a wide range of frequencies, all of which have
energies proportional to ν. A narrow part of that spectrum, the visible range, is detectable by human eyes. Under other conditions,
electromagnetic radiation behaves as a stream of photons (small packets of energy). This situation-dependent dual behavior
(wave-like and particle-like) is common in the atomic world, where energy is granular rather than continuous. The emission of light
from material objects (black-body radiation) and the light-induced ejection of electrons from metal surfaces (the photoelectric
effect) provide windows on the quantum nature of that microscopic world.
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01-7 Spectroscopy
The study of how photons of light interact with matter is called spectroscopy. Light may be transmitted or reflected by the
system under study, typically without energy being exchanged, or it may be absorbed or emitted by the system in processes that
convert one form of energy into other forms. These energy conversion processes are quantized, providing an opportunity to probe
the internal structure of matter. Visible light spectroscopy illustrates how the intensity of light of all colors is affected when the light
beam passes through a sample, with some wavelengths being absorbed by the molecules in its path. In general, the results of
light absorption depend strongly on the energy of the absorbed photons. Typically, highly energetic photons (gamma, X-ray,
ultraviolet) ionize atoms and molecules, and break bonds whereas lower energy photons (visible, infrared, microwave, radio
waves) do not damage the molecules, but in all cases most of the delivered energy eventually leads to increased kinetic energy of
the sample, i.e. it is converted to heat.
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01-1 Atomic structure
Atoms are composed of protons and neutrons held in the nucleus
surrounded by an electron cloud
Ordinary human senses perceive the world on a macroscopic scale (down to 10–3 m sizes), but the actual chemistry
happens on a nanoscopic scale (10–9 m or less). On this scale, the basic building blocks of matter are atoms. The kinds
of atoms present (composition) and their arrangements (structure) are responsible for the observed macroscopic
properties and behavior of matter. Chemists must think in terms of miniscule and directly invisible atoms to be able to
understand the workings of the tangible substances of the macroscopic world.
Atomic sizes and masses are very small. Atoms are 100 to 500 pm in diameter (1 to 5 Å where 1 Å = 100 pm = 10–
m), with the heaviest having masses on the order of 10–22 g. They are built from even smaller subatomic particles, only
three of which have bearing on chemical behavior: the proton, the neutron, and the electron. Protons and neutrons
reside in the extremely small nucleus of the atom (ca. 10–14 m) and account for virtually all of the mass of the atom. The
vast majority of volume of the atom, on the other hand, is essentially empty space occupied by a “cloud” of rapidly moving
electrons, which contribute negligibly to the mass of the atom.
10
Figure F01-1-1. The structure of an atom
showing the nucleus (composed of protons
and neutrons) surrounded by the electron
cloud. In all drawings the nucleus is shown
out of proportions to the size of the atom. If
drawn on scale, it would be unnoticeably
small.
Electrons and protons are held together by an attractive electrostatic force
The electrons do not fly away from the atom because they are attracted to the protons in the nucleus by an
electrostatic force that is proportional to the magnitude of the charges (Q1 and Q2) on the interacting particles and
inversely proportional to the square of the distance (r) between them. That relationship, known as Coulomb’s law, is
shown below, where k is just a proportionality constant:
Fel =
kQ1 Q2
r
E01-1-1
2
An electron has a negative charge of –1.602 × 10–19 coulombs (C). The charge of the proton is equal in magnitude
to that of an electron, but has the opposite sign (+1.602 × 10–19 C). Neutrons, as the name indicates, have no charge (are
neutral). Atoms, as a whole, have no net charge, as the number of electrons is equal to the number of protons.
Because both the masses and charges of the elementary particles comprising atoms are so small, for convenience
atomic units have been defined in such a manner that electric charge (au) is expressed as a multiple of the electron
charge, and the atomic mass unit (amu or just u) is 1.66054 × 10–24 g; exactly 1/12 of the mass of carbon-12 which
contains 6 protons and 6 neutrons in its nucleus (see below). The charge and masses of the components of atoms are
collected in Table T01-1-1.
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Table T01-1-1. Atomic components
Particle
Charge (C)
Charge (au)
Mass (kg)
Mass (amu)
proton
p+
+1.602 × 10−19
+1
1.6726 × 10−27
1.0073
neutron
n0
0
0
1.6749 × 10−27
1.0087
electron
e−
−1.602 × 10−19
−1
9.1094 × 10−31
5.486 × 10−4
The chemical identity of an atom is designated by an atomic number, and is determined by the number of protons in
the nucleus. Any one element from the periodic table is a collection of one or more atoms with the same number of
protons in their nuclei. The variety of material objects in our world is the result of various combinations of any of just about
100 different elements.
Isotopes are atoms with the same atomic number but a different number of neutrons
Atoms of a given element can differ in the number of neutrons in their nuclei and consequently in their mass. Such
atoms are called isotopes of one another. For example (Figure F01-1-2), in addition to the most abundant isotope of
hydrogen, protium (99.98%), two other isotopes are known: deuterium, with one neutron in the nucleus (0.0156%
abundance), and unstable (radioactive) tritium, with two neutrons in the nucleus (4 × 10–15% abundance). To distinguish
all the possibilities, chemists have introduced a special notation. The mass number superscript lists the number of protons
and neutrons, while the atomic number subscript lists the number of protons.
Figure F01-1-2. The naturally occurring isotopes of hydrogen: protium, deuterium (2H = D), and tritium (3H = T). For
illustration purposes, nuclei are out of proportion to atomic sizes.
Since the atomic number corresponds to the atomic symbol, the use of both is redundant, and the atomic number is
typically omitted (Figure F01-1-3). For example, both 12C and carbon-12 are unambiguous representations of the
dominant isotope of carbon (atomic number 6).
Figure F01-1-3. Examples of isotopes of carbon, oxygen and uranium. For readability nuclear sizes are out of proportion
to atomic sizes.
Essentially all elements have multiple isotopes that occur naturally or can be synthetically produced in nuclear
reactions, but the majority of them are unstable. When an element occurs in nature as a mixture of isotopes, its atomic
weight (AW) is the average of the masses of all isotopes present (E01-1-2).
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AW = ∑ [(isotope mass) × (f ractional natural abundance)]
E01-1-2
For example, naturally occurring carbon is composed of 98.93% of the carbon-12 isotope with atomic mass of 12
amu (exactly ) and 1.07% of the carbon-13 isotope with atomic mass of 13.00335. The atomic weight of carbon, as found
in the periodic table, can be calculated as follows (E01-1-3):
AW (C) = (0.9893)(12 amu) + (0.0107)(13.00335 amu) = 12.01 amu
E01-1-3
The atomic masses and natural abundances of various isotopes can be very precisely determined in this day and
age using mass spectrometry. In this technique, electrons are “kicked out” of atoms in a vacuum and the resulting ions
travel in a magnetic field that bends their paths according to their mass/charge ratio. Special ion detectors are then used
to calculate the ratios of ions with different masses.
It is important to note that in chemical reactions the nuclei of atoms remain unchanged. Only the electron clouds are
modified. Thus, no atoms can be transformed into other atoms, and, of course, no atoms may be destroyed or created.
This observation is the basis of the law of conservation of mass.
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01-2 Avogadro’s number
Avogadro's number is a conversion factor for relating grams to atomic mass units
Let’s return for a moment to the idea of the atomic mass unit (amu). The unit is defined as exactly 1/12 of the mass of
one atom of the carbon-12 isotope. From precise measurements we now know that 1 amu = 1.66054 × 10–24 g.
We can “invert” that number, 1 g = 6.02214 × 1023 amu, to learn how many atomic mass units there are in one
gram. That number should look very familiar. Indeed, it's Avogadro’s number (NA)! This number, which is fundamental to
chemistry, was originally defined by studying gases (see Lesson 12-3) and trying to connect macroscopic measurements
(such as the mass of the sample) with the nanoscopic world (such as the number of atoms or other matter particles in that
sample).
Imagine that you are asked to count identical small yellow plastic spheres in a large container. It would take a long
time, and the chance of making errors would be large. How would you do it if the spheres were of atomic size? It turns
out that we can use a balance for counting purposes. If the mass of one sphere were known, then a simple division of the
total mass (obtained by weighing all of the spheres together) by the mass of one sphere would give a count of the
spheres. By analogy, if the mass of an individual atom is known, then the atom count of any element's sample can be
obtained by measuring its total mass.
Figure F01-2-1. Weighing of yellow spheres. How many
spheres are there in the Erlenmeyer flask, if the total mass of
the spheres is 336.336 g? Click on the image for a close up.
Playing along with our imaginary example, let’s assume that the yellow sphere (an atom) has a mass of 1 amu, and
we have separate collections of red spheres (also atoms) with a mass of 7 amu each, and green spheres weighing in at
19 amu each. We can now prepare samples of yellow, red, and green spheres containing exactly the same number of
spheres in each by weighing samples with total masses in a 1:7:19 ratio, respectively, for example, 1 g of yellow spheres,
7 g of red spheres, and 19 g of green spheres.
How many spheres would these samples contain? We don’t need a calculator to determine this; each sample would
contain Avogadro’s number of spheres (E01-2-1):
1 g
7 g
=
1 amu
19 g
=
7 amu
= 6.02214 × 10
23
E01-2-1
19 amu
A mole is a sample containing Avogadro's number of objects
A collection (or a sample) containing Avogadro’s number of objects (spheres, atoms, electrons, molecules, or
anything else ) is called a mole. It is a convenient measuring unit (not unlike a baker’s dozen) of nanoscopic objects that
can be employed on the macroscopic scale. You may notice that for elements, one mole of atoms will be contained in a
sample weighing the same number of grams as its atomic mass in amu units. The same is true for groupings of atoms
called molecules, but the mass of the sample in grams must now match the molecular mass (molecular weight) in amu.
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Figure F01-2-2. Samples of various compounds: (a) the
large flask in the back contains hydrogen gas, H2 (2.016 g,
22.4 L), (b) the medium sized bottles from left to right:
sucrose (a.k.a table sugar, C12H22O11 (342.30 g)), copper
sulfate (CuSO4 (159.60 g)), sulfur (S8 (256.51 g)), aluminum
oxide (Al2O3 (101.96 g)), copper sulfate pentahydrate
(CuSO4•5H2O (249.68 g)), and (c) small vials in the front from
left to right: water (H2O (18.02 g)), aluminum (Al (26.98 g)),
zinc (Zn (65.39 g)), iron (Fe (55.85 g)), and copper (Cu
(63.54 g)).
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01-3 Ions and molecules
Atoms gain or lose electrons to form anions or cations
which are attracted to each other by electrostatic forces
Even if atoms are the simplest representatives of an element, under typical conditions only the noble gas atoms exist as
individual separate entities in unchanged form. In other forms of matter atoms are converted into ions or exist as
collections of atoms bonded together into molecules or extensive networks. The formation of ions or assemblies of atoms
is the result of changes in the electron arrangements around atomic nuclei while the nuclei remain unchanged. Atoms
may change identity (changes in proton or neutron count) only through nuclear reactions.
Structurally, the simplest chemical changes in atoms are the formation of ions. Atoms may gain or lose electrons to
become ions, charged species with an excess or deficiency of electrons as compared to the number of protons in the
nuclei. An ion with a net negative charge (excess of electrons) is called an anion (AN-ion) and an ion with a net positive
charge (deficiency of electrons) is called a cation (CAT-ion).
Figure F01-3-1. Ions are formed by the gain or loss of an electron. The chlorine atom gains one electron, resulting in an
anion with 17 protons in the nucleus and 18 electrons in the electron cloud. The sodium atom loses one electron, yielding
a cation with eleven protons in the nucleus and 10 electrons in the electron cloud. Other atoms may gain or lose multiple
electrons, yielding ions with higher charges (for example, 2–, 3–, or 2+, 3+). These net charges on ions are represented
by corresponding superscripts.
Ions have different chemical properties than neutral atoms. Their behavior is dominated by strong electrostatic
interactions with other charged particles. Ions with opposite charges are strongly attracted to each other, and form
extended ionic solids (salts). For example, sodium cations and chloride anions combine to form a highly ordered
(crystalline) solid of sodium chloride, known as table salt (F01-3-2). The formula of this solid, “NaCl,” makes the empirical
ratio of sodium to chlorine obvious, but it gives us no clues as to the actual structure of the crystal they form. In general,
ionic compounds are 3-dimensional arrangements of ions extending in space, which can reach macroscopic sizes—as
illustrated by the crystals of sodium chloride found in everyday saltshakers.
Figure F01-3-2. Sodium cations and chloride anions form ionic bonds when sodium atoms transfer an electron to chlorine
atoms. When multiple such interactions take place, the result is an extended 3-dimensional arrangement (organized
lattice) of alternating ions where each cation is electrostatically attracted to multiple anions (and vice versa). The 3dimensional arrangements may reach macroscopic sizes, as seen in NaCl crystals (click for a closeup).
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Ions may also take the form of more complex multi-atom entities (see below); in addition to simple, single-atom
derived ions, there are polyatomic ions composed of multiple atoms. Simple examples of such ions include NH4+
(ammonium ion), HO– (hydroxide ion), or SO42– (sulfate). In such ions, the total number of electrons does not match the
total number of protons in all nuclei. The numerical excess or deficit of electrons defines their overall charge.
Atoms combine to form molecules or extensive solids
Atoms may also combine with other atoms without forming ions. Atoms of the same element or of two or more
elements may bond together to form molecules or extensive solids. For example, the simplest element, hydrogen, exists
as a diatomic molecule, H2. The most stable form of oxygen is diatomic oxygen, O2, but another form is the triatomic
version O3, which is called ozone. These different elemental forms, called allotropes, have different structures and
different physical and chemical properties. Carbon has a variety of allotropes, from molecular buckminsterfullerene (C60)
to extended solids of diamond and graphite where carbon atoms form exquisite 3-dimensional networks, reaching
macroscopic sizes (Figure F01-3-3). Formation of extended solids made of many atoms that share electrons is common
for the metallic and metalloid elements.
Figure F01-3-3. Examples of allotropes of carbon: a molecular version composed of 60 carbon atoms
(buckminsterfullerene), a diamond network where the whole solid is one huge molecule, and graphite with its hexagonal
“chicken-wire” layers. To appreciate the nuances of these 3D structures, you may interact with these models using your
mouse.
Molecules constructed with more than one type of atom are called molecular compounds. The number of possible
combinations is practically infinite, and millions of compounds have either been isolated from natural sources or
synthesized and characterized. The atoms in these compounds bond together following well-defined chemical rules that
we will explore in depth in later Lessons. For now, it suffices to notice that when elements combine they have a preferred
number of bonds to other atoms. For example, hydrogen and all halogens (F, Cl, Br, I) typically form just one bond, oxygen
tends to form two bonds, nitrogen prefers to form three bonds, and carbon does four. The number of bonds formed is
called the valence of the atom. Atoms can be “connected” via single or multiple (double or triple) bonds. Chemists
commonly use a notation where each bond is represented by a line between elements to make structural drawings. Such
drawings show atom connectivity, but ball-and-stick models are usually employed in order to truly appreciate the shapes of
molecules. These models show bonds as sticks and identify atoms (balls) through color differentiation. Alternatively,
space-filling models are used to show the space occupied by the electron clouds surrounding the nuclei of all atoms
participating in formation of the molecule (Figure F01-3-4).
You may have noticed that a large proportion of molecules in Figure F01-3-4 contain carbon. This is not an accident.
Carbon is able to form four bonds to other carbon atoms or to atoms of many other elements, resulting in a large variety of
diverse and occasionally very complex structures. Indeed, life on Earth is based on compounds of carbon — these are
called organic molecules. A paragon example of such molecules is a small DNA fragment, shown much scaled down in
the central panel of Figure F01-3-4. For now we can just admire its beauty, helical symmetry, and complexity while we
begin our journey to comprehend molecular structures.
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Figure F01-3-4. Examples of simple molecules in different representations. The molecular formula lists all atoms present
in the molecule and their proportions. The structural formula shows how the atoms are connected, with the lines
representing the bonds between atoms. In perspective formulas, the bonds may be shown as solid wedges indicating that
the bond is pointing out of the page, or with dashed wedges showing a bond extending "behind" the plane of this page.
Ball-and-stick models are well suited to show bond angles, while space-filling models illustrate relative sizes of atoms.
Atoms are colored according to a standard scheme (check it in our interactive periodic table) that we are going to follow
consistently throughout the course. You may interact with the 3D models in the central panel by clicking on stick-and-ball
or space-filling pictures in the figure (the models are not to scale; they were set to fill the viewing window).
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01-4 Introduction to energy
Energy is the capacity to do work and transfer heat
The concepts of matter and energy are very familiar to most of us in the context of everyday life or in science. While the
concept of matter is relatively easy to define and grasp (as we can see and touch material objects), energy is much more
abstract and harder to precisely define. In the broadest sense we can say that energy is the capacity to do work and
transfer heat. Or, in colloquial terms, it is the ability to make something happen.
Energy exists in two basic forms: potential and kinetic. Potential energy is related to the positioning of an object in
relation to other objects, while kinetic energy is related to the motion of objects. The two forms of energy can interconvert,
but no energy can be lost or created in any chemical or physical process. The law of energy conservation is known as the
first law of thermodynamics.
The SI unit of energy is the joule, (1 J = kg·m2/s2). Since energy is an extensive property (i.e. it depends on the
amount of substance present), kilojoules (1 kJ = 103 × J) are often used on the molar scale (kJ/mol). Alternatively, calories
(1 cal = 4.184 J) or kilocalories (kcal) are employed. There is also a nutritional Calorie unit (with a capital C), equal to 1
kcal.
The kinetic energy of particle motion and electrostatic potential energy dominate chemical
processes
Potential energy can be thought of as “stored” energy and can manifest itself in many forms, such as electrostatic,
nuclear, or gravitational energy. These types of energy are directly linked to the corresponding fundamental physical
forces. On the other hand, kinetic energy (Ek) is determined only by the mass (m) and velocity (v) of a moving object (E014-1):
1
Ek
=
2
2
mv
E01-4-1
In terms of potential energy, chemistry is completely dominated by electrostatic forces, where the relative positioning
of charged particles (nuclei, electrons, or ions) decides their energy. We already know that the force of interaction
between two charged particles (with charges Q1 and Q2) is conveyed by Coulomb’s law (E01-1-1). From basic physics, we
also know that work (w) is equal to force (F) multiplied by distance (r), and that energy is "equivalent" to work (w). Thus,
potential electrostatic energy (Eel) can be expressed by equation E01-4-2, with the charges Q1 and Q2 in coulombs (C).
The proportionality constant is called Coulomb's constant, k = 1/4πε0 = 8.99 × 109 J·m/C2 where ε0 (= 8.86 × 10−12 F/m) is
vacuum permittivity.
Eel =
kQ1 Q2
E01-4-2
r
Since electrons within atoms, atoms themselves, atoms within molecules, and whole molecules are in constant
motion, kinetic energy is another energy component crucial to the understanding of chemical interactions and
transformations. In fact, atomic and molecular motions constitute the “thermal” component of energy; the average kinetic
energy of particles in a sample is directly related to the temperature of that sample. A “hotter” sample has faster moving
particles (on average) than a “cooler” one. Heat (designated by the symbol q) may be transferred from a hotter sample to
a cooler one by transferring some of the kinetic energy, for example by collisions between particles.
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The change in internal energy of a system results from the exchange of heat or work
with the surroundings
The internal energy of a system (E) is the total energy associated with the system, the sum of all sources of kinetic and
potential energy. In most situations, chemists do not care about “absolute” internal energy of a sample, but instead are
more interested in the internal energy changes (ΔE) directly connected to a physical or a chemical process under
consideration (Figure F01-4-1). Since energy cannot be created or lost, chemists define the system and the surroundings
and observe the energy flow between the two to determine the energy changes from the initial state to the final state. The
system is defined as a collection of particles of interest, examples of which could be an atom emitting light energy, or a
mole of molecules undergoing a chemical reaction. The surroundings are defined as everything else.
The energy flow is always referenced with respect to the system, in a way analogous to the balance in a bank
account. Thus, if the energy of the system is lowered (by transferring some of it to the surroundings) the energy change
has a negative sign (energy has been “withdrawn” from the system). On the other hand, if energy is added to the system
(from the surroundings) the energy change has a positive sign (energy has been “deposited” into the system).
Figure F01-4-1. Energy flow (a) from and (b) to the system. The lower the final energy, the more stable the system is in its
final state. Processes where ΔE < 0 are described as being spontaneous (we will define that concept more precisely in
later Lessons).
Let’s consider a system constructed of two charged particles with charges Q1 and Q2, as shown in Figure F01-4-2.
At infinite separation the electrostatic energy of the system is zero, as the two particles do not experience any electrostatic
forces from each other. If the two particles have like charges (+/+ or –/–), bringing them closer requires work against a
repulsive force. That work is done on the system (by the surroundings) and it results in an increase in the energy of the
system. Conversely, if the charges are opposite (+/–), the work of bringing them closer by the attractive force is done by
the system, and the result is a transfer of energy to the surroundings and a lowering of the energy of the system.
Figure F01-4-2. Electrostatic energy of two interacting charges as a function of separation between them.
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We have now discussed the two major ways in which energy is transferred between a system and its surroundings,
heat (q) and work (w). Adding heat (+q) to the system and doing work on the system (+w) both increase the overall energy
of the system.
E01-4-3
ΔE = q + w
Conversely, removing heat from the system (–q) and having the system do the work (–w) lowers the overall energy
of the system. The sign convention followed in such energy transfer processes are illustrated in Figure 01-4-3.
Figure F01-4-3. Illustration of sign conventions for system where (a) the heat is added to the system and work is done on
the system and (b) the heat is removed from the system and work is done by the system.
The internal energy of a system is a function of state
The internal energy of the system does not depend on the path or method (i.e., the mechanism) used to get to the current
state. It is a state function; it depends only on the existent state of the system (its temperature, volume and pressure).
Let’s look at a simple example in Figure F01-4-4.
The internal energy is lowest for ice, and highest for hot
water, but the internal energy of the samples at 25 °C is the
same, regardless whether prepared by cooling the hot water or
by heating and melting the ice. In a simple analogy, the overall
change in altitude when traveling from State College, PA (370
m above sea level) to Boulder, CO (1650 m above sea level) is
always 1280 m, regardless of the road chosen for our trip. In
this example, the altitude behaves as a state function.
Figure F01-4-4. Samples of identical masses of H2O in different
states: ice at 0 °C, water at 25 °C, and water at 99 °C, close to
a boiling point. The internal energy of the samples at 25 °C is
the same.
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We use capital letters to signify state functions like P (pressure), T (temperature), E (internal energy), and H
(enthalpy). However, heat and work (q and w) are not state functions. Depending on how we set up a process, we can
release lots of heat and do little or no work, or we can force the process to do more work and release less heat. Consider
the combustion of a gallon of gasoline. We could light a gallon of gasoline on fire and it will generate a great deal of heat
as it burns. However, if we burn the gasoline in a car engine we can use the released energy to power the car. Some heat
will still be produced, but a substantial fraction of the energy will be used to do work. In each case the total amount of
energy released by the burning of the gasoline will be the same, but the relative amount of heat and work produced is
different (ΔE = q + w).
Figure F01-4-5 This figure depicts a process where the internal energy of
the system is lowered by two different paths. Path 1 (blue) may correspond,
for example, to the situation where a reaction produces heat, but does not
do any work. Path 2 (orange) may represent (for example) the process
where the system does work and produces a small amount of heat. Despite
the different paths, the change in internal energy is the same.
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01-5 Enthalpy
Enthalpy is the heat exchanged with the surroundings under constant pressure
Let's assume we have two flasks that start in the same state, and contain the same amount of dry ice (solid CO2) at room
temperature. One flask is closed with a stopper and one has a deflated plastic bag over the mouth of the flask. Let's
further assume that the same amount of heat (q) enters each of these systems at the same rate. Room temperature (25
°C) is higher than the temperature of the dry ice (−78 °C), and the solid will change into a gas as it warms up. The internal
energy of both systems will increase as the samples warm up, but will the two systems end up with the same amount of
internal energy? The one with the stopper remains at constant volume because it cannot expand; the pressure rises
(F01-5-1a). The system with the bag remains at constant pressure (at least initially) because its volume can expand (F015-1b), while performing work against the ambient pressure.
Figure F01-5-1. (a) The system absorbs heat from the surroundings at constant volume. No work is done by the system.
(b) The system absorbs heat from the surroundings at constant pressure (until the bag is fully inflated). The work done by
the system on the surroundings, against ambient pressure (P), is w = PΔV.
At any point in time (before the bag fully inflates) after an equal amount of heat is absorbed by both systems, which
one will have the higher internal energy? The system that absorbs heat at constant P (F01-5-1b) does work on the
surroundings that lowers its internal energy. The system that absorbs heat at constant V does no work and therefore will
have a higher internal energy. Anyone watching the flasks will be waiting for the cork to pop off the flask in Figure F01-51a. And it does! Eventually, when the bag is fully filled, it breaks as well under the increasing pressure of CO2.
In chemical and physical changes occurring under constant pressure, it is indeed quite common that the only work
done by the system is due to volume change (ΔV). This P-V work term is particularly important when the transformation
leads to a significant change in the volume of a system, as often happens for processes involving gases. Since P and V
are state functions, P-V work is also a state function, and under constant pressure, when only heat and P-V work
contribute to energy changes, we can write (E01-5-1):
ΔE = qp + w = ΔH − P ⋅ ΔV
E01-5-1
The heat exchanged with surroundings under such conditions is called enthalpy (qp = ΔH). Like internal energy,
enthalpy is a state function. It is a very useful quantity as it is relatively easy to measure (we will learn how in later
Lessons), and for many reactions in liquids or solids where change in volume is small or zero, it directly matches the
changes of the internal energy of the system. Internal energy and enthalpy are both extensive properties, as they depend
on the quantity of the matter in the system.
Transfer of heat to and from the system during chemical and physical processes is so common that chemists use
special names to indicate the direction of the heat flow. Processes where ΔH > 0 (i.e., the heat flows into the system) are
called endothermic (endo means “into”), while processes where ΔH < 0 (with heat transfer to the surroundings) are called
exothermic (exo means “out of”).
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01-6 Light energy
Light is an electromagnetic wave
Light is the “purest” form of energy, as it may exist “outside” of matter. On one hand it is an electromagnetic wave
traveling with enormous speed (approximately 3 × 108 m/s). On the other hand, it interacts with matter as a stream of
energy packets (or photons). This dual behavior (wave-like and particle-like) is dependent on the situation, and is
characteristic of the quantum world we are about to enter in order to learn more about the detailed structure of atoms and
molecules.
An electromagnetic wave propagates in a direction that is at right angles to the vibrations of both the electric and
magnetic oscillating fields, carrying energy from its source. The two fields are mutually perpendicular. Their amplitudes
oscillate through a repeating pattern of peaks and troughs.
Figure F01-6-1. A schematic representation of an electromagnetic wave. The magnetic and electric fields oscillate in
planes perpendicular to each other and to the direction of propagation. The wavelength (λ) is the distance between two
adjacent peaks (or troughs). The maximum amplitude of the waves is a measure of intensity of the radiation.
Waves have characteristic wavelengths (λ), as measured by a peak-to-peak distance. Regardless of their
wavelengths, all electromagnetic waves move with the same speed, c (c ≈ 3 × 108 m/s in a vacuum). Thus, instead of
using wavelengths, the waves can be characterized by the frequency ν, which is the number of cycles that pass a given
point per second (E01-6-1). The frequency is usually expressed in cycles per second (s−1) or in hertz (1 Hz = 1 s−1).
c = λν
E01-6-1
The properties of waves depend on their wavelengths. The wavelengths of electromagnetic radiation span an
enormous range, from kilometer-long radio waves to subatomic-length gamma rays. The "visible" light spectrum, a very
narrow range of electromagnetic radiation detectable by human eyes, has wavelengths of 400—750 nm (1 nm = 10−9 m),
with different wavelengths perceived as different colors of the rainbow. Figure 01-6-2 shows the whole spectrum of
electromagnetic waves arranged by their wavelengths with the visible part of the spectrum expanded for emphasis.
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Figure F01-6-2. The spectrum of electromagnetic radiation with the visible spectrum emphasized. Both wavelength and
frequency ranges are shown. For the sake of convenience, different wavelength units are commonly used for various
regions of the spectrum (top).
Electromagnetic radiation exhibits typical wave behavior. The light waves diffract when they encounter small
obstacles or small openings to pass through, generating interference patterns in a way similar to those commonly
observed in water waves (Figure 01-6-3) wherein overlapping wave peaks increase in intensity and overlapping peaks
and troughs diminish the wave amplitudes. .
Figure F01-6-3. Diffraction of light waves, producing an interference pattern of alternating dark and light regions on the
screen. The "arriving" wave (left) passes through two slits and two propagating waves form. The propagating waves'
crests are shown as arcs (partial circles).
Most sources of light in the visible range produce collections (spectra) of different wavelengths with different relative
intensities. For example, the unique combination of all “rainbow” colors delivered from our sun is perceived as “white”
light. There are some sources that produce just a few discrete wavelengths, and there exist monochromatic sources (such
as lasers) that emit just one wavelength of light. Soon we will learn how we can use the interactions of light with matter to
gain an understanding of the fine structures of matter. The study of these interactions is called spectroscopy.
Light is also a stream of photons which are quantized energy packets
Not all light-involving phenomena can, however, be explained in terms of wave characteristics alone. For example, the
distribution of frequencies of radiation emitted by a heated body, so called black-body radiation, cannot by accounted for
by classical wave physics. This distribution depends on the temperature of the solid. At lower temperatures infrared
frequencies are emitted, but with increasing temperature the heated body starts emitting visible light, brown at first, then,
red, orange, yellow, and white at high enough temperatures. Only in 1900 did German physicist Max Planck realize that
objects couldn’t gain or lose energy in arbitrary or continuous amounts, but could only transfer energy in discrete “packets”
of some minimum size (or its multiple). The packets of energy were named quanta, and thus quantum physics and
quantum chemistry were born.
Planck proposed that the energy, E, of a single quantum is proportional to the frequency of the radiation, with the
proportionality constant, h (now known as Planck’s constant) equal to 6.626 ×10–34 J·s.
E = hν
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In quantum theory, matter can emit or absorb energy only in whole-number multiples of hν (hν, 2hν, 3hν, …. or NAhν,
for a mole of quanta). This limitation means that on a microscopic (nanoscopic) scale the energy transfer is “granular”
rather than continuous. On a macroscopic scale, we do not perceive this granularity simply because the quanta of energy
are so small. Even so, the quantization of energy rules at the atomic level.
With this new insight we can now understand how wavelengths govern the properties of electromagnetic waves.
Long waves (large λ or, alternatively, small v) carry small amounts of energies while short waves (small λ or large v) are
very energetic. Radio waves are low on the energy scale; they barely deliver any energy and are unable to affect
chemistry in any significant way. At the other extreme, gamma rays, X-rays, or even ultraviolet radiation (UV) carry a big
punch, with energies equal to or exceeding the energies typically involved in chemical interactions. These high-energy
waves can be quite destructive.
We have just learned that matter (as exemplified by the black body) can only emit electromagnetic radiation in a
quantized fashion. Is the same true when material objects absorb light waves?
Einstein provided the first affirmative answer to this question when he explained the photoelectric effect, the
phenomenon in which only light of proper frequency can eject electrons from an illuminated metal surface. The effect was
accounted for by the fact that the light, despite its apparent continuous nature, can only deliver energy to a metal’s
electrons in a quantized form. A packet (quantum) of light energy, called a photon, has to be energetic enough to at least
match the energy holding the electron within the metal in order to be able to eject the electron.
These phenomena, black-body radiation and the photoelectric effect, illustrate that light (electromagnetic radiation)
can exhibit a “corpuscular” nature in its interaction with matter. We also seen light behaving like a classical wave when it
diffracts and forms interference patterns of light and dark areas. This wave-particle duality turns out to be the governing
phenomenon in the quantum word, and as we will see shortly, it also applies to elementary particles such as electrons.
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01-7 Spectroscopy
Spectroscopy is the study of how light interacts with matter
We have just learned about light carrying energy as an electromagnetic wave or a stream of photons. When light
interacts with matter its energy may be converted to other forms of energy, and energy contained within matter may be
converted into light. We have already described two examples of such energy conversions, the photoelectric effect and
black-body radiation. In general, such energy-conversion processes are quantized (the energy exchanged is delivered in
discrete "packages"), and therefore provide a great opportunity to study energy levels within the material to be
investigated, probing its internal structure. To do it effectively, we need precise information about the incoming and
outgoing light energy. What wavelengths (energies) are involved? Do they all interact with our sample to the same
extent?
Let us consider a very familiar part of the electromagnetic radiation, the visible spectrum (Figure F01-7-1). The
range of wavelengths from around 400 nm to 750 nm is called visible light because it is detected by human eyes. Any
photons in that range are detected by the cone and rod cells in the retina, the detected signal is processed by the optic
nerve in the back of the eye, and then interpreted by our brain. If all wavelengths in the visible range are represented, we
perceive it as white light. Isaac Newton (in 1665 at age of 23!) using prisms and mirrors showed that the white light can
be split into colors that can then be recombined back into the white light. He also arranged colors into the color wheel
(Figure F01-7-1), discovering their "complementary nature". Typically, the colors we see are the result of some sort of
filtering of portions of the visible spectrum that reaches our eyes. For example, if the orange light is filtered, we see the
blue color from the opposite side (180°) of the color wheel while filtering red results in green color. Thus, our color
perception most often depends on the part of the spectrum of white light that is "missing".
Figure F01-7-1. The visible spectrum can be represented as a color wheel. If the range of wavelengths represented by a
certain color are filtered, the complementary color (across the wheel) is observed.
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When visible white light interacts with macroscopic transparent objects it can be transmitted through it (the light is
usually refracted as well). If the object is clear (like glass), all wavelengths of visible light are transmitted. If the medium is
colored (such as a colored solution), then light of that color is transmitted while the wavelengths of light corresponding to
the complementary color are absorbed by the medium (Figure F01-7-2a). Similarly, an opaque material reflects the
wavelengths of light that correspond to the observed color which is complementary to the color being absorbed by the
sample. For example, a white piece of paper reflects all wavelengths of visible light, a black piece of paper absorbs all
wavelengths of light, and a green piece of paper reflects green light while absorbing (mainly) red (Figure F01-7-2b).
Finally, when light is emitted by matter, e.g., the radiation from a light stick (Figure F01-7-2c), the perceived color matches
the wavelengths given off by the system. If an object can be seen when it is otherwise completely dark, then the object is
emitting visible light.
Figure: F01-7-2. Macroscopic objects appearing green: (a) a solution that transmits green light, absorbing red, (b) an
object that is reflecting green light (and absorbing red), and (c) a light stick that is emitting green light.
The absorption of light is measured by a spectrophotometer
To measure the energy exchanged with the material under study we need an instrument that can quantify the portion of
light absorbed at all wavelengths of interest. The instrument that accomplishes that task is called a spectrophotometer. In
our example, it is used to measure the absorption and transmission of light by the sample over the the visible range of
wavelengths. In a way, our eyes are nature's spectrophotometers (see above); they can distinguish wavelengths (colors)
with 3-10 nm resolution and detect light with intensity differences of 10 orders of magnitude. What our eyes lack are the
measurement scales.
A simplified diagram of a spectrophotometer is shown in Figure F01-7-3. A light source is used to produce
wavelengths needed to examine the sample. A monochromator (such as a prism or a diffraction grating) is used to
separate the wavelengths of light, and a movable slit can be used to select the specific wavelengths to be measured in
turn while exploring the full spectrum. The light then passes through a sample cell and is detected on the other side. The
detector measures the intensity of light transmitted through the sample and determines the decrease in its amplitude for
each wavelength as compared to the original intensity at the source (measured with a "blank" sample). The absorption
spectrum obtained is a plot of absorbance (the logarithm of fraction of light absorbed) as a function of the wavelength of
the light (see Figure F01-7-3). The absorbance at a given wavelength depends on how many molecules are in the path of
the light beam (i.e. concentration of the sample), the length of the path (i.e. the sample "width"), and the probability that
the molecules absorbs a photon of this wavelength.
Figure F01-7-3. Schematic representation of a simple spectrophotometer.
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As explained above, in visible spectroscopy, the wavelengths of light transmitted correspond to the complementary
color of the light absorbed. Figure F01-7-4 shows the spectrum of two dye solutions. The spectrum on the left represents
a solution of a blue dye. The maximum absorbance is in the range of 620 – 640 nm (corresponding to orange and red).
The complementary color (across the color wheel from this range of wavelengths) is blue, so the sample appears to be
blue. In the spectrum on the right the maximum absorbance is in the range of 480 – 540 nm (mainly green). The
complementary color for this range of wavelengths is red. Not surprisingly, this solution appears to be red.
Figure: F01-7-4. Absorption spectra of a blue (left) and red (right) dyes. The blue dye absorbs mainly in the orange and
red region, while the red dye absorbs mainly in the green region.
Light energy absorbed by atoms or molecules is converted into various forms
of kinetic or potential energy
What happens on a molecular level when light interacts with matter? When light is transmitted or reflected both the light
and the medium are mainly unchanged (sometimes the light is bent by the medium, as a result of its speed being
changed). The outcome is dramatically different when the light is absorbed by the material: the energy is transferred and
changes occur at the atomic and molecular level. The specific changes depend on the energy of the incoming photons
and the structure of the molecules in the absorbing medium. Indeed, much of what we know about the structure of matter
is the result of studies of how atoms and molecules change when photons are absorbed.
When an atom or molecule absorbs energy, the photon energy is converted into different forms of kinetic or potential
energy. In most absorption processes the light energy eventually is converted into kinetic energy, referred to as
translation, increasing the velocity of the particles. This type of energy is directly related to the temperature of the
collection of molecules, so an increase in translational kinetic energy is perceived as an increase in heat.
High-energy photons such as gamma, X-rays and ultraviolet rays can damage or break molecules by causing
ionization or bond scission when a photon is absorbed. In these situations, the energy of the absorbed photon is
converted into potential energy by creating high-energy ions or molecules with unpaired electrons (free radicals). The
absorption of visible light also results in the photon energy being converted into potential energy when electrons in atoms
or molecules move to higher energy states called excited states (a concept we will explore more in Chapter 2). Photons of
lower energy (infrared and microwave) do not damage molecules, but increase amplitudes of their vibrations and speed of
their rotations, respectively. Absorption of photons in all of these situations increases the total energy in the system and
much of the energy is eventually converted to heat through particle collisions. Table T01-7-1 describes the possible
molecular changes upon absorption of energy.
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Table T01-7-1. The types of changes that occur at a molecular level when photons are
absorbed.
Radiation
λ
gamma
< 0.01 nm
X-ray
1 − 10 nm
ultraviolet
10 − 400 nm
visible
400 − 750 nm
infrared 750 nm − 1 mm
microwave 1 mm − 1 m
radio
>1m
Interaction with matter
Ionizes atoms and molecules
Breaks chemical bonds
Ionizes atoms and molecules
Breaks chemical bonds
Ionizes atoms and molecules
Breaks chemical bonds
Promotes electrons to higher energy (electronic transitions)
Promotes electrons to higher energy (electronic transitions)
Increases amplitudes of vibrations (kinetic energy increases)
Increases speed of molecular rotations (kinetic energy increases)
Flips the nuclear spin
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02 Hydrogen Atom
Based on everyday experience, we often assume that nature is continuous and precisely measurable. Given a good
enough scale, we think, we could measure any amount of graphite (carbon), or equipped with a powerful enough
microscope we would be able to observe electrons in atoms with ease equal to watching bouncing ping-pong balls in a
lottery machine in slow motion. On the atomic and molecular scale, however, these assumptions break down. We find
that not only mass and charge, but also energy all come in discrete packets. We cannot have a half of a carbon atom (it
would not be a carbon atom anymore), or an imperfect proton with just ⅔ positive charge, and the energy of an electron in
an atom can only change in a stepwise manner. Moreover, we learn that there is a limit to the precision of our
measurements. We cannot know simultaneously the momentum and position of an electron in an atom with a satisfactory
certainty. Instead of the neat deterministic Newtonian mechanics we enter a weird granular and probabilistic quantum
world.
In that unintuitable world, light shows particle-like behavior and small particles display wavelike properties in their
motion. The light and matter interact by exchanging quantized amounts of energy, and probability replaces certainty. In
this Chapter we begin to explore some basic ideas of quantum chemistry in order to understand the electronic structure of
atoms and their chemical properties.
02-1 Line spectra
Atoms emit characteristic colors of light when excited with some forms of energy. For atoms in the gas phase, these colors
correspond to light with discrete individual wavelengths whose collections are called line spectra. Each line represents uniform
packets of energy sent out when electrons within an atom transition from one energy level to another with lower energy. The
observation of a number of sharp spectral lines indicates that an electron can have only precisely defined energy in the atom and
that it can change energy only in a stepwise manner.
02-2 Bohr's model
To explain line spectra of hydrogen atom, Bohr proposed a “planetary” model wherein the electron is allowed to circle the
nucleus only in specific orbits with set (quantized) energy described by the principal quantum number (n). If an electron changes
its orbit (change in n), energy is absorbed or emitted in the form of a photon (quantum of electromagnetic radiation) matching the
energy difference between the orbits. Bohr’s model works very well for the hydrogen atom and ions with just one electron, but fails
for atoms or ions with more electrons.
02-3 Matter waves
On the atomic scale, matter has dual particle-like and wave-like properties. Positions and momentum of small particles,
including electrons in atoms, cannot be simultaneously determined with sufficient precision (Heisenberg’s uncertainty principle).
Instead, electrons are described by standing waves around atomic nuclei. The squared values of these wavefunctions at any
given point of space correspond to the probability of finding the electron in an immediate vicinity of that point. That probability
density is called electron density. For atoms (and molecules) electron densities and allowed energies of electrons are found by
solving the Schrödinger equation that describes the potential energy of all electrostatic interactions between electrons and nuclei,
and the kinetic energy of electrons.
02-4 Quantum numbers
The solutions of the Schrödinger equations for atoms are called orbitals. The orbitals have discrete energies and they
mathematically describe the spatial distribution of electron density (where the electrons are likely to be in the relation to the
nucleus). These orbitals can be distinguished in a short-hand notation by a set of quantum numbers that include the principal
quantum number (n), the angular momentum quantum number (ℓ), and the magnetic quantum number (mℓ). In the hydrogen atom,
only n determines the energy of the orbital, while ℓ and mℓ control the shape and spatial orientation of the orbital, respectively.
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02-5 Orbitals
The atomic orbitals have characteristic shapes, sizes and energies associated with their quantum numbers. The
wavefunctions, their squares (electron densities), and radial probabilities are used to visualize the electron distribution around the
nucleus. These graphical representations allow us to see what the most probable spaces for electrons are, and where electrons
are not allowed to be (nodes). In a simplified pictorial version, we present only artificial “skins” (isosurfaces) selected in such a way
that the probability of finding an electron inside the demarcated space is 90%. These boundary surfaces may be labeled with the
algebraic sign of the wavefunction from which they are derived. Such qualitative graphical representations are in most cases
satisfactory for understanding of the basic electronic structure of atoms (or molecules).
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02-1 Line spectra
Excited atoms in the gas phase emit light of discrete wavelengths
Imagine that you are given a sealed black box and you would like to find out what’s inside. You do not have any tools to
open it, yet curiosity pushes you to explore. You might lift the box to see how heavy it is, or shake it and see if something
rattles inside. Perhaps you may bang it with your hand, or even kick it if it is too large to lift, to see how it responds. When
probing matter, scientists tend to do the same, although in a bit more sophisticated way. For example, they may provide
some energy to the sample they want to explore to see if any information comes back. We have seen how in the
photoelectric phenomenon the light energy sent to the sample resulted in release of electrons with different kinetic
energies. We have briefly explored visible light spectroscopy to see which colors (energies) are absorbed. We have also
encounter a reverse situation in black-body radiation. When given energy in the form of heat, the sample responded by
emitting light, which could be studied for its distribution of frequency and intensity. These experiments teach us that light
when it interacts with matter, behaves as a "stream" of photons and that matter sends out energy in quantized forms.
Similar experiments can tell us about the internal structure of matter. The analysis of conversion of thermal and
electromagnetic forms of energy within samples under study opens a window on the inner workings of atoms and
molecules. For heated solid samples (black-bodies) continuous spectra are observed (see F01-4-2) without any fine
features, and therefore, without much detailed structural information. We can suspect that strong interactions between
closely-packed atoms in solids complicate the analysis. However, when separate atoms of various elements are excited in
the gas phase, the observed emission patterns are much simpler.
Figure F02-1-1. Neon lights made out of discharge tubes filled with Ne (left image) and glowing discharge tubes
containing different gases (from left to right): H2, Ne, N2, O2, and Hg (vapor).
We are all familiar with the neon lights commonly encountered in the commercial and entertainment districts of our
cities (F02-1-1). These lights are sealed glass tubes fitted with metal electrodes at the ends and filled with various gases,
with neon (Ne) being a common one because of the attractive color it generates when excited. When a high-enough
electric potential is applied to the electrodes, an electric discharge takes place inside the tube, and the energy is emitted
in the form of radiation with characteristic colors and wavelengths. Figure F02-1-1 shows glowing tubes of several different
gases. A closer analysis of the emitted light with the help of a diffraction grating reveals that the light sent out by the
atoms is composed of several distinctive wavelengths, resulting in what we call a line spectrum. Each element produces a
unique set of these spectral lines (F02-1-2), although one (or more) lines can be more intense than the rest and dominate
the spectrum, giving the glowing light its characteristic color.
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Figure F02-1-2. Line spectra of several elements. The lines have been enhanced for better visibility. For an expanded
view of more elements click on the image. You may explore authentic, unenhanced line spectra of all of the elements in
our interactive periodic table (click on the periodic table button above this page, then click on an element).
These characteristic atomic emission colors are not limited to elements that are gases under normal conditions.
Color spectra may be obtained even for metallic elements, provided that enough metal atoms get vaporized and excited.
The flame tests (F02-1-3) are a simple illustration: many salts inserted into a hot, reducing flame have flame colors
characteristic of the metallic atoms. The most spectacular demonstration of atomic emissions are fireworks (F02-1-4)
where the reds are characteristic of strontium or lithium, greens are due to barium, blue are due to copper, orange are due
to calcium, and yellow is due to sodium.
Figure F02-1-3. Alcohol burning over salts of calcium (left), lithium (center), and barium (right). The Fourth of July firework
shows get their colors from various metal salts.
Wavelengths of light emitted by hydrogen atoms follow a simple mathematical formula
called the Rydberg equation
Let's take a closer look at the line spectra produced by excited hydrogen atoms, which are the simplest of them all. The
four lines observed in the visible range (called the Balmer series) are just a part of the story. It turns out that a similar
series of lines are detected in the ultraviolet range (the Lyman series) and several sets of lines (the Paschen, Brackett and
Pfund series) are found in the infrared if proper spectroscopes are employed. Indeed, these spectra sets were discovered
in the second half of 19th century and are named after their discoverers. In 1885, Johann Balmer, a Swiss school teacher,
realized that the wavelengths of the lines follow an unusually simple mathematical formula. That formula was easily
extended to cover all series and it is called the Rydberg equation (E02-1-1). In this equation λ is the wavelength of the
spectral line (in m), RH is the Rydberg constant (1.097 × 107 m–1) and n1 and n2 are positive integers, with n2 being larger
than n1.
1
λ
= RH (
1
2
n
1
−
1
2
)
E02-1-1
n
2
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Despite the apparent simplicity of the hydrogen atom’s spectra, it took about 30 years before progress was made on
interpreting the line pattern in terms of an atomic model that accounts for the observations. Based on what we have
described so far, what kind of model would you propose to account for the Rydberg equation?
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02-2 Bohr's model
In the Bohr model the electron can only be in orbits that have quantized radii and energies
In 1913, Niels Bohr proposed a theoretical model that accounts very well for the emission lines observed for hydrogen
atoms (and one-electron ions such as He+, Li2+, and Be3+). Bohr followed up on Rutherford’s planetary model, in which an
electron circles the nucleus, but added three important postulates:
1. Only orbits of certain radii, corresponding to discrete energies, are allowed for the electron in a hydrogen atom.
2. The electron in the allowed orbit (allowed energy state) does not radiate energy and, therefore, does not spiral into
the nucleus, as was predicted by classical physics.
3. Energy is emitted or absorbed by the atom only as the electron transitions from one allowed orbit to another. That
energy is emitted or absorbed in the form of a photon of matching energy E = hν, with a characteristic frequency
("color").
The model was based on the simple idea that a speeding electron can remain bound to the nucleus only if the
centripetal and electrostatic forces acting on it are balanced. If the electron moved too slowly in its orbit, it would be
attracted by the positive charge of the nucleus and would collapse on it; if it moved too fast the attractive force would not
be able to hold it within the atom and it would fly away. The groundbreaking contribution was the postulate that only
certain radii of orbits are allowed and therefore the total energy (kinetic and potential) of the electron is quantized, i.e., it
can only have specific values, determined by the orbit. This quantization is expressed by the principal quantum number n,
a positive integer specifying the orbit number, with the orbit closest to the nucleus having n = 1, the next one n = 2, and so
on. The results derived from the Bohr postulates (as shown on a separate page) are very instructive and will help us
tackle more sophisticated quantum models in the future.
The first important corollary of the model is the existence of discrete orbits whose distance from the nucleus
increases in a stepwise manner. In one-electron species, the radii of the allowed orbits (E02-2-1) are proportional to the
square of the quantum number and inversely proportional to the charge of the nucleus as expressed by the atomic
number, which indicates the number of protons (Z = 1 for H, Z = 2 for He+, Z = 3 for Li2+, etc). The radius of the smallest
orbit, a0, is used as the scaling factor.
2
n
rn
=
Z
a0
E02-2-1
n = 1, 2, 3
In fact, a0 is a convenient unit of length on the atomic scale called the Bohr radius; it has a numerical value of 5.29 ×
10–11 m = 0.529 Å. For historical reasons, it is marked with subscript "0", even if it corresponds to the orbit with n = 1.
The kinetic energy of the electron is determined by its velocity (v), which is orbit-dependent (E02-2-2). The velocity is
inversely proportional to the quantum number, with v1, the velocity in the smallest orbit, serving as a scaling factor:
vn
=
Z v1
E02-2-2
n = 1, 2, 3
n
The result tells us that electrons move the fastest in the smallest orbit (n = 1) with velocity v1 = 2.2 × 106 m/s (just
about 1% of the speed of light!). Such a large velocity keeps it from "falling" on the nucleus, as the electrostatic attraction
is the strongest because of small orbit radius. The electron moves slower in the larger orbits as the attraction to the
nucleus decreases with the orbit size.
Finally, the most important result relates the allowed values for the total energy of the electron to the quantum
numbers (E02-2-3), where E1 represents the energy of the electron in the smallest orbit (n = 1):
2
Z
En
= − E1
2
n
= −(2.18 × 10
−18
2
Z
J)
2
n = 1, 2, 3
E02-2-3
n
For the same reason that we introduced the Bohr radius, it is convenient to express energies of atomic levels in
units of rydberg (1 rydberg = 1 Ry = hcRH = 2.18 × 10–18 J, where RH is the Rydberg constant in E02-1-1).
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The emission or absorption of photons by hydrogen atoms result in electron transitions
between orbits
The energies of the allowed orbits in the hydrogen atom are all negative and inversely proportional to the square of n (the
principal quantum number). The lower the energy (the more negative), the more stable the atom is, and n = 1 is the lowest
energy orbit possible. When the electron occupies the orbit closest to the nucleus (n = 1), the hydrogen atom is in the
ground state (the lowest energy state). As n gets larger, the energy increases by becoming less negative in steps
proportional to 1/n2. In a way, we have a peculiar ladder with rungs representing the different allowed energy levels (F022-1a). The rungs are far apart at the bottom of the ladder and are increasingly closer to each other as we climb. The
higher we climb, the higher the energy of the atom.
Figure F02-2-1. Energy levels for the hydrogen atom. The reference point (E = 0) has the electron and proton at rest and
at infinite separation. As the distance between them gets smaller, the attraction increases and the energy gets lower and
more negative. In this case, negative energy is arbitrarily chosen as attractive energy. The energy of individual levels is
inversely proportional to n2, resulting in unequal and decreasing spacing between energy levels at higher n values. For
more details click on the pictures in Figures F02-2-1a, b, c. a) Energies and energy differences for several low n values
are illustrated on a per atom and per mole basis. b) Electron transitions from a higher (less negative) energy orbit (larger
n) to a lower (more negative) orbit result in the emission of photons with matching energies. Transitions to the lowest orbit
(n = 1; the Lyman series) give off UV light, transitions ending at n = 2 produce photons in the visible region (the Balmer
series), and transitions down to higher orbits (n ≥ 3) result in IR photons being emitted. c) Absorption of photons of the
appropriate energy promotes electrons from a lower orbit to a higher one. The resulting absorption line-spectrum in the
visible region is illustrated. That spectrum is complementary to the emission line-spectrum in F02-2-1b . An ultraviolet light
with λ = 91 nm has sufficient energy to eject the electron from a ground-state hydrogen atom. That photon energy
corresponds to the hydrogen ionization energy (IE).
At the limit (n = ∞) the energy is zero. Indeed, this state is our energy reference point. We always compare the
energy of the hydrogen atom to the energy of a proton and an electron that are at rest and at infinite separation.
We can now return to our line spectra (F02-1-2) and illustrate what happens when the electron transitions from one
energy level to another. In the emission process, energy is given off by the atom as the electron moves from a higher orbit
to one with lower energy that is closer to the nucleus. The Balmer (visible light) series corresponds to transitions down to
the orbit with n = 2 (F02-2-1b). The higher-energy ultraviolet Lyman series involves transitions down to the ground-state (n
= 1), while the Paschen, Bracket and Pfund lower-energy infrared series have orbits with n = 3, 4, or 5, respectively, as the
final electron destinations.
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The energy difference between the final orbit (Ef) and the initial orbit (Ei) can be easily calculated using E02-2-3.
This energy corresponds precisely to the energy of the emitted photon.
ΔE = Ef − Ei
= (−2.18 × 10
−18
J) ⋅ (
1
1
−
2
2
n
E02-2-4
) = hν
n
f
i
For example, ΔE for the lowest-energy line in the Balmer series (nf = 2, ni = 3) is a negative number. Since this is an
emission line, the negative sign is fully consistent with our convention that the system (hydrogen atom) is giving off energy
to the surroundings (F01-4-1).
ΔE = Ef − Ei
= (−2.18 × 10
−18
1
J) ⋅ (
1
−
2
2
2
) = −3.03 × 10
−19
J
E02-2-5
3
This energy is, of course, also the energy of the emitted photon. To calculate the wavelength of that emission we
use the standard frequency-to-wavelength conversion, ΔE = hν = hc/λ.
λ =
hc
(6.63 × 10
−34
J ⋅ s)(3.00 × 10
8
m
)
s
=
ΔE
3.03 × 10
−19
= 6.56 × 10
−7
m
E02-2-6
J
Notice that we have not included the negative sign of the energy in this calculation. Wavelengths and frequencies
are always expressed as positive quantities. The direction of the energy flow (from the atom to the surroundings) in such
cases is indicated by stating that the photon was emitted. You may compare the calculated wavelength with one
experimentally observed (F02-1-2).
The Bohr model predicts that we can also excite hydrogen atoms with light. For example, if we direct a beam of
white light at a sample containing hydrogen atoms and collect the light after it has passed through, we notice that some
wavelengths are missing (black lines). Such an absorption spectrum is shown in F02-2-2.
Figure F02-2-2. Absorption spectrum of hydrogen. Missing wavelengths (black lines) correspond to wavelengths absorbed
by the sample, resulting in the promotion of electrons to higher orbits.
Only the wavelengths that exactly match the energy differences between orbits are absorbed and, therefore, no
longer detected by the spectroscope. This experiment illustrates nicely the complementary nature of emission and
absorption spectra (compare with F02-1-2).
Sufficiently energetic photons can eject electrons from atoms causing ionization
We can carry out the absorption experiment to an extreme. If we use light that is energetic enough, we should be able to
remove the electron completely from the hydrogen atom. The smallest amount of energy to accomplish that for the
ground-state hydrogen atom corresponds to the electron jump from ni = 1 to nf → ∞. It is called the ionization energy (IE),
since we are producing a separate ion (proton, H+) and an electron. It is easy to figure out (E02-2-4) without doing any
calculations that the value of ionization energy for the hydrogen atom is +2.18 × 10–18 J. We may calculate the IE value
for one mole of hydrogen atoms by multiplying the per-atom energy by Avogadro’s number.
IE = (6.02 × 10
23
atom
mol
) (2.18 × 10
J
−18
atom
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kJ
) = 1310
E02-2-7
mol
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This prediction agrees very well with the experimentally determined value (see L04-2) and adds to our confidence
that the model is at least a good approximation of the internal structure of the simplest atom. Unfortunately, that simple
model fails to accurately explain the spectra of atoms and ions having more than one electron, except at the most
rudimentary level. The underlying weakness of the model is that it relies on the idea that an electron behaves in a
deterministic fashion as a small particle circling the nucleus in an orbit. As we will learn next, this corpuscular description
of electrons does not apply on the atomic scale. Nevertheless, Bohr’s model has taught us some valuable lessons about
electrostatic interactions between electrons and nuclei, the kinetic energy of electrons, quantized energy levels, sizes of
orbits, and interactions of light with atoms (atomic spectroscopy).
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02-3 Matter waves
Subatomic particles and atoms exhibit dual wave-particle behavior
We have already seen examples of the dual wave-photon behavior of electromagnetic radiation. For example, the
photoelectric effect was explained by considering light as a stream of packets of energy called photons that may transfer
their energy through collisions, quite like material particles. How about the reverse situation? Can electrons, or any small
particles of matter behave as waves when in motion?
In 1924 Louis de Broglie first proposed that a beam of electrons can be diffracted just like a beam of light or a water
wave. Such matter waves, referred to as de Broglie waves, are at the center of quantum theory and another example of
wave-particle duality. Photons, electrons, neutrons, and even atoms all have both wave-like and particle-like
characteristics. How we observe or measure them decides which aspect they display. Of course, we do not have to worry
that a baseball thrown 100 miles per hour will “wave” its way to the catcher’s glove. Ordinary-size objects have incredibly
small and undetectable wavelengths. Indeed, the de Broglie wavelengths (λdB) may be used as a guide to distinguish the
classical behavior of the macroscopic objects from the quantum world. Only objects for which λdB is comparable to the
size of the object will show these dual particle-wave characteristics.
Just a few years after de Broglie’s proposal, the wave properties of electrons were demonstrated through a
diffraction experiment on a crystal (F02-3-1a). You may appreciate the similarity of that pattern to the one made by laser
light waves passing through crossed diffraction gratings. Since then, the technique of electron diffraction has been
highly developed to obtain images on the atomic scale. The electron microscope (F02-3-1b) can magnify objects by 3 ×
106 times (electron waves may have wavelengths comparable to X-rays), much more than can be achieved with visible
light (103 magnification).
Figure F02-3-1. From left to right: (a) Electron diffraction pattern from a silicone crystal, (b) electron microscope, and (c)
electron-microscope image of a flea.
De Broglie postulated that in a one-electron atom the quantization may have the same matter wave origin, but in
this case the electron behaves as a standing wave. In the hydrogen atom the wave would have to be circular and to be a
persistent, well behaved wave, an integral (quantized) number of wavelengths would have to fit into the circumference of
the circular orbit (Figure F02-3-2).
Figure F02-3-2. Standing circular wave (left) and a non-integer wave structure (right). No persistent wave can form if an
integral number of wavelengths do not match the circumference of the circle such as the structure on the right.
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The inevitable consequence of the de Broglie standing wave description of the electron in an atomic orbit is that we
cannot simultaneously know the precise position and the momentum of the electron. After all, even if the momentum (mv)
is known precisely, the position of the electron is “spread” around the circle. (F02-3-2). That “uncertainty” was shown
mathematically by Werner Heisenberg to be an unavoidable consequence of the dual nature of matter. The Heisenberg
principle (called the uncertainty principle) puts a fundamental limit on the precision with which we can simultaneously
determine the momentum and position of atomic-scale particles.
Electrons in atoms are described by quantized wavefunctions
whose squares represent electron density
Even if we cannot know the exact position and momentum of the electron, we can talk about the probability of it being
within certain volumes of space around the nucleus. That was the approach taken by Erwin Schrödinger, who in 1926
formulated the equation (E02-3-1) bearing his name that opened a new way of dealing with subatomic particles, now
known as quantum mechanics. The applications of the Schrödinger equation require advanced calculus, but its solutions
and some important basic ideas may be presented graphically in a qualitative but useful manner.
HΨ = EΨ
E02-3-1
At first glance the equation looks quite simple, but the “quantum devil” is in the detail. It all starts with de Broglie’s
idea that electrons in atoms are described by standing waves. The mathematical expressions of these waves are called
wavefunctions and are usually represented by a Greek Ψ (psi). The waves cannot be measured directly (they have no
physical meaning), but their squares (Ψ2) represent the probability of an electron residing in a given tiny volume of space;
this probability is called the electron density. The wavefunctions for an atom (or a molecule, or any quantum system) are
found by solving E02-3-1. In that equation, “ℋ” hides a set of mathematical operators (mathematical instructions on how
to manipulate the wavefunction) that calculate all energies of the system including all electrostatic interactions between all
nuclei and electrons and the kinetic energies of all electrons. The results of the calculations are the energies of the
quantum systems, E, and the wavefunctions describing them.
Before we look at the specific solutions for a hydrogen atom, let’s play with some waves and wavefunctions (Figure
F02-3-3). As you will discover, some experience at the beach may help us to conquer the quantum world.
Figure F02-3-3. (a) Water wave produced by a bounced droplet and (b) the cross-section of the water wave. When
"frozen" in the picture, the shape sufficiently illustrates the concept of a "standing" wave, analogous to standing electron
waves in atoms.
A familiar wave is created when a water drop hits the water surface and bounces back as immobilized in the picture.
Here is our standing wave: tall in the middle, then dropping below the level of the undisturbed water surface, to climb
again, and drop again in a series of concentric hills and valleys, frozen by the camera to be examined in detail. If we
produce a cross-section trace of the water’s surface on graph paper, with the height of the water’s surface represented on
the vertical axis (z) and the separation from the center on the horizontal axis (x), we get a mathematical function that has
positive algebraic values in the upper part of the plot and negative values below the x axis.
The function in F02-3-3b is two dimensional, for each point [x] we have a value of the wavefunction as read off of the
z-axis [Ψ(x)]. If we spin our function around the z axis we can recreate the water’s surface. That surface is threedimensional; for each point [x,y] in the xy plane the function has a value expressed on the z axis [Ψ(x,y)].
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To transition to atomic wavefunctions we have to add another dimension. For each point in space [x,y,z] around the
nucleus we have to specify the value of our wavefunction [Ψ(x,y,z)]. Unfortunately, displaying things in four dimensions is
rather difficult. We will rely on various cross-sections to pare the dimensionality (as we did in our water-wave example
here) or use other graphical “tricks” to simplify the presentation.
The probability of finding an electron in a miniscule volume of space around a point [x,y,z] is equal to the square of
the value of the wavefunction at this point. We need to use these tiny volumes (dxdydz cubes, for those mathematically
inclined) as points themselves have no volume. This electron density (probability per unit volume) can be visualized as a
fraction of an electron in that minuscule volume cube. Notice that squared values are never negative, as probability must
be positive, even if the wavefunctions may be negative in certain regions of space.
Electron density is a directly observable quantity. Many modern techniques allow us to probe it and have
successfully verified the predictions of quantum theory.
Now we have completed the transition of our understanding of electrons in an atom from a particle in an orbit to a
probability wave. Next, we will explore these electron density distributions, called orbitals, in more detail.
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02-4 Quantum numbers
Orbitals identified by sets of quantum numbers, n, ℓ, and mℓ,
represent electron density distributions in atoms
Solving the Schrödinger equation for a hydrogen atom gives us a series of wavefunctions with their associated specific
energies describing the ground-state and excited states of the atom. These solutions are called orbitals in order to
distinguish them from the deterministic orbits of Bohr’s model. Each orbital has a characteristic shape and size reflecting
the distribution of electron density. Since there is only one electron in the hydrogen atom, that electron, depending on its
energy, can be described only by one of these wavefunctions at any given time. We say that the electron occupies that
orbital and has the energy of that orbital. Nevertheless, for convenience, we talk about all orbitals as objects in existence
all the time. Strictly speaking, the orbital "exists" (i.e. represents electron density) only if occupied, but if its electron’s
energy changes (by absorption or emission of light, for example) the unoccupied orbitals "are there" ready to serve as a
new “home” for that electron (now of different, “new” energy).
The Bohr model arbitrarily introduced the principal quantum number n to describe an orbit. The solutions to the
Schrödinger equation are characterized by three quantum numbers, which result naturally from the wave mathematics
used to calculate them.
1. The principal quantum number n can have positive integral values (n = 1, 2, 3...) as in Bohr’s model. As n increases,
the orbital becomes larger, and on average the electron spends more time farther away from the nucleus. Because of
the diminished electrostatic attraction, the energy of the orbital increases with n and it does so in exactly the same
manner as it did in Bohr’s model (E02-2-3). The energy En increases proportionally to 1/n2; En = RyZ2/n2 (the
Rydberg equation), where Z is the atomic number and also the nuclear charge (Z = +1 for the hydrogen atom).
2. The angular momentum quantum number, ℓ can take on any integral value from 0 to (n–1). This quantum number
defines the shape of the orbital. For historic reasons, specific values of ℓ have letter designations as shown below.
For higher values of ℓ, the alphabetical order is followed with g for ℓ = 4 and so on.
value of ℓ
0
1
2
3
letter designation
s
p
d
f
3. The magnetic quantum number, mℓ can have integral values between –ℓ and ℓ, including zero. This quantum number
describes the spatial orientation of the orbital.
Orbitals are grouped into shells of the same n and subshells of the same ℓ
Although there are an infinite number of orbitals possible for the hydrogen atom, we will be mainly interested in those that
have low energy, i.e. those with low values of n. The table below collects some of the possible orbitals to illustrate their
characteristic pattern.
Table T02-4-1. The allowed values of quantum numbers for n = 1, 2, and 3
n
1
2
ℓ
0
0
mℓ
0
0
subshell designation
1s
2s
2p
3s
3p
3d
number of orbitals in a subshell
1
1
3
1
3
5
number of subshells in a shell
1
2
3
number of orbitals in a shell
1
4
9
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1
−1
0
0
+1
0
1
−1
0
2
+1
−2
−1
0
+1
+2
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The collection of orbitals with the same value of n is called an electron shell. The set of orbitals that have the same n
and ℓ values is called a subshell. Each subshell is designated with a number (n) and a letter corresponding to the value of
ℓ (shown in bold in the table). There are (2ℓ+1) orbitals in each subshell. The total number of subshells for each shell is
equal to n, and the total number of orbitals in each shell is equal to n2.
You may continue to develop this pattern with n = 4, to find 4 subshells (4s, 4p, 4d, and 4f) and a total of 16 orbitals,
the last 7 of them belonging to the 4f subshell. The total number of orbitals in consecutive shells multiplied by two (2, 8,
18, 32) hints tantalizingly at the structure of the periodic table (as these numbers match the number of elements in various
rows). Of course, that relationship is not accidental, as we will discover in Lesson 03.
The relative energies of hydrogen orbitals exactly follow the pattern of Bohr’s model (compare to F02-2-1). In Figure
F02-4-1, each box represents an orbital and the boxes are grouped according to their subshell designations. As already
indicated, the energies of hydrogen orbitals depend only on the value of n. Thus, all orbitals of a given shell have the
same energy. Such orbitals of the same energy are called degenerate orbitals. The 1s orbital has the lowest energy. If it is
occupied by the electron, the hydrogen atom is in its ground state. If the electron occupies any of the other orbitals
(including those not shown in the figure with n > 3) the hydrogen atom is in an excited state. At ordinary temperatures,
essentially all hydrogen atoms are in the ground state, unless energy is absorbed by the atom resulting in a transition.
Transitions between orbitals (absorption or emission) follow exactly the same pattern as that described for Bohr’s model
(F02-2-1).
Figure F02-4-1. Energies of atomic orbitals in the hydrogen
atom (s, p, d, etc.) follow the pattern of the energy levels of Bohr's
orbits. Each box represents one orbital, and orbitals (boxes) of
the same shell have the same color. The orbital energies are
inversely proportional to n2, and the energy differences between
them diminish with increasing n. For n ≥ 2, each energy level is
represented by several degenerate (equal energy) orbitals.
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02-5 Orbitals
Orbitals describe the probability of finding electrons in defined areas around the atomic nucleus
Imagine that we find very special 3D cameras on sale. We get a few hundred of them and give them to several sections of
CHEM 110 students for a weekend project. Each student gets a camera and just one hydrogen atom in a special box.
Their task is to take as many photographs of their hydrogen atom as possible. You know that in the real world this would
not be possible for many reasons, but there are few limits to our imagination.
We collect all the digital pictures taken and add them together in "Photoshop 3D" in such a way that all nuclei are
perfectly aligned at the point where the Cartesian axes intersect. In our special pictures, the nucleus is invisible, but
electrons show up as dots. Our congregate image (F02-5-1) represents a measure of the probability of finding the electron
in different volumes of space around the nucleus of the ground-state hydrogen atom.
F02-5-1. Pointillist representations of hydrogen orbitals obtained in an imaginary experiment, where students are
"photographing" electrons (in 3D) with a special camera. Thousands of dots illustrate the probability of finding one electron
in different spaces around the hydrogen nucleus located at the origin. Click on the orbital pictures to explore them in the
3D viewer.
You may easily discern the probability distribution pattern. It has spherical symmetry, with high electron density
toward the center (nucleus) and low and diminishing densities at the outer edges of our observation box. By taking
thousands of pictures of thousands of electrons in thousands of individual positions in thousands of atoms, we have
learned about the probability of finding one electron in the different regions of space surrounding the nucleus.
Interestingly, the collections of pictures from students from some class sections showed different patterns (Figure
F02-5-1). It turns out that some students were given imaginary boxes with an extra energy supply (batteries included!) that
kept the hydrogen atoms in various excited states. Some of these probability patterns (shown here as slices) are still
spherical, but they have larger spreads than that observed for the ground state hydrogen atom. They also have some
white spaces in their interiors, illustrating regions where electrons are not allowed to be (these are called nodes). Other
probability distributions are a bit more complicated. They have “directionality” and more complex node patterns.
These pictures are our first introduction to the different shapes and sizes of hydrogen orbitals. All orbitals with
spherical symmetry are s-type orbitals: 1s, 2s, and 3s with n and ℓ defined ( ℓ = 0). The dumbbell shapes are p-type
orbitals, 2p and 3p (ℓ = 1) with nodes at the nucleus. They are representatives of a 3-piece set in each case. The last one
illustrated in F02-5-1, 3d, belongs to a 5-member club (ℓ = 2) of even more diverse profiles. In hydrogen atoms, orbitals
with the same n are degenerate, so 3s, 3p, and 3d would all have the same energy.
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Spherical s orbitals increase in size as the value of the principal quantum number increases
In reality, the shapes, sizes and directions of orbitals are obtained by solving the Schrödinger equation (E02-3-1). These
mathematical functions (i.e., wavefunctions) for s-type orbitals are plotted in Figure F02-5-2 in various representations.
1s
2s
3s
Figure F02-5-2. The wavefunctions (top), electron densities (middle), and radial density distribution (radial probabilities,
bottom) for 1s, 2s, and 3s hydrogen orbitals as they spread out from the nucleus positioned in the center of the plots (r =
0). The distance from the nucleus (r) is given in bohrs (ao) and Å. Click on the plots to explore them in more detail.
The first set of plots illustrates the wavefunctions for 1s, 2s, and 3s hydrogen orbitals. These functions have the
highest value at the origin (which is at the nucleus) and they drop off equally in all directions as the distance from the
center (r) grows; they have spherical symmetry. The wavefunctions for 2s and 3s extend further from the center than 1s,
have regions of positive and negative values, and have nodes in spots where the wavefunctions change their sign. These
functions have no physical meaning, but you can compare the 3s function to our water wave (F02-3-5b) to appreciate the
analogy.
The squares of these wavefunctions represent electron density. They show the probability of finding an electron in a
tiny volume of space around each point of atomic space. These are given as fractional numbers (or probabilities) per unit
volume. The squares of the wavefunctions are always either zero or positive; probabilities must be positive. The electron
density is the highest at the nucleus and drops off as r increases. In a fashion parallel to the wavefunctions, the number of
nodes increases with the value of n.
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The relationship between these wavefunctions and our imaginary photographs (F02-5-1) is shown in the cutout 3D
representations ("onion-layer" spheres) where the color intensity varies with values of electron density. These are indeed
some of the “tricks” of visualizing four-dimensional constructs that we mentioned earlier.
The total probability of finding an electron anywhere within the orbital must be unity, so the sum of all probability
densities for all volume units must equal 1. As you may have noticed, the wavefunctions and electron density functions
stretch to infinity, even though there are extremely low values of electron density at larger r values. To facilitate graphical
presentations of orbitals, an arbitrary cut-off point is chosen in such a way that the probability of finding the electron inside
the demarcated volume is 90%. That cutoff point corresponds to a very low value of electron density (anyway, what’s 10%
difference between friends?). For orbitals, it represents an artificial isosurface that serves as an imaginary border or "skin",
making the outside onion-layer smooth in our pictures, instead of the fuzzy outline shown in pointillist representations in
Figure F02-5-1.
We may ask what the most probable separation distance is between the electron and the nucleus. With the help of a
pointillist representation (Figure 02-5-1) we may divide the space around the nucleus into narrow spherical shells ("onion
layers") and count the number of dots in each layer. The results of such a count carried out on electron density values are
shown by the last set of functions in Figure F02-5-2, which are called radial probability distributions or radial probability.
Close to the nucleus we have high values of electron density, but the "onion layers" have small volumes. At larger
distances the electron density drops off, but the volume of the layers increases As a result, the most probable distance for
the electron in the ground state hydrogen atom is 0.529 Å; this is the same number as the Bohr radius, ao (E02-2-1). Bohr
was almost right after all! The most probable distance increases for the other s orbitals. As we will see later, such patterns
of distribution of electron density will help us to understand the chemical properties of atoms.
For shells with n > 1, there are three degenerate dumbbell-shaped p orbitals
perpendicular to each other
The p orbitals come in sets of three, all having the same overall shape but each pointing in a different direction in space.
For reasons beyond the scope of our course , we cannot assign a specific value of mℓ to each member of the set, but we
will call them px, py, and pz to indicate their spatial orientation. Examples of wavefunctions and radial probability for p
orbitals are shown in Figure F02-5-3. The two lobes of p orbitals have opposite signs for the wavefunction, and there is a
node at the nucleus. The squares of the wavefunctions are all positive, and the radial probability distributions show the
most likely distance from the nucleus for the electron.
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2p
3p
Figure F02-5-3. The wavefunctions (top), electron densities (middle), and radial density distributions (radial probabilities,
bottom) for 2p and 3p hydrogen orbitals as they spread out from the nucleus positioned in the center of the plots (r = 0).
The distance from the nucleus (r) is given in bohrs (ao) and Å. Click on the plots to explore them in more detail.
If we spin the p wavefunction around the corresponding axis, we get a 3D representation of the shape of this type of
orbital. As we have done with the s orbitals, we will only show the isosurfaces ("skins") selected in such a way as to have
90% of electron density inside the contained volume. Additionally we will paint (or otherwise label) the isosurface of the
orbital with a color representing the sign of the underlying wavefunction. In fact, we do not even know which lobe is
positive and which is negative (we only know the values of their squares), but that distinction is not important as long as
we remember that they have opposite algebraic signs (F02-5-3). In other words, we will simplify the graphical
presentations of orbitals to show only the 90% probability volumes covered by the skin marked with the sign of the
wavefunction. We will leave all the detailed information of electron distribution inside the orbitals (in the form of contour
plots, or other mathematical functions) only to those who want to explore it in detail. Any volunteers?
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Figure F02-5-4. The boundary surfaces and nodal planes for 2p orbitals. The lobes are colored to indicate the opposite
algebraic signs of their wavefunctions. The volumes encapsulated by the isosurfaces contain 90% probability of finding an
electron within. Note that the sum of the three p orbitals gives an overall spherical distribution of electron density. Click on
an orbital picture to explore it in 3D (small delays are possible as orbitals are calculated).
There are five degenerate d orbitals for shells with n > 2, each having four major lobes
The shapes and directions in space of d-type orbitals are shown in Figure F02-5-5. To keep our promise of simplification,
we will only look at their 90% probability lobes colored according to the algebraic sign of their wavefunctions. We will skip
here the formal presentation of f orbitals (coming in sets of 7 ) as they are the least important for the chemistry covered in
our course.
All d orbitals in the hydrogen atom with the same value of n have the same energy, despite their diverse shapes.
Again we are not able to assign individual m ℓ values to specific d orbitals, but use Cartesian designations to show their
spatial orientations.
Figure F02-5-5. The boundary surfaces and nodal planes for 3d orbitals. The lobes are colored to indicate the opposite
algebraic signs of their wavefunctions. The volumes encapsulated by the isosurfaces contain 90% probability of finding an
electron within. Note that the sum of the five d orbitals gives an overall spherical distribution of electron density. Click on
an orbital picture to explore it in 3D (small delays are possible as orbitals are calculated).
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To help you to recognize the orbitals, you may notice that s orbitals have 1 main lobe ("blob"), p orbitals have 2, and
d orbitals have 4 (the "doughnut" lobe counts as two). For p orbitals, the direction along the axes decides their names.
The d orbitals with lobes between the axes are designated by these axes' labels, and those with lobes along the axes
have the axes' labels "squared" (the "doughnut " is left out of labeling). Except for the 1s orbital, all other orbitals have
characteristic nodal surfaces (nodes), which could be spherical, conical, or planar. The number of nodes is (n-1), where n
is the principal quantum number.
It turns out that the shapes of orbitals of other atoms are identical to those of hydrogen with only an adjustment for
size. These atoms have larger effective nuclear charges (see Lesson 03-2) which attract the electron closer to the
nucleus on average. Thus, the corresponding orbitals have smaller lobes. Additionally, some adjustments in orbital
energies are required. Nevertheless, the overall pattern of orbital shapes and spatial orientations remain the same for all
atoms. As you can see, we have arrived at the point where the intricate components of even more complicated atoms,
ions, and molecules are in our possession. Spend some time familiarizing yourself with these orbitals; it is good to know
your tools well.
After this discussion, the orbitals (or their graphical representations) may seem very real. Indeed, it is convenient to
talk about them as real objects, like empty containers or boxes ready to be packed with electrons. The colored surfaces
we use to visualize them give a feeling of walls behind which we can keep the electrons, such that they act in a
predictable way around nuclei. However, it is important to remember that these are only visualization models and that
there are no walls to keep electrons in or out of certain spaces. These orbitals do not even really exist until we have at
least one electron in them.
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03 Multi-Electron Atoms
Hydrogen orbitals provide us with a blueprint for the electronic structure of multi-electron atoms of other elements.
Nuclear charges and electron-electron repulsion vary among the different elements and affect their relative orbital
energies. In turn, the manner in which the orbitals are filled with electrons dictates the chemical properties of elements.
03-1 Electron spin
Electrons have an additional quantum number that is not associated with the orbital that they occupy. This value, called the
spin quantum number, denotes the magnetic dipole moment of the electron. The spin quantum number can only be one of two
values, ms = ½ or ms = –½. The Pauli exclusion principle dictates that a maximum of 2 electrons can occupy the same orbital
simultaneously. Such electrons are referred to as being paired, and their spins (and thus their spin quantum numbers) must be
opposite (antiparallel).
03-2 Effective nuclear charge
In a multi-electron atom, the electrostatic interactions between electrons and the nuclear charge are complicated by electronelectron repulsion and mutual shielding. The various interactions are summed up by the effective nuclear charge (Zeff), which
represents the average charge experienced by an electron interacting with the nucleus after being influenced by all other
electrons. Differences in the effective nuclear charges account for the lifting of the degeneracy of shell orbitals (seen in hydrogen
atoms) and establish the relative energies of orbitals. In general, for a given n, the subshell energies follow the pattern of ns < np <
nd < nf.
03-3 Electron configuration
We typically establish the electronic structure of an atom using the aufbau principle of adding electrons to the available
lowest-energy orbitals to generate the ground state atoms. An orbital can hold a maximum of two electrons with opposite spins
(and different spin quantum numbers). For degenerate subshells, according to Hund’s rule, the configuration that maximizes the
number of electrons with the same spin has the lowest energy. Irregularities in orbital filling patterns crop up when the orbitals
have similar energies and subtle factors may influence their ordering. The electrons in an atom are divided into low-energy, inner
electrons (core electrons) and high-energy, outer electrons (valence electrons).
03-4 Periodic table
The patterns electrons follow when filling subshells results in a periodic trend wherein the atoms of different elements can be
arranged according to the configuration of their valence electrons. The main groupings of elements include the s- and p-block
elements, the d-block elements (called transition metals), and the f-block elements (called lanthanides and actinides). The position
of an element in the periodic table reflects its valence electron structure as well as its physical and chemical properties.
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03-1 Electron spin
The magnetic moment of an electron can only have two values, ms = +½ or ms = ‒½
Consider a beam of hydrogen atoms being sent through a non-homogeneous magnetic field produced by an
unsymmetrically shaped magnet. The hydrogen atoms are neutral and they should pass through the field without any
perturbation to their path. Instead, we find that the beam is split into two beams bent in opposite directions. Since the
atoms are neutral, this bending can only occur if the atoms are behaving like small magnets; in other words, they have a
magnetic dipole moment. Because the beam is being split into just two beams (as opposed to being spread out in all
possible directions) we can conclude that the magnetic dipoles are only aligning in the magnetic field in two opposing
directions. For this to happen, the magnetic moment must be quantized.
Figure F03-1-1. In the Stern-Gerlach experiment a beam of neutral
atoms is deflected and split into two beams as it passes through an
inhomogeneous magnetic field. The original experiment was carried
out with neutral silver atoms.
The result of this experiment is explained by the existence of a fourth quantum number: the spin quantum number
(ms). The spin quantum number refers to the orientation of the magnetic dipole moment of a specific electron; it is not an
orbital quantum number, as it does not allocate an electron to any orbital. This number may only take on two values, +½ or
–½. We visualize the spin quantum number as the "up" or "down" magnetic dipole moment of a spinning electron.
Figure F03-1-2. The electron is visualized as a spherical
charge spinning about its axis and generating a magnetic
dipole moment (shown as arrows). The two directions of
the magnetic dipole correspond to the two possible values
of ms(+½ and –½). The picture of a spinning charged
sphere is only an artificial representation as we know that
electrons have wave-like properties.
The Pauli exclusion principle limits the number of electrons in any orbital to two,
having opposite spin
Electron spin is crucial to understanding the structure of atoms and the bonds between them as it imposes certain
constraints on electron configurations. It turns out that no two particles with half-integer spin may have the same quantum
state. This restriction, known as the Pauli exclusion principle, means that no two electrons in an atom may occupy the
same space and have all the same quantum numbers. When orbital quantum numbers are specified (n, ℓ, m ℓ ), the only
variability left is in the spin quantum number, which can only have one of two possible values (ms = +½ and ms = –½).
Thus, a maximum of two electrons may reside in a given orbital. If the two electrons are to occupy the same orbital, they
must have opposite spins. Such electrons are called paired and their spins are antiparallel. If an orbital is singly occupied
(by one electron), that electron is called an unpaired electron.
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03-2 Effective nuclear charge
Outer electrons experience diminished nuclear charge
as they are screened by the internal electrons and by each other
Atoms bigger than hydrogen have a larger number of electrons, and more protons in their nuclei. As the atomic number
rises by one, the proton and electron counts also rise by one. The orbitals available to hold these electrons are very much
like those in the hydrogen atom. They are described by the same quantum numbers, and have similar shapes.
Taking into account the Pauli exclusion principle, we can easily visualize the electronic structure of any atom by
apportioning all its electrons to hydrogen-like orbitals, starting with the lowest energy orbital and placing no more than two
electrons in each. This “building up” construction method (called the aufbau principle, from German) assures that the atom
will have the lowest possible total energy and will be in its ground state.
To see how that scheme may work, let's start with hydrogen. Hydrogen’s electron is in the 1s orbital. The case for
helium (2He, atomic number of two) seems straightforward. It has one more electron than hydrogen, but that electron can
be simply added to the 1s orbital. We now have a pair of electrons with opposite spins (filling up 1s completely) trying to
minimize their mutual repulsion by staying as far from each other as possible within the orbital. This task is harder than
one might anticipate because the helium 1s orbital is smaller than the hydrogen 1s orbital. This orbital “shrinking” is due to
the increase in the nuclear charge (Ζ). At first glance it seems that each of the two electrons in the helium atom is
attracted to a +2 nuclear charge. Using Bohr’s model as a guide (E02-2-4), we would then conclude that the average orbit
size in helium should be half of that in hydrogen, as the orbit size is inversely proportional to Z.
The situation is a bit more complex, however, as we also need to account for electron-electron repulsion. In general,
in multi-electron atoms, we cannot analyze all the attractions and repulsions exactly, but we can approximate how any one
electron interacts with an average electrostatic force field generated by the nucleus and all other electrons. The resulting
net attraction is viewed as if it were an interaction of that electron with a point charge at the nucleus, called the effective
nuclear charge, Zeff. For helium, Zeff = +1.69, meaning that each of the two electrons is attracted, on average, to a +1.69
positive charge, rather than +2.00, as we had first assumed (see above).
Moving on to lithium (3Li), its 3rd electron has to be placed in the next available orbital, which is 2s. The attraction of
this electron to the nuclear charge is significantly modified by the presence of the electrons in the 1s orbital. The two 1s
electrons are on average closer to the nucleus than the 2s electron, screening (shielding) it quite effectively from the full
electrostatic attraction expected from the +3 charge on the nucleus. This shielding can be expressed by a positive number
S, called a screening constant, and we can write (E03-2-1):
Zef f = Z − S
E03-2-1
Again, at first approximation we might assume S = 2 for the 2s electron in Li and imagine that the two 1s electrons
“neutralize” +2 of the +3 nuclear charge (Figure F03-2-1). In reality, the situation is again more complex. On the one hand,
the 1s electrons are shielding each other the same way they did in helium. On the other hand, the 2s electron can
penetrate to the nucleus, since it has a non-zero probability of being in the vicinity of the nucleus. When we take all this
into consideration, we find that Zeff for the 1s electrons in Li is +2.69 (less than the +3 value expected without any
screening), and Zeff for the 2s electron is +1.28 (more than the +1 charge we anticipated when we assumed maximum
screening for Z = +3 and S = 2).
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Figure F03-2-1. Effective nuclear charge of the Li atom. The nuclear +3 charge is screened by the two inner 1s electrons.
Effective charge of the outer 2s electrons is approximately, Zeff = 3 – 2 = 1. A peak between 0 and 1 Å on the plot of radial
probability versus distance (right) shows that the outer 2s electrons can penetrate close to the nucleus. Therefore, the 1s
electrons are not fully effective at screening the 2s electrons from the nucleus. Precise calculations show that Zeff(1s) =
2.69 and Zeff(2s) = 1.28.
Orbitals in the same shell of multielectron atoms are not degenerate
because the effective nuclear charge is different for each subshell
The concept of effective nuclear charge is fundamental to our understanding of many atomic properties. The precise
values for Zeff for various orbitals in an atom can only be obtained through sophisticated quantum calculations. However,
for the highest-energy electrons (which to a large degree dictate the chemistry of atoms) an approximation where S is
taken as the number of all inner core electrons is quite adequate for many qualitative considerations. We will use the
concept of Zeff quite often.
The effective nuclear charge is responsible for the lifting of the degeneracy of subshell orbitals. In the hydrogen
atom, the 2s and the three 2p orbitals all have the same energy (see F02-4-1). Based on that information, we could have
considered placing the 3rd electron of lithium in one of the 2p orbitals. It turns out, however, that although all three 2p
orbitals remain degenerate, they all have higher energy than the occupied 2s orbital in Li. Plots of radial probability
(compare F02-5-2 and F02-5-3) are best suited to illustrate the relative penetrations of the orbitals to the nucleus (F03-22). A small peak close to the nucleus is present in the diagram for the 2s orbital, but absent in the diagram for the 2p
orbital; this shows how the electrons in the 2s are sometimes found closer to the nucleus, lowering their energy and
partially screening an electron in the 2p orbital.
Figure F03-2-2. From left to right: (a) Radial probability for the 2s and 2p orbitals showing the penetration of 2s electrons
closer to the nucleus and their subsequent partial screening of the 2p electrons. (b) Similar radial probability plots for 3s,
3p, and 3d orbitals. The small peaks in radial probability close to the nucleus illustrate that regions of greater electrostatic
attraction contribute to lowering of orbital energy in that region, and to the partial screening of electrons in orbitals that are
in the same shell (in this case, shell 3).
This pattern is general: for a given n, the ns orbitals are lower in energy than the corresponding np orbitals, which in
turn have lower energy than the nd orbitals. The same trend is apparent in Zeff values. For example, for boron (5B)
Zeff(2s) = 2.58 and Zeff(2p) = 2.42. For scandium (21Sc) Zeff(3s) = 10.34, Zeff(3p) = 9.41, and Zeff(3d) = 7.12.
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With so many readjustments of orbital energies due to differences in Zeff from atom to atom (and even from orbital to
orbital in the same atom), one may despair that the simplistic order found in the shell-subshell structure of hydrogen
orbitals is lost. However, despite the complications (which do at times provide intriguing outcomes) the “onion-layer”
electronic structure does survive with a few modifications. The orbital energies are still proportional to (Zeff/n)2, and the
most probable nuclear-electron separations are still proportional to (n2/Zeff), just as they were in hydrogen orbitals (or even
in the simple Bohr model). The major difference is that now Zeff is a complex function of both n and ℓ.
The energy ordering scheme for multi-electron atoms is shown in Figure F03-2-3. The subshells are no longer
degenerate within their shells, and there is some intermingling of shells. For example, there are times when the 4s and 5s
orbitals have lower energy than 3d and 4d, respectively. Indeed, the similarity in energy levels between these subshells
results in occasional irregularities and exceptions. Now that we better understand the ordering of energy levels for the
orbitals, we are ready to start exploring the details of the electronic structure of atoms.
Figure F03-2-3. Ordering of orbital energies for a typical multi-electron atom (right) as compared to the energy levels in
the hydrogen atom (left, compare to F02-4-1). The subshells are no longer degenerate (as they are in the hydrogen atom.
Now the energies follow the ns < np < nd pattern. Even though there are some ns subshells that are lower in energy than
(n-1)d subshells (for example 4s has lower energy than 3d, and 5s is lower than 4d), the layered structure is quite
apparent.
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03-3 Electron configuration
Electrons fill orbitals from the lowest energy to the highest with up to two electrons per orbital
Armed with the freshly-acquired knowledge of the previous two sections, we can now build the periodic table! The rules
governing the construction process can be summarized as follows:
1. The atomic orbitals are ordered in a layered structure of shells and subshells based on the energies of
interactions of electrons with effective nuclear charges (Zeff).
2. The aufbau principle is applied to fill the orbitals with available electrons, starting with the lowest-energy
orbitals and moving up. This filling strategy results in the ground state electronic configurations for atoms.
3. The Pauli exclusion principle restricts the maximum number of electrons per orbital to two (with opposite
spins).
4. Hund’s rule (a new addition, see below) governs electron distribution among degenerate orbitals.
We can translate the energy ordering scheme of Figure F03-2-3 into an orbital-filling “roadmap” to facilitate this
process (F03-3-1). Each box represents a subshell: s subshells have one orbital each, p subshells have 3 orbitals, d
subshells have 5 orbitals, and f subshells have 7 atomic orbitals. Any of the orbitals can hold a maximum of two electrons.
We start with 1s, and follow the arrows diagonally down through 2s, 2p, 3s, 3p, 4s, 3d, etc., until we place the number of
electrons equal to the atomic number of a given atom.
Figure F03-3-1. The “road map” for filling atomic orbitals with electrons in multi-electron atoms. The order follows the
diagonal arrows, from the top to bottom.
We have already analyzed the electronic configuration of H, He, and Li (Section 03-2). We can present their
configurations in a graphical fashion where boxes represent orbitals and up and down arrows represent electrons with ms
= ½ and –½, respectively (Table T03-3-1). Alternatively, we can write that information in a long-form notation, indicating
the subshell label and its electron occupancy with a superscript number. Thus, H would be 1s1, He 1s2, Li 1s22s1, etc. The
continuation of the pattern is straightforward (Table T03-3-1). Beryllium has one more 2s electron (1s22s2), and boron
starts to fill up the 2p subshell (1s22s22p1).
Carbon raises an interesting issue. Where should the 2nd p electron be placed? The three 2p orbitals have the same
energy, but should the new arrival be paired with the other p electron in the same 2p orbital, or should it be placed in
another 2p orbital? Situations like this are governed by Hund’s rule, which states that the lowest energy is obtained, if the
number of electrons with the same spin is maximized. In other words, when electrons are added to orbitals of equal
energy, a single electron enters each orbital of the degenerate set before any are paired. This distribution of electrons
among the orbitals of the degenerate set with parallel spin arrangements helps to minimize electron-electron repulsion,
lowering the energy of this configuration.
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The same principle reveals that nitrogen (1s22s22p3) has three unpaired electrons. Oxygen (1s22s22p4) and fluorine
(1s22s22p5) continue filling up the 2p subshell, with consecutive electrons added to already half-filled orbitals, pairing their
spins. Neon (1s22s22p6) concludes the filling of the 2nd shell, and sodium starts the 3rd one, with one electron in 3s (Table
T03-3-1).
Table T03-3-1. Electron configuration for the first 11 elements
Z 1s
2s
2p
3s
core valence
H
1
↑
1s1
He
2
↑↓
1s2
[He]
Li
3
↑↓
↑
1s22s1
[He] 2s1
Be
4
↑↓
↑↓
1s22s2
[He] 2s2
B
5
↑↓
↑↓
↑
1s22s22p1
[He] 2s22p1
C
6
↑↓
↑↓
↑ ↑
1s22s22p2
[He] 2s22p2
N
7
↑↓
↑↓
↑ ↑ ↑
1s22s22p3
[He] 2s22p3
O
8
↑↓
↑↓
↑↓ ↑ ↑
1s22s22p4
[He] 2s22p4
F
9
↑↓
↑↓
↑↓ ↑↓ ↑
1s22s22p5
[He] 2s22p5
Ne
10 ↑↓
↑↓
↑↓ ↑↓ ↑↓
1s22s22p6
[Ne]
Na
11 ↑↓
↑↓
↑↓ ↑↓ ↑↓
1s22s22p63s1
[Ne] 3s1
↑
1s1
Note on degenerate orbitals
The valence electron configuration is the same for elements in the same column of the periodic
table
Already, we can recognize a certain periodicity in the electronic configuration of atoms. The noble gases helium (2
electrons) and neon (8 electrons) have their respective shells completely filled. Any element following a noble gas starts
over with one electron in the s orbital, repeating the pattern. Indeed, hydrogen, lithium, and sodium have one s electron
(ns1) as their highest energy electron. Looking at the highest energy electrons, it is apparent that we should have a set of
s-type elements (2 of them for each n) followed by a set of p-type elements (6 of them for each n). Indeed, this pattern
becomes our first draft of the periodic table.
The full-shell configuration is particularly stable as it has a completely spherical electron-charge distribution (F02-54). The electrons making up this noble gas configuration that has valence orbitals filled are called core electrons.
Electrons in the outermost shell that comes after the core shells are called valence electrons. All elements in the second
row of the periodic table (between Li and Ne) have a helium core, and all elements in the third row (between Na and Ar)
have a neon core. Instead of listing these core electrons explicitly, we can use a short-hand notation by specifying the core
in square brackets. For example, carbon’s configuration can be denoted [He]2s22p2, and sodium’s configuration is [Ne]3s1
(Table T03-3-1).
This division of electrons into low-energy core electrons and high-energy valence electrons is not just for
convenience of notation. It reflects the basic chemical principle that low-energy inner electrons are not involved in
chemical processes. They experience high Zeff and are strongly attracted to the nucleus. On the other hand, the highenergy outer electrons are shielded by the core electrons, experience a lower Zeff, and are readily involved in chemical
processes such as bond formation.
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Anomalies in electron configurations are caused by subtle differences in subshell energies
For elements of higher atomic number than argon ([Ne]3s23p6), a simplistic approach would predict that the highest
energy electron in potassium should end up in the 3d orbital as the d-subshell becomes available for n = 3. However, our
energy diagram (Figure F03-2-3) and the road map we prepared (Figure F03-3-1) indicate that the 4s subshell is lower in
energy and therefore filled first. Consequently, K has a configuration of [Ar]4s1, and Ca is [Ar]4s2. Only after calcium do
we place the next electron in the 3d subshell. Scandium’s electron configuration is [Ar]3d14s2, initiating a series of 10
elements that gradually fill all of the five 3d orbitals with their highest energy electrons. Therefore, as we start the 4th row,
our periodic table must be expanded to include d-block elements.
A very similar situation occurs in the 5th row with 4d orbitals and the 6th (n = 6) and 7th (n = 7) rows of our periodic
table when the f-subshells become accessible. Each f-subshell has 7 degenerate orbitals, which may accept 14
electrons. This gives us a new f-block with 14 elements per row. Here, the ns, (n-1)d, and (n-2)f subshells are very close
in energy. As you may have guessed, the ns orbitals are filled before (n-1)d or (n-2)f, but what may surprise you is that (n2)f will fill before (n-1)d. The closeness in energy of these orbitals leads to numerous anomalous electronic configurations
(Figure F03-4-2). Most of these deviations are of limited chemical significance and we are going to leave them
unexplored.
One of the significant anomalies that illustrates a broader trend is what happens when the orbitals are close in
energy and there are enough electrons to precisely half-fill or completely fill the subshells. Let’s consider only the 4th row
of the periodic table. We would predict that chromium would have the electronic configuration [Ar]3d44s2, but in fact the
correct configuration is [Ar]3d54s1. One filled subshell and one partially filled subshell are “traded” for two half-filled
subshells. Half-filled subshells have perfectly spherical electron charge distribution (F02-5-5) and can better minimize
their electron-electron repulsions, which in turn provides extra stability. Similarly, copper has the configuration
[Ar]3d104s1 (with one filled and one half-filled subshell), rather than [Ar]3d94s2, which would be higher energy. It should be
noted that the electronic configurations of d-elements can be written by ordering the subshells by n ([Ar]3d104s1) or by
filling order ([Ar]4s13d10). Both ways are equally acceptable.
With this new understanding of d-block and f-block elements, we can now return to the concept of valence
electrons. Formally, the d-electrons and f-electrons are outer electrons. If the d-subshell or the f-subshell is not completely
filled, we consider these outermost electrons to be valence electrons; they participate in chemical reactions, which involve
ion or bond formation. However, if these subshells are fully filled, these electrons do not participate in bond formation, and
behave as if they were core electrons. In such cases we do not consider these electrons to be valence electrons.
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03-4 Periodic table
The s, p, d, and f blocks of elements in the periodic table are due to the periodicity of the
electron configuration of atoms
The relative energies of hydrogen-like subshell orbitals, coupled with the Pauli exclusion principle, are responsible for the
repeating patterns we see in the organization of the atoms of various elements. The result of these patterns is the
formation of periods, where any atom has the same valence electron configuration as an atom of the element directly
above it in the periodic table (Figure 03-4-1). For n = 1, the only orbital available (1s) can hold up to two electrons. Thus,
the first row of our table can only hold two elements. For n = 2, the 2s orbital is available (two elements), along with three
2p orbitals (6 elements); there are a total of 8 elements in the second row. The pattern gets a bit more complicated in the
third row: the 3s and 3p subshells repeat the 2s/2p arrangements, but the 3d subshell drops down because its energy is
higher than that of 4s, which therefore fills first. In the 4th row we gain a 10-element-wide d-block, where electrons
gradually fill the five available d orbitals. Here, too, the 4f subshell’s energy is higher than that of 5s, so the f-block drops
down in our table. The 5th row repeats the pattern of the 4th. In the 6th row, the f-block elements (14-elements wide) finally
become available, bringing the seven f-subshell orbitals into play.
Figure F03-4-1. The periodic table is constructed based on the relative energies of atomic orbital subshells, where each
orbital can be filled by at most 2 electrons, based on the Pauli Exclusion Principle. The s-block elements have ns valence
electrons and the p-block elements add np electrons to the valence shell. The d-block and the f-block electrons are
considered valence electrons only if their subshells are not full.
Figure F03-4-2. The common version of the periodic table, with the f-block dropped below the rest of the table. The
members of a vertical column are said to form a group or belong to a family of elements. In groups, elements have very
similar properties and exhibit clear trends in their properties down the group. The groups are numbered from 1 to 18 (the
current international convention), or from 1 to 8 with A or B designations (an older scheme used in the United States).
Elements with electronic configurations that deviate from the standard pattern are marked.
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The exceedingly wide periodic table in Figure F03-4-1 is often converted to a narrower form (F03-4-2) where the f-block
elements are pushed below the rest of the table, as shown in Figure F03-4-2. The s- and p-block elements are usually
called the main group elements. The d-block elements are called the transition elements (or transition metals), and the
two rows of f-block elements are called the lanthanides and actinides, respectively.
Metals occupy most of the periodic table whereas nonmetals are located in the upper right
corner
We will next explore how the position of an element in the periodic table (and therefore its electronic configuration),
determines its physical and chemical properties. In the most general sense, most elements in the periodic table are metals
(Figure F03-4-3). They occupy the left-side and the center of the periodic table. The nonmetals are assembled in the
upper-right section of the periodic table, with a few metalloids (of intermediate character) on the borderline between the
two. More details on the elements can be found in our interactive periodic table which you can activate by clicking the
button above the top of this page.
Figure F03-4-3. The periodic table with metals, nonmetals, and metalloids marked. The elements in the same column
(with the same valence electron configuration) exhibit similar chemical properties and are given "family" names. Alkali
metals (group 1) are all highly reactive and readily become +1 ions, while alkaline earth metals (group 2) are less reactive
and form +2 ions. Although the electronic configuration of transition metals is not the same within the block (groups 3 12), they share many characteristics that set them apart from other elements. Transition metals in a single group are
similar because of their incompletely filled d-subshells. Similarly, lanthanides and actinides resemble each other due to
their partially filled f-subshells. Among nonmetals we have the chemically inert noble gases (group 18 or 8A), the
halogens (group 17 or 7A), the oxygen family (chalcogens, group 16 or 6A), and the nitrogen family (group 15 or 5A). We
will learn more about the chemical properties of these elements in future Lessons.
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04 Periodic Properties
The periodic table’s ordering of atoms according to their electronic configurations allows us to systematically explore
the trends in their properties that dictate their chemical behavior. Atomic size dictates how close atoms will come to each
other when making covalent bonds in molecules, while ionic size does the same for ionic solids. As we will soon learn,
distance plays a crucial role in determining the strength of a bonding interaction. Ionization energies give us a quantitative
measure of how easily valence electrons may be removed from atoms (thus creating positively charged cations), while
electron affinities tell us how easy it is to add electrons to atoms (creating negative anions). Understanding these
properties and the trends that they follow allows us to better appreciate the periodic nature of an element’s characteristics,
and sets us up to explore bonding in future lessons.
04-1 Atom sizes
Nonbonding radii measure how closely the atoms approach each other during collisions, while bonding radii determine the
distance between the nuclei of bonded atoms. The bonding radius is dependent on the Zeff of the outermost (valence) electrons
and on the number of electron shells that the atom has. As we move down any column of the periodic table the atomic radii
increase significantly due to the addition of one electron shell per each row. From left to right, the atomic radii decrease due to the
systematic increase in Zeff. These trends are strongly pronounced among the main group elements, but are weaker among
transition metals.
04-2 Ionization energies
The ionization energy (IE) of an atom is the minimum energy required to remove an electron from a ground state atom in the
gas phase. The consecutive ionization energies measure the energies needed to remove additional electrons. The values of the
ionization energies increase dramatically when the electron to be removed comes from a filled shell; this trend makes it quite
simple to distinguish between valence and core electrons. Trends in IE1 (removal of the first electron) are dictated by atomic size
(bigger atoms are easier to ionize) and Zeff (larger nuclear charge increases IE). Thus, the ionization energies generally increase
from left to right in each row of the periodic table (with noble gases having the largest IEs), and decrease going down the columns,
since the largest atoms have the smallest ionization energies. Small deviations from the general trends are observed in atoms with
filled or half-filled subshells, as these electron configurations are more stable and result in a larger IE. The patterns for transition
metals are less pronounced than those for the main group elements.
04-3 Electron affinities
The electron affinity (EA) is the energy released when an atom in the gas phase accepts an electron. Most atoms in the gas
phase can accept one electron to produce stable anions of charge −1. The trends are similar to those observed for ionization
energies, but are less pronounced and there are some shifts in column trends. In general, EAs become more negative as we
move from left to right in any row of the periodic table, with halogens having the most negative EAs. The trend down the columns
is less defined, but the bigger atoms usually have more negative values. Atoms with filled or half-filled subshells have small
negative or unmeasurable (and unfavorable) electron affinities, while atoms that are one electron short of having filled or half-filled
subshells have highly favorable EAs.
04-4 Ion sizes
Cations are always smaller than their parent neutral atoms: this is the result of removing the outermost electrons and having
diminished electron-electron repulsions. Anions are always larger than their parent neutral atoms as they have increased electronelectron repulsions. Ion sizes follow the same periodic trend as bonding atomic radii: they increase down the column and
decrease from left to right on the periodic table (for the same ion type). Ions most commonly have the configuration of the noble
gas nearest to the parent atom. Ions with the same electronic configuration are called isoelectronic. Ion sizes decrease
systematically with increasing Z, regardless of the ion type. When cations are formed, the electrons are first removed from the
occupied orbital having the largest principal quantum number, n. If there is more than one occupied subshell for a given value of n,
the electrons are first removed from the orbital with the highest ℓ value. In such cases the cation may end up with an electron
configuration that differs from a noble gas configuration.
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04-5 Periodic trends
We can sort the elements into metals, nonmetals, and metalloids simply by examining the IE1 values; our boundary line
dividing metals from nonmetals is just under 1000 kJ/mol for the ionization energy. Metals have low IE1s (below the line) and
easily form cations. Nonmetals have large IE1s (above the dividing line); they do not form cations readily, but some of them form
anions. The metalloids have intermediate properties and IE1 values close to the boundary line. The metals occupy the left and
center part of the periodic table. Their properties are related to the presence of high energy valence electrons which are easily
removable. The nonmetals occupy the right side of the periodic table. Their valence electrons are of lower energy and are held
more tightly. The metalloids occupy the narrow space between the metals and nonmetals in the periodic table. These patterns
dictate how these atoms will behave when bonding with other atoms of the same element, or with atoms of other elements.
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04-1 Atom sizes
Effective nuclear charges for valence electrons increase significantly from left to right
within each period
We learned in Lesson 03-2 how the effective nuclear charge (Zeff) influences the sizes and the energies of atomic
orbitals. Let’s examine the trends in Zeff in more detail, focusing on the valence electrons. As you may recall, we can
simply approximate Zeff by subtracting the number of core electrons from the number of protons in a given atom, Zeff = Z –
S (E03-2-1). Z is the atomic number. For more precise answers, we must rely on sophisticated quantum calculations. We
compare the results by plotting them against their atomic numbers in Figure F04-1-1.
Figure F04-1-1. The approximate effective nuclear charge for
valence electrons calculated using equation E03-2-1 (orange circles),
and precise values obtained by quantum methods (blue circles). The
open circles represent transition metals (d-block elements). Zeff
increases significantly from left to right in the individual periods, but
increases only slightly down the individual columns (blue circles).
Plots of properties versus atomic numbers (Z) are an excellent way to observe recurring trends that repeat for every
row of the periodic table. Indeed, the "saw-tooth" pattern is easily recognizable in our Zeff plots in the figure. This pattern is
emphasized by the labeling of the alkali metals and noble gases which start and end each period, respectively.
At first, one may be discouraged by the divergence of the readily available approximate values (orange circles) from
the more realistic numbers that can only be obtained from the quantum calculations (blue circles). However, as long as
we are interested in trends rather than precise values, we can be satisfied that the slopes of the blue segments are just
slightly smaller than the slopes of the orange segments.
A major exception is observed for the transition metals, which show a much smaller slope than expected. This
“flattening” of the curve is indeed an indication that transition metals are much more similar to each other in their
properties than are elements of other adjacent groups (families), as we have previously stated.
We can safely conclude that Zeff values for elements steadily increase from left to right in individual rows of the
periodic table. Also, the effective nuclear charge experienced by the valence electrons increases somewhat going down
the columns.
The effective charge experienced by the valence electrons influences chemically-significant atomic properties. The
larger the charge, the more strongly and closely the electrons are attracted to the nucleus, the harder they are to remove,
and, for a given n, the smaller the atom is. It is as though the outermost electrons set the atom “boundaries”.
Nonbonding and bonding atomic radii are two measures of atomic sizes
We have learned that electron clouds are quite fuzzy, so defining the atomic size requires some explanation. First
imagine two identical atoms in the gas phase colliding with each other, with typical kinetic energy (at room temperature).
They will fly apart like billiard balls because of the electrostatic repulsion of their electron clouds. The closest distance
between the two nuclei in the instant of their collision corresponds to twice their nonbonding radii. Such a radius, also
called the van der Waals radius, is a measure of atomic sizes for “free” atoms. We will explore it, and its molecular
analogue, in more detail when we discuss intermolecular interactions. You may explore the van der Waals radii trends
for many elements in our interactive periodic table (the button on the top right side of the page).
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Figure F04-1-2. The nonbonding radius (rn) is half the closest distance between nuclei in the instant of the collision of gas
phase atoms. It is also called the van der Waals radius. The bonding radius (rb) is half the distance between bonded
nuclei in a compound. Bonding requires larger interpenetration of electron clouds.
In contrast, when atoms form molecules, their electron clouds penetrate each other much deeper than during a
simple collision. We will explore bond formation in much greater detail in Chapter 6, but the inter-nuclear distances in
molecules provide another measure of the atomic size called the bonding atomic radii. These distances have been
measured by a variety of scientific techniques for a large number of molecules, and then averaged for individual atoms.
For example, the carbon-carbon distance in the structure of diamond is 154 pm (1.54 Å), which translates to a bonding
radius of 77 pm (0.77 Å) for the carbon atom. The inter-nuclear distance in an I2 molecule is 266 pm (2.66 Å), which gives
an atomic radius of 133 pm (1.33 Å) for an iodine atom.
Figure F04-1-3 Determination of the bonding radius for iodine in an I2 molecule, and for carbon in a diamond lattice.
Atomic radii can be added to estimate bond lengths as illustrated by the molecule of methyl iodide, H3C−I.
As an example, we can now predict the C-I bond length in methyl iodide (H3C–I) to be 210 pm (77 pm + 133 pm),
which is in perfect agreement with the experimentally measured bond length of 212 ± 4 pm. The benefit of understanding
such data is clearly apparent; an atomic property can be used to predict molecular properties, even for molecules not yet
studied. The bonding atomic radii are collected in Figure 04-1-4. For atoms that do not form molecules (such as some
noble gases) or when experimental data are not available (such as for short-lived radioactive atoms) the values are
estimated.
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Bonding atomic radii decrease from left to right across the periods as Zeff increases
and increase down the columns as n increases
Figure F04-1-4. Bonding atomic radii for main group and d elements. The standard atom's colors are used and the size of
the sphere reflects the relative size of the atom. Click on the picture to see the numerical data for the main group elements
and for the transition metals. All values are in picometers (100 pm = 1 Å). Values in parentheses are estimated.
Even without detailed analysis, the trends in atomic size are very apparent. . For the main group elements, the
atoms “shrink” on average by about 40% from left to right across each row. The transition metals, on the other hand, are
much more uniform in their sizes: they differ, on average, by less than 25% across any row, with the larger atoms
distributed at the beginning, the middle, and the end of each row. Overall, these size patterns match those seen in Zeff
values (discussed above). This similarity is perfectly understandable, since valence electrons that experience a greater
charge are kept closer to the nucleus.
For main group elements, the bonding radius significantly increases from top to bottom within each column, doubling
on average from the top to the bottom (excluding H and He). A similar but much less pronounced trend is observable
even in the transition metal columns. At first glance, the growing atomic size down the columns may appear to be contrary
to the Zeff trend discussed above (F04-1-1). However, as we mentioned previously the increase in Zeff down the column
is relatively small and is easily overwhelmed by the overall increase of orbital sizes that accompanies the increase in the
principal quantum number. Increasing n by 1 adds one extra electron shell to the size for each row of the periodic table.
Thus, the atomic sizes increase down each column mainly due to the increasing principal quantum number, and shrink as
we move to the right in each row because of the increasing effective nuclear charge. These trends are summarized in
plots in Figure F04-1-5.
Figure F04-1-5. From left to right: the trends in bonding atomic radii (100 pm = 1 Å) versus (a) Zeff and (B) versus atomic
number (Z). Both plots show the periodic trends of increasing atomic size as n varies. In (a) each line represents a
different n. In (b), a peak indicates a new n and these peaks increase in height as n increases. Each peak is followed by a
smooth decline in size across the row (constant n), reflecting the increasing effective nuclear charge. The transition metals
are shown as open circles, and f-elements are omitted. The saw-tooth pattern characteristic of periodic properties is
apparent in (b). Click on a plot for a larger version.
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In Figure F04-1-5a each line corresponds to one row of the periodic table. The increasing height of the lines is due
to the addition of one electron shell per each increase in n. Within each row, the atomic radii decline, following the trend
of increasing effective nuclear charge. In the "periodic" plot (F04-1-5b) each "tooth" corresponds to one row of the
periodic table. The sharp "spikes" happen when a new shell is added (after each noble gas), and then within each row the
atomic size shrinks in a smooth, predictable pattern that coincides with increasing Zeff There are some expected minor
deviations for transition metals.
Since the energy of electrostatic attraction for valence electrons is directly proportional to the effective nuclear
charge and inversely proportional to their average separation from the nucleus, the atomic size and Zeff trends we have
just discussed are crucial to understanding the chemical behavior influenced by these electrons. The weaker the
electrostatic attraction, the easier it is to remove these electrons from the atom, and the more readily they participate in
bond formation with other atoms. In fact, we will next learn how to directly quantify that attraction by measuring the
energy necessary to remove electrons from atoms.
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04-2 Ionization energies
Successive ionization energies demonstrate that valence electrons are
much easier to remove than core electrons
The ionization energy (IE) of a chemical species (an atom, an ion, or a molecule) is the minimum energy required to
remove an electron from the ground state of that species in the gas phase. In other words, it is a measure of how strongly
that electron is held within the atom, ion, or molecule. The first ionization energy, IE1, is the energy needed to remove the
first electron. That electron is the highest in energy and easiest to remove. The second ionization energy, IE2, is the
energy required to remove the second electron, and so forth, for successive removal of additional electrons. For example,
the consecutive ionization processes for a magnesium atom are shown below:
+
Mg(g) ⟶ Mg
+
Mg
2 +
(g) ⟶ Mg
2 +
Mg
−
(g) + e
3 +
(g) ⟶ Mg
−
(g) + e
−
(g) + e
I E1 = 738 kJ/mol
C04-2-1
I E2 = 1451 kJ/mol
C04-2-2
C04-2-3
I E3 = 7733 kJ/mol
The more strongly the electrons are held (the lower their energy within the species), the harder it is to remove them
and the higher the ionization energy. In our example of Mg, the second electron is twice as difficult to remove as first,
while the 3rd is about five times(!) more difficult to remove than the second electron. An increase in the energy required to
remove each successive electron is expected since the electron is being pulled away from an increasingly positive ion.
However, the large jump in the energy required to remove the 3rd electron in our example signifies another phenomenon
in action. The electron configuration of Mg is [Ne]3s2, so the first two electrons are removed from the 3s subshell. The 3rd
electron is not removed from the valence shell, but must come out of the 2p orbital of the noble gas core of the Mg2+ ion
([Ne] = [He]2s2p6). The large amount of extra energy required to remove core electrons is a general property of all atoms,
as illustrated in Table T04-2-1.
T04-2-1. Successive ionization energies (in MJ mol−1) of the first 21 elements
Z
Element
IE1
IE2
IE3
1
2
3
H
He
Li
1.31
2.37 5.25
0.52 7.30 11.82
4
Be
5
B
0.90 1.76
0.80 2.43
6
C
7
N
8
IE4
IE5
IE6
IE7
14.85
21.01
25.03
32.83
1.09 2.35
3.66
4.62
47.28
4.58
6.22
7.48
37.83
1.40 2.86
53.27
64.36
O
1.31 3.39
5.30
7.47
9.44
10.99
71.33
9
F
1.68 3.37
6.05
8.41
11.02
13.33
15.16
10
Ne
2.08 3.95
6.12
9.37
12.18
15.24
17.87
20.00
11
Na
9.54
13.35
16.61
Mg
7.73
10.54
13.63
18.02
13
Al
0.50 4.56
0.74 1.45
0.58 1.82
6.91
12
14.84
14
Si
0.79 1.58
2.74
3.23
11.58
15
P
1.01 1.91
2.91
4.36
4.96
16
S
1.00 2.25
3.36
17
Cl
1.25 2.30
18
Ar
1.52 2.67
19
K
20
Ca
21
Sc
0.42 3.05
0.59 1.14
0.63 1.24
IE8
IE9
IE10
84.08
92.04
106.43
115.38
131.43
20.12
23.07
25.50
25.66
28.93
31.65
141.36
21.71
18.38
23.33
27.46
31.85
35.46
38.47
16.09
19.80
23.78
29.29
33.88
38.73
21.27
25.43
29.87
35.90
40.95
4.56
6.27
7.01
31.72
36.62
43.18
3.82
5.16
6.54
8.50
9.36
27.11
38.60
43.96
5.77
7.24
8.78
11.02
11.99
33.60
3.93
40.76
46.19
4.42
5.87
7.98
9.59
11.34
13.84
1494
4.91
6.49
8.15
10.50
12.27
14.21
16.96
18.19
2.39
7.09
8.84
10.68
13.31
15.25
17.37
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Indeed, one can readily recognize how many valence electrons each atom has by locating the specific ionization
energy that shows the first dramatic jump in the consecutive IE values (as marked in the table). Every element shows a
large increase in IE when the electrons start being removed from the noble gas core. This observation validates our
division of electrons into valence and core electrons. Only the weakest-bound valence electrons participate in ion
formation or give rise to chemical bonding and reactions. The core electrons are too low in energy, and too tightly held by
the nucleus, to be lost or shared. The consecutive IEs also support the image of an onion-layer atomic structure that we
introduced when building our periodic table and examining electron configurations.
The logarithmic-scale plot shown below (F04-2-1) for potassium clearly demonstrates the shell structure of the
atom. You may further explore ionization energies for all atoms in our interactive periodic table (button on the top of the
page).
Figure F04-2-1. Consecutive ionization
energies for the 19K atom presented on a
logarithmic scale for better visibility. The
individual shells are marked on the plot.
Ionization energies (IE1) increase from left to right across the periods and decrease down
the columns since the further the electron is from the nucleus, the easier it is to remove it
As we have previously posited, the outermost, highest-energy electrons are the most readily available for participation in
chemical processes such as ion or bond formation. The values of IE1 provide us with a simple and quantitative measure of
that availability, and the trends in their magnitudes give us insight into relative chemical reactivity of atoms. At the most
basic electrostatic level, electrons are the easiest to remove if they are far away from the nucleus and experience the
smallest effective nuclear charge. We have already explored the atom size and Zeff trends in some detail in this Chapter
(see 04-1). With that knowledge, we can predict with confidence that IE1 should decrease as we move down any column
of the periodic table, following the trend of increasing atomic radii. On the other hand, IE1 should increase as we move
from left to right across any period, tracking the increasing effective nuclear charge and diminishing atomic size. We may
also expect that variations for transition metals will mimic the main group trends, but in a muted fashion. All of these
predictions are indeed true, as shown in plot F04-2-2a.
Figure F04-2-2. From left to right: a) The first ionization energies and b) atomic radii as functions of atomic number (Z).
The open circles correspond to d- and f-block elements. Both plots have the characteristic saw-tooth periodic character,
but with "up-side-down" teeth, mirroring the inverse relationship between IE1 and atomic sizes. The dark blue points and
green points in (a) mark full- and half-filled subshell exceptions, respectively.
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The IE1 plot (F04-2-2a), has an unusual "upside-down" relationship to the analogous atomic radii plot (Plot F04-1-5b
is repeated here for easier comparison). Note that noble gases and alkali metals have reversed their relative valley and
peak positions. The peaks now decline in size with increasing Z (they are increasing in F04-2-2b), and the inter-peak
points are on an upward trajectory (whereas they decline in F04-2-2b). These "inverted' characteristics of the plots are
nothing more than the manifestation of the inverse relationship between the atomic radius and IE1 (bigger atoms have
smaller IE1), and of the fact that increasing Zeff shrinks the atomic size and increases IE1.
We can also see some irregularities within the periods of the saw-tooth pattern, appearing as secondary, smaller
teeth, which are marked as green and dark blue points in F04-2-2a. Although quite subtle, these deviations can be
rationalized in terms of subshell structure and the extra stability of filled and half-filled subshells.
As a general rule, the completely filled subshells provide some added stability to the electron configuration. Thus,
slightly larger IE1 values are commonly observed for atoms with a completely filled shell (ns2, nd10, and np6). This extra
stability can be attributed to the perfectly spherical electron distribution in the completely filled subshells. These "filledsubshell" deviations are marked in dark blue on our plots (F04-2-2a and F04-2-3). A very similar trend is observed for halffilled subshells (marked green in F04-2-2a and F04-2-3). For example, nitrogen, with its half-filled p subshell (T03-3-10), is
harder to ionize than oxygen (which has one doubly occupied p orbital ). This is true even though the latter has a larger
Zeff (4.45 for 2p on oxygen vs 3.83 for 2p on nitrogen). In addition to the symmetrical charge distribution, such half-filled
electronic configurations obey Hund's rule of maximum multiplicity. As previously stated, their maximum-spin
configurations have lower energy, since they help to minimize electron-electron repulsion.
The measured IEs are really the enthalpies (ΔH) of the ionization processes under a different name. Their positive
signs unequivocally indicate that liberating electrons from the electrostatic grasp of the nuclei costs energy and is
endothermic. The IE1 value is a measure of the stability of the electron configuration of the neutral atom: the bigger the
value, the more stable the atom, with the highest stability exhibited by the noble gases (F04-2-3). On the other hand, the
consecutive IEs measure the stability of the corresponding ions, again reaching the highest stability at the nearest noble
gas configuration (with a smaller Z). Returning to our example of a Mg atom, we can easily deduce that in forming positive
ions, this element would strongly prefer to form +2 cations (Mg2+), reaching the noble-gas configuration of Ne. Any further
oxidation (electron removal) would be energetically prohibitive. Next, in an obvious extension of our quest to understand
atomic properties, we shall explore how well atoms can accept additional electrons (i.e. undergo reduction).
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04-3 Electron affinities
Most atoms can accept one electron in the gas phase, yielding stable anions
The electron affinity (EA) is the energy released when an atom in the gas phase accepts an electron. For example, in the
case of a chlorine atom, the process is exothermic (C04-3-1):
−
Cl(g) + e
⟶ Cl
−
(g)
EA = −349 kJ/mol
C04-3-1
Most atoms, but definitely not all, can accept one electron to produce stable –1 anions in the gas phase. Atoms that
have positive electron affinity cannot exist as gaseous anions; they would immediately eject the extra electron in an
exothermic, spontaneous process. No atom can accept more than one electron in the gas phase, so we do not need to
consider successive electron affinities. Note, however, that negative ions that are unstable in the gas phase may exist in
liquid or solid states if some additional stabilization is provided (for example, electrostatic interactions with positive ions or
other molecules).
The EA values are collected in Figure F04-3-1. Missing values indicate that the anions are unstable in the gas phase
and have unmeasurable positive EA values. The most exothermic values are found on the right side of the periodic table.
Figure F04-3-1. The electron affinities (in kJ/mol) of main group and transition elements. Elements without values do not
form stable anions in the gas phase (EA > 0). The color gradient (yellow → green) goes in order from atoms with least
affinity to electrons (yellow) to atoms with most affinity (green).
Electron affinities become more negative toward the right of the periodic table,
but are near zero or positive for atoms with filled or half-filled subshells
If considered as absolute values, (|EA|) the periodic trends in electron affinity largely follow those in ionization energies
(IE1); the absolute numbers increase from left to right across the rows and decrease from top to bottom along the columns
of the periodic table. Although the trends are much less pronounced for the electron affinities than they are for ionization
energies, one interesting contrast is evident. The absolute values have their most dramatic changes (and peaks) one
column earlier than what we found for IE1 values. For example, the largest IE1 values were found for noble gases (Figure
F04-2-2a); now the most negative values belong to halogens (group 17).
It might seem counterintuitive that the trends would be similar for energetics of removing an electron from an atom
(IE1) and that of adding one (EA), but both processes are governed by the same underlying principles of changing Zeff,
changing atom sizes, and shell and subshell filling patterns. In general, electron affinities do not change greatly as we
move down the group. The extra electron is added to bigger and bigger orbitals as we travel down the column, and
although it is more separated from the nucleus with each increase in n, that trend is compensated by the slight increase
in Zeff and the fact that the bigger, more diffuse orbitals help to reduce repulsions between electrons.
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In general, the electron affinities are more favorable (more negative) as we move toward the right side of the periodic
table. This trend follows the increasing Zeff in the same direction. As shown in F04-3-1, the near-zero negative or
unfavorable (positive) values of EA are found for atoms with ns2 (group 2), nd5 (group 7), nd10 (group 12), np3 (group 15)
and np6 (group 18) electron configurations. In all of these cases we are dealing with filled or half-filled subshells that are
particularly stable. Adding one more electron would destroy that stability, so these processes are unfavorable; the
highest positive affinities are found for filled-shell noble gases. On the other hand, atoms in columns just one before the
filled or half-filled configurations need just one extra electron to reach that extra-stable configuration. For those atoms,
EAs will be favorable (negative) and become more exothermic the further the atom is to the right of the periodic table (as
Zeff increases). These trends, concluding with halogens having the most negative electron affinities, are illustrated in our
now-standard periodic plot (F04-3-2).
Figure F04-3-2. The electron affinity (EA) trends across a period.
Although the trends are less clear than for other atomic properties, the
saw-tooth pattern culminates in halogens (far right of a period in the
periodic table) with most favorable (negative) electron affinities. The 1st
ionization energy (IE1) plot on bottom illustrates the "mirror-image"
relationship between IE1 and EA, where the plots show a peak for every
change in period and show a small decrease in IE1 and EA down a
column of the periodic table.
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04-4 Ion sizes
Cations are smaller than their parent neutral atoms, and anions are bigger
We have learned about the trends in atomic radii (04-1) and energetics of making both positive ions (IE) and negative ions
(EA). When cations are formed from neutral atoms electrons are removed from the outermost (valence) orbitals, usually in
the number required to reach a stable noble gas electron configuration. Electron-electron repulsion is reduced as the total
number of electrons is diminished. The removal of the most spatially remote electrons (the outer shell) and diminished
repulsions result in significant size shrinkage; cations are always smaller than their parent neutral atoms.
On the other hand, when electrons are added to atoms (again, usually in numbers required to reach the stable
electron configuration of a noble gas) they end up in the outermost valence orbitals and electron-electron repulsion is
increased. The effect is a significant increase in size; anions are always bigger than their parent neutral atoms.
These trends in ion size changes are illustrated in Figure F04-4-1. Only the atoms with the most negative EAs are
capable of forming stable anions, even with electrostatic stabilization from other ions in a crystalline ionic solid.
Therefore, it is not surprising that the majority of ions shown in the summary are cations. For the main group elements,
most of the ions included in our chart have a noble gas configuration.
The ionic sizes follow the same trends as those observed for neutral atoms (04-1). Ions of the same charge increase
in size down the column, mainly due to the new electron shell added with each increase in n. The ions shrink from left to
right for both the cations and anions (in separate trends). As can be seen by comparison with the sizes of the neutral
atoms, this trend is a combination of the increasing nuclear charge and the effect of the increasing ion charge for the
cations or diminishing ion charges for the anions (see below).
For d-block elements (click on F04-4-1 to see the transition metal ions), the ions are quite similar in size and there
are no clear trends. As we have noted previously, the neutral atoms of transition metals do not differ very much in their
sizes. Additionally, they most often make cations of similar charges. The result is a small variance in the ionic radii, with
no pronounced trends.
The ionic radii provide useful information for evaluating bonding arrangements in ionic crystals, much as the bonding
radii allow us to predict bond lengths in molecules of interest. As we will learn, such information is beneficial even if only
applied in a qualitative way based on the observed trends. You may be surprised, for example, to learn that the trend in
ionic radii can be translated into a ranking of the melting points of ionic solids! We will return to that topic when we
address ionic bonds in Lesson 06-3.
Figure F04-4-1. Ion sizes (gray hemispheres) as compared to the neutral parent atoms (shown in their standard assigned
colors). For main group elements, mostly ions with noble-gas configurations are shown. For transition metal ions, only
some of the most common ions are shown (click on the picture to see the d-block elements). All sizes are in pm (1 pm =
100 Å).
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The size of the ions in isoelectronic series decreases as nuclear charge increases
As mentioned above, most of the ions in Figure 04-4-1 were selected because they have filled shells, which matches the
electron configuration of the noble gases. Let's examine this choice a bit closer by concentrating on the most commonly
encountered ions of the main group elements (F04-4-2). In the first row Li+ and Be2+ have the electronic configuration of
He, but N3–, O2–, and F– have the electron configuration of Ne. The same [Ne] configuration is found in Na+, Mg2+ and
Al3+ in the next row. Thus, we have a nicely ordered isoelectronic series: a group of ions that has exactly the same
number of electrons, with the exact same ground state electron configuration.
— increasing nuclear charge→
N3−
146 pm
O2−
140 pm
F−
133 pm
Na+
102 pm
Mg2+
72 pm
Al3+
54 pm
— decreasing radius →
Such ordering makes predicting their size trends quite simple: their radii decrease with increasing nuclear charge, Z.
Notice that in this order, the anions of the higher row precede the cations of the lower row.
Figure F04-4-2. Isoelectronic series for the most common
ions of the main group elements. All ions on a common
color background have the same number of electrons in
the same ground-state electron configuration. Their
sizes shrink as their nuclear charge Z increases. Click on
the picture for a larger version.
Main group elements tend to form ions with noble gas configurations,
whereas transition metals can form a number of different cations
Although our table of ions (Figure 04-4-1) lists specific ions for all main group elements and transition metals, in reality, the
situation is a bit more complex. The number of common mono-atomic stable anions is quite limited. You may remember
that in the gas phase only anions with –1 charge are thermodynamically accessible, and not even for all elements.
Anions with larger charges (–2 and –3) exist only in the liquid or solid phases when they can gain electrostatic stabilization
from cations or other molecules. Stable examples are strictly limited to the anions of nonmetals on the right side of the
periodic table, with all anions having stable noble gas configurations (Figure F04-4-3).
The mono-atomic cations are much more diverse, since they can be formed by all metals from the left and central
part of the periodic table. The main group metals form cations with noble gas configurations (group 1 and 2) or with filled
subshells (group 14 or 15). Transition metals form multiple ions with various electron configurations. A hydrogen atom
may become a cation (proton, H+) or an anion (hydride, H−).
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Figure F04-4-3. The most common stable ions. Metals (red and orange elements) tend to form cations, while nonmetals
(green) tend to form anions. The anions shown all have noble gas configurations. Metalloid elements (brown) tend to form
either cations or anions. Transition metals may have noble gas electron configuration but may form non-noble gas
electron configurations as well. The hydrogen atom may become a cation (proton) or an anion (hydride). Although
hydrogen is not a metal at standard conditions, experiments have proven that hydrogen forms metallic bonds at very high
pressures (on order of GPa).
In general, when cations are formed from neutral atoms, the electrons are always first removed from the occupied
orbital having the largest principal quantum number, n. If there is more than one occupied subshell for a given value of n,
the electrons are first pulled out of the orbital with the highest value of ℓ. Let's analyze some examples:
2
2 +
Mg ([Ne]3 s ) ⟶ Mg
6
2
Fe ([Ar]3d 4s ) ⟶ Fe
Fe
2 +
6
([Ar]3d ) ⟶ Fe
C04-4-1
−
([Ne]) + 2 e
2 +
3 +
6
C04-4-2
−
([Ar]3d ) + 2 e
5
C04-4-3
−
([Ar]3d ) + e
In the case of magnesium (C04-4-1), an ns2 main group element, the valence electrons from the 3s subshell are
removed resulting in a cation with a noble gas configuration. In the case of iron, a transition metal, the first two electrons
are removed from the 4s subshell (C04-4-2), and only the next electron to be removed (forming Fe3+) would come out of
the 3d subshell (C04-4-3). This may seem confusing if you remember first filling the lower energy 4s subshell and then
the 3d subshell when we were determining electron configurations for the transition metals in the periodic table.
However, this imaginary process of adding electrons one at a time as we move from element to element does not take into
account the change in effective nuclear charge from atom to atom (and from orbital to orbital). We must recall that in
transition metals the (n-1)d and ns orbitals are close in energy, and d-electrons actually have lower energy than s
electrons once the d orbitals are occupied. When occupied, the 3d orbitals shield the 4s electrons from the nucleus so
that the 4s orbital is higher in energy.
Sn ([Kr]4d
2 +
Sn
10
2
2
2 +
5s 5p ) ⟶ Sn
([Kr]4d
10
2
4 +
5s ) ⟶ Sn
([Kr]4d
([Kr]4d
10
10
2
−
5s ) + 2 e
−
)+ 2e
C04-4-4
C04-4-5
In the case of atoms with more than one occupied subshell, such as in tin (C04-4-4), the first two electrons are removed
from the subshell with the highest n and ℓ (5p), and only the next two come out of the 5s subshell (C04-4-5). Note that in
this case the electrons in the completely filled 4d subshell are not considered valence electrons, so the oxidation of Sn,
stops at +4, which is not a noble gas configuration (although it has no open subshells).
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04-5 Periodic trends
Ionization energies are the dominant factor influencing the metallic or nonmetallic
properties of elements
We are now well versed in the properties of individual atoms and their ions in the gas phase and can understand the
trends governing these properties. Yet, with the exception of noble gases, none of the elements exist in the form of
separated, individual gaseous atoms. We need to develop a way to translate the properties of individual atoms and ions
into the properties of bulk elements. We need to figure out how atoms combine together into larger assemblies such as
molecules or extended networks, and how the individual molecules interact with each other to determine the properties of
bulk materials. And we will eventually need to understand how atoms of different elements come together to form the
variety of substances that we find in the surrounding world or make in our laboratories and factories.
We started our journey already (03-4) when we drew a map with a dividing line between different regions to explore.
We reproduce that "map" below (F04-5-1). At the time our "divide-and-conquer" strategy might have seemed arbitrary, but
now we are ready to support it with some objective data.
Figure F04-5-1. The periodic table with metals, nonmetals, and metalloids marked. Alkali metals (group 1) are all highly
reactive and readily become +1 ions, while alkaline earth metals (group 2) are less reactive and form +2 ions. Transition
metals form many different ions, but share characteristics that set them apart from other elements. They are similar
because of their incompletely filled d-subshells. In the same way, lanthanides and actinides resemble each other due to
their partially filled f-subshells. Among the nonmetals we have the chemically inert noble gases (group 18), the halogens
(group 17), the oxygen family (chalcogens, group 16), and the nitrogen family (pnictogens, group 15).
Let's revisit the basic atomic property that tells us how strongly electrons are attracted to the nucleus, IE1 (F04-5-2).
We now draw another dividing line in the plot at IE1 slightly below 1000 kJ/mol. Essentially all elements below this line are
identified as metals in the periodic table above, all elements above the line are nonmetals, and the few elements in close
proximity to the line are metalloids.
The cleanness of the break (although there are a few minor outliers) is dazzling in its simplicity, yet it underscores
how electronic configuration, an abstract atomic property, governs the tangible properties of elements in bulk. The metals
have low IEs, easily give up their relatively weakly bound electrons and form cations. The nonmetals have high IEs, do not
give up their electrons readily, and a select few (with highly favorable electron affinities) form stable anions. The
metalloids are somewhere in between, but as their name indicates, they are more like metals than nonmetals on average
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Figure F04-5-2. The first ionization energies as a function
of atomic number (Z). The dividing line drawn at IE1 just
below 1000 kJ/mol, neatly divides the elements into
metals (orange circles) that fall largely below the line,
nonmetals (green squares) which appear mainly above
the line, and metalloids (brown triangles), in close
proximity to the line.
Metals are lustrous, malleable and ductile (easily shaped), and they conduct heat and electricity very well. Most are
solids; Hg is the one exception, and is a liquid at room temperature. All of these properties are related to the availability of
high-energy electrons that can roam freely in the solid structures of bulk samples. The group 1 alkali metals form +1
cations most easily (they have the lowest IEs), with the largest atoms being the most reactive. The group 2 alkaline earth
metals are somewhat less active in formation of +2 cations, and they are followed in reactivity by the other elements that
form multiple cations with various charges (F04-4-3).
Nonmetals are usually brittle solids, liquids or gases at room temperature. Some nonmetals are monatomic (noble
gases), while others are diatomic (O2, N2, F2, Cl2, I2, Br2, H2), tetratomic (P4), or even octatomic (S8) at room temperature
and pressure. Generally, they are poor or non-conductors of heat or electricity. Their properties are related to their
unwillingness to let their valence electrons go (they have high IEs). Instead, some of them tend to accept extra electrons
and form anions (−1 for group 17, −2 for group 16, and −3 for group 15). The noble gases are an extreme example of
stability and are very unreactive. With their closed shell electron configurations, they do not readily give up their electrons,
nor do they accept any; they have the highest IEs in each row. They do not form stable ions, and the two smallest (He and
Ne) never participate in any bonding, forever destined to be unattached.
You may explore all the elements, including their properties, appearance, and history, in our interactive periodic
table. The table can be launched by clicking on the button "Periodic table" on the right upper corner of any web page.
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05 Molecular Composition
In the previous Lessons, we learned a lot about the properties of individual atoms. The substances we encounter in
the world around us, however, are very rarely in atomic form. Your experience with atomic species is most likely limited to
the helium in party balloons, or to the trace amounts of other noble gases in our atmosphere. The substances we
encounter most often are not atomic species, but assemblies of atoms, joined by strong bonding forces. The assemblies
are quite diverse, from pure elements in various allotropic forms, to compounds built of atoms of different elements formed
into molecules or extended networks. They differ in composition and structure. In our quest to understand how atoms join
together, we will first analyze the composition of chemical compounds. After we learn what atoms are present, and in what
ratios, we will be ready to probe their bonding arrangements and structure in future Lessons.
05-1 Assemblies of atoms
Pure elements may exist in atomic form (noble gases only), as homonuclear diatomic molecules (hydrogen, oxygen,
nitrogen, and halogens), as homonuclear polyatomic molecules (phosphorus and sulfur), or as extended networks (such as
metallic solids or carbon and silicon). Compounds that are assemblies of at least two types of atoms can exist as molecules or
extended networks. The latter includes ionic solids, wherein atoms exist in the form of ions. The participating atoms and their
ratios are described by their empirical or molecular formulas which list the atoms with subscripts indicating their relative ratios.
05-2 Formulas
The empirical formula is the simplest formula that gives the correct relative number of atoms of each kind. It is usually
obtained experimentally from elemental analysis. It defines the smallest repeating unit of the composition (but not necessary the
smallest repeating structural unit). The molecular formula is the formula that specifies the number of each atom type in one
molecule. The molecular formula can be defined only for substances that consist of well-defined molecular structures. The
molecular formula is always a multiple of the empirical formula, with "1" being a possible and even common multiplier.
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05-1 Assemblies of atoms
Atoms combine into compounds composed of molecules, ions, or extended networks
In previous Lessons we learned about individual, isolated atoms and their properties, and how properties are determined
by an atom’s electron configuration. Yet, with the exception of noble gases, atoms are encountered in assemblies, where
the individual participants are strongly bonded together within molecules or extended networks. In our quest to
understand how and why atoms bond together, we start with the simple question of composition: how many atoms and
what type of atoms come together to form the assembly?
In general, we will encounter assemblies of atoms of just one element or two or more elements. Various structural
forms in which pure elements can exist are called allotropes. Allotropes have the same composition, but different
structures, i.e., different atom connectivity. In such cases there is really no issue of composition, except perhaps for
relative isotopic abundances, as all atoms present are of the same type. Assemblies of atoms of two or more elements
are called compounds. The majority of substances around us are pure compounds or are mixtures of compounds.
The simplest assemblies are composed of just two atoms. If the atoms are of the same element, they are called
homonuclear diatomic molecules. If they are composed of two different atoms they are called heteronuclear diatomic
molecules. To express their compositions, we just list the atoms present. For example, H2 is the smallest molecule built of
the two simplest atoms, F2, Cl2, Br2, or I2 are the halogen molecules, and N2 and O2 are the most abundant gases in our
atmosphere. Examples of heteronuclear diatomic molecules include HF (hydrogen fluoride), HCl (hydrogen chloride), HBr
(hydrogen bromide), CO (carbon monoxide), and NO (nitrogen oxide).
Figure F05-1-1. Examples of homonuclear (Cl2, O2) and heteronuclear (HCl, CO) diatomic molecules. At this stage we are
interested only in the composition of atomic assemblies. We will analyze these structural details in later Lessons.
The vast majority of atomic assemblies contain more than two atoms. These may include collections of identical
atoms bonded into polyatomic molecules such as O3 (ozone), P4 (white phosphorus), or S8 (cyclooctasulfur), or extended
networks containing huge numbers of bound atoms such as diamond (an allotrope of C) or a metal (such as Fe or Au).
For illustration purposes, a 1 carat diamond (F05-1-3) contains 0.017 mole or 1022 atoms of carbon, while a cubic
centimeter of gold (F05-1-3) has about 0.1 mole or 6 × 1022 atoms of gold.
Figure F05-1-2. Examples of the simple elemental molecules: ozone (O3), white phosphorus (P4), and cyclooctasulfur
(S8). For now, we are focusing only on the composition of these molecules. You may explore them in 3D by clicking on
the picture, although the bonding details will be discussed in later Lessons.
,
Figure F05-1-3. Examples of the extended atomic networks of carbon (diamond) and gold. Click on structural pictures to
interact with 3D models.
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The most often encountered assemblies, called compounds, are composed of atoms of different types. Some form
polyatomic molecules consisting of just a few atoms, such as CH4 (methane), H2O (water), NH3 (ammonia), HNO3 (nitric
acid), or H2SO4 (sulfuric acid). However, they may also be built of hundreds of millions of atoms, as in the DNA molecule
of a single gene, or anything in between in size.
Figure F05-1-4. Examples of polyatomic compounds methane (CH4), water (H2O), ammonia (NH3) and nitric acid
(HNO3). You may click on the molecules to explore them in 3D. We will explore the details of these structures in later
Lessons.
Atoms may also participate in compounds in the form of ions, forming extended ionic solids. These solids can be
composed of only atomic ions, such as NaCl (sodium chloride), MgBr2 (magnesium bromide), or CaO (calcium oxide).
Ionic solids can also be composed of both atomic and molecular ions, as in NaNO3 (sodium nitrate), KOH (potassium
hydroxide), or NH4Cl (ammonium chloride). Additionally they can be composed completely of molecular ions, such as in
(NH4)2SO4 (ammonium sulfate). Such ionic solids may contain a huge number of ions, depending on their crystal size. A
1 mm3 crystal of NaCl from a salt shaker contains about 2 × 1019 ions of each type. Ionic solids may also contain water
molecules incorporated in a definite ratio in the crystal structure. Such water molecules usually surround metallic ions and
often can be removed from the crystals with heat. A notation "•nH2O" is used to indicate the number (n) of water
molecules per empirical formula, for example CuSO4•5H2O shows that there are 5 water molecules per each Cu2+ ion.
Such water-containing salts are called hydrates
CaO
ScN
NaNO3
KOH
Figure F05-1-5. Examples of ionic solids from left to right: CaO, ScN, NaNO3, and KOH.
Compounds are characterized by their composition and structure
As you may have noticed reading the formulas above, we express the atomic composition by listing all the atoms
participating in the assembly and their relative amounts (as subscripts). Both of these pieces of information are critical in
describing the composition of the atomic assembly, but are not sufficient to fully describe its structure. Just listing of
atoms does not tell us anything about the structure, for example, dioxygen (O2) and ozone (O3) are both made of only
oxygen atoms but they have different molecular structures. Carbon (C) may exist as C60 molecules, or as extended
networks of diamond or graphite, each with its unique structure.
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Similarly in assemblies that contain multiple types of atoms, without specifying the atom ratio, we could not
distinguish between CO (carbon monoxide) and CO2 (carbon dioxide), both of which are composed of carbon and oxygen,
but in different ratios and connectivity. Even with the atoms and their ratio specified, we may still not be able to uniquely
identify the assembly, For instance, for a formula such as CH2, even though we know that the compound is made of
carbon and hydrogen atoms in a 1 : 2 ratio, we still need to know how many of these CH2 units are present in the entire
assembly. For example, there are two CH2 units in ethylene (C2H4) five in cyclopentane (C5H10) and six in cyclohexane,
(C6H12) and each of these compounds has a different structure. Even when the molecular formula is defined, numerous
structural variations are possible. For example, there are 12 different stable structures (called isomers) possible for a
compound with a molecular formula of C5H10 (Figure F05-1-6).
Figure F05-1-6. Some molecular structures with an empirical formula of CH2. Click on the structures to explore their 3D
shapes. Click on the cyclopentane link to see the slide show of all 12 stable isomers with the formula of C5H10. The details
of these structures and their nomenclature will be explored in future Lessons.
In the atomic assemblies listed above, atom ratios are expressed with small integers (whole numbers). This
observation is in total agreement with atomic theory, which postulates that atoms are the smallest building blocks of matter
from the chemical point of view and that they do not change their identity in chemical reactions. Indeed, the basis for the
atomic theory first proposed in the early 19th century at the beginnings of modern chemistry was the observation that the
proportion of elements in a given compound are the same regardless of its source, and that they are expressed by small
integers.
We will tackle the issue of structure (how the atoms are bonded and why) in later Lessons, but for now we will
concentrate on various ways to express and understand composition. We will define and apply the concepts of empirical
and molecular formulas.
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05-2 Formulas
Empirical formulas give the smallest relative number of each kind of atom whereas
molecular formulas specify the number of each type of atom in one molecule
According to our observations, each atomic assembly or compound may be described by a formula that specifies the
relative number of atoms of each element present in that compound. The empirical formula is the simplest formula that
gives the correct relative number of each kind of atom. In other words, the empirical formula defines the smallest
repeating unit of the composition (but not necessarily the smallest repeating structural unit).
The molecular formula is the formula that specifies the number of each atom type in one molecule. The molecular
formula can be defined only for substances that consist of well-defined molecular structures. All gases (except for noble
gases) consist of molecules, and so do most liquids and a number of solids. The molecular formula is always a multiple of
the empirical formula (with "1" being a possible and even common multiplier).
When the molecular formula is known, it is clearly a preferable way to describe the composition of the compound, as
it precisely defines the molecular unit. However, it is impossible to provide the molecular formula for many substances that
are not composed of molecules; in those cases the empirical formula is the only option.
Let's look at some examples. The empirical formula of carbon dioxide is CO2. Since the carbon dioxide molecule is
built of one carbon and two oxygen atoms, the molecular formula is also CO2 (the multiplier is one). The empirical formula
of glucose is CH2O, but its molecular formula is C6H12O6 (the multiplier is six). The empirical formula only tells us about
the ratio of atoms, while the molecular formula identifies that one molecule of glucose is built of 6 carbon atoms, 12
hydrogen atoms, and 6 oxygen atoms.
Figure F05-2-1. Molecular structure of glucose, C6H12O6. Click on the
image for a 3D model.
In contrast, sodium chloride (table salt) has the empirical formula NaCl, indicating that the ratio of sodium to chlorine
is 1:1 (the structure actually contains ions of Na+ and Cl−). As we have previously illustrated, NaCl forms an extended
ionic solid with a 3 dimensional structure of alternating ions. In this case, no individual "NaCl" molecules can be identified;
all of the ions are interacting equally with their oppositely charged neighbors. We can only use the empirical formula in
such cases.
The empirical or molecular formula specifies the ratio of all atoms present in the compound, so now it is possible to
specify the formula mass (or molecular mass) by adding the masses of all the atoms in the formula. For example, in
carbon dioxide there are two oxygen atoms per each carbon, and the molecular mass is 44.01 amu (12.01 amu + 2 x
16.00 amu). We can also specify the mass of one mole of molecules. This quantity, called molecular mass, molecular
weight, MW, or formula weight, FW, (especially if it is based on the empirical formula) can be calculated by adding the
atomic weights of all atoms present, and using units of g/mol. One mole of CO2 molecules contains one mole of carbon
atoms and two moles of oxygen atoms, and thus FW(CO2) = 44.01 g/mol. Note that we measure mass by weighing it,
and in Earth's gravitational field weight and mass are synonymous; in general, the same mass would have different
weights in different gravitational fields.
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Figure F05-2-2. The relationships between number of particles, number of moles, and masses of samples.
Empirical formulas are determined experimentally using percent composition data
We can convert the molecular composition into mass percentage of its constituent elements. The calculation can be
performed in amu or in g/mol, and it is equally applicable to the empirical formula and the molecular formula, with both
giving the same results since they represent the same ratio of elements. In this formula n is the number of atoms of an
element, and AW is its atomic weight, while FW is the molecular weight (both must be in the same units, whether amu or
g/mol).
% of element =
n × AW
E05-2-1
× 100%
FW
For CO2, the %C in carbon dioxide is 27.3% and the %O is 72.7%. Since carbon and oxygen are the only elements
present in CO2, the mass percentages of the two must add up to 100%. It is always advisable to carry out that internal
check, especially for more complex formulas with multiple elements. So, given a formula, one can determine the mass
percent of each element in the molecule.
Mass percent calculations can also be inverted; one can determine the formula given the mass percent of each
element in the molecule. Imagine, for example, that we know that the substance named butyric acid has an
experimentally-determined composition by mass of %C = 54.2%, %H = 9.15%, and %O = 36.6%. What would be the
formula?
A simple way to accomplish this task is to assume that we have a 100 g sample; since we are dealing with %, that
simplifies the calculations. Our sample would then contain 54.2 g of carbon, 9.15 g of hydrogen, and 36.6 g of oxygen
atoms. We can calculate the number of moles of each element:
moles of C =
54.2 g
= 4.52 mole
=>
9.15 g
moles of H =
= 9.08 mole
=>
= 1.97 per mole of O
9.08
= 3.97 per mole of O
2.29
1.01 g/mol
36.6 g
moles of O =
4.52
2.29
12.0 g/mol
= 2.29 mole
=>
16.0 g/mol
2.29
= 1.00 per mole of O
2.29
Knowing the number of moles of each atom, we can calculate the ratio of atoms in the formula. We divide all the
numbers of moles of each atom by the smallest number of moles (which is oxygen in this example). Thus, we have
determined the ratio of carbon to oxygen and hydrogen to oxygen.
Since the % composition values were obtained experimentally, we do not expect to get perfect integer ratios for all
atoms. The deviations are very minor, however, and are easily rounded to yield a formula for butyric acid of C2H4O. This
calculated formula is an empirical formula; it gives the smallest integer ratio for the atoms present. This is also why such
a formula is called "empirical"; it comes from the experimental measurement of the mass percentages of the elements
present. Such measurements are called elemental analysis, and are routinely carried out to establish the atomic
composition of compounds.
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Empirical formulas augmented by molecular mass information give molecular formulas
Returning to our results for butyric acid, we still need to decide if the empirical formula is actually the molecular formula of
the compound. What is the multiplier? To answer that question we need additional information; we need to know how
many "formula units" are contained within the structure of butyric acid. The simplest way to do that is to establish the
molecular weight of the compound by another independent method. We have learned before that mass spectrometry can
provide precise molecular weights. This technique is based on forming ions from atoms or molecules and sorting them by
mass with the help of a magnetic field. Using mass spectrometry, we can determine that butyric acid has a molecular
mass of 88.10 g/mol. The empirical formula of C2H4O that we calculated has a formula weight that is too small;
FW(C2H4O) = 44.05 g/mol. Clearly our empirical formula has to be multiplied by 2. Therefore the molecular formula of
butyric acid is C4H8O2..
Figure F05-2-3. Molecular structure of
butyric acid, C4H8O2. The structure is
based on independent determination.
There are hundreds of isomeric structures
possible with this molecular formula. Click
on the image for a 3D model.
Even though we have determined the empirical formula and the molecular formula of butyric acid, we still really don’t
know anything about the structure of the compound. Many different structures that contain these component atoms are
possible. Our next goal is to figure out how the atoms are connected to each other and by what type of bonds. This
description will be called a structural formula (as in F05-2-3), and will account for all the valence electrons of all atoms
involved in bond formation
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06 Bonding
The vast majority of substances around us are composed of assemblies of atoms held together by strong
electrostatic forces called bonds. To form such bonds, atoms redistribute their valence electrons to achieve a noble gas
configuration. The bonding that results lowers the energy of the system.
06-1 The octet rule
Filled-shell noble gas configurations, with their perfectly spherical electron densities, are especially stable. Other atoms may
achieve this configuration only by gaining, losing, or sharing electrons. With the exception of the hydrogen atom, which only needs
to be surrounded by two electrons to achieve the same configuration as [He], all atoms need eight electrons in their valence shell
to have a full shell, or noble gas configuration. This is called the octet rule; fulfillment of the octet rule is the driving force for bond
formation. Atoms achieve this octet by forming ionic or covalent bonds. In ionic bonding, ions with noble gas configurations and
opposite charges are electrostatically attracted to each other and come together to form extended ionic solids. In covalent bonds,
atoms share electron pairs that are attracted to two nuclei at once. For nonmetals, the number of covalent bonds that an atom
prefers (called valence) is 8 − N, where N is the number of valence electrons on that atom. Metals and metalloids often do not
follow the octet rule when bonding to nonmetals. Metal atoms do not have enough electrons to satisfy the octet rule and they bond
with each other through metallic bonding wherein all valence electrons are shared equally by an extended network of metallic
cations and are free to move throughout the metal.
06-2 Ionic bonding
Metal atoms can lose electrons, and nonmetal atoms can gain electrons; both form ions with noble gas configurations. The
formation of such ions in the gas phase is endothermic, but the electrostatic attraction between ions in close proximity in the solid
state compensates for the energy required to form gaseous ions. As a consequence, ionic compounds form extended networks or
lattices, which are organized to maximize the electrostatic attractions between ions of opposite charge and minimize repulsions
between ions of like charge.
06-3 Lattice energy
The strength of ionic bonding is gauged by the lattice energy, i.e., the energy necessary to convert one mole of an ionic solid
into individual ions in the gas phase. Lattice energy is not directly measurable, but can be calculated based on thermodynamic
cycles. If the electrostatic interaction between the ions is stronger, the lattice energy is higher. The energy of these electrostatic
interactions is proportional to the product of the charges on the ions, and inversely proportional to the separation between them
(taken as the sum of their ionic radii). Lattice energy accounts for all attractions and repulsions in an ionic solid, and is therefore
affected by the packing of ions in the crystal. Ionic solids generally have high melting points, a property which correlates with the
magnitude of their lattice energy.
06-4 Covalent bonding
Covalent bonds form between nonmetal atoms when electron pairs are shared between adjacent atoms. The shared
electrons are attracted to both nuclei. These energy-lowering electrostatic interactions are most effective if the electron density
increases along the internuclear axis between the bonded atoms. A shared pair of electrons constitutes a covalent bond; if only
one pair is shared between two atoms, that bond is referred to as a single bond. To account for electron sharing schemes, we use
Lewis structures, which are diagrams of lines and dots surrounding atomic symbols. Shared pairs are drawn as lines (representing
the connectivity of atoms within the molecule), and the nonbonding lone pairs are represented by dots.
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06-5 Multiple bonds
In some molecules the atoms share two or three electron pairs, forming double or triple bonds, respectively. The number of
electron pairs shared is called the bond order. Bond orders of 2 or 3 are normally found between nonmetal atoms of the second
row of the periodic table. Multiple bonds are shorter and stronger than corresponding single bonds, although the second or third
bonds are typically weaker than the first. Bond lengths between a specific pair of atoms usually change very little from compound
to compound and average values provide an approximate measure of interatomic distances. For single bonds, bond lengths are
interpreted as the sums of the atomic bonding radii, and the trends that govern atomic size may be used to predict trends in bond
lengths. In general, smaller atoms form shorter and stronger bonds than bigger atoms. Bond strength is measured by the bond
dissociation energy (BDE), i.e., the energy needed to break that specific bond (while leaving the other bonds in the molecule
intact). In general, BDEs follow the same trends as bond lengths and do not vary drastically from molecule to molecule. Average
bond energies provide a good measure of the stabilizing effects obtained when electron pairs are shared.
06-6 Polyatomic ions
So far we have discussed ions formed by atoms accepting or giving up their valence electrons. Covalent molecules may also
form ions, for example, by losing or gaining protons (H+). The polyatomic ions so created have internal covalent bonding, but form
ionic solids with other atomic and polyatomic ions of opposite charge. Usually, such salts exhibit ordered crystalline structures, but
with weaker electrostatic attractions between ions than those found in atomic ions. The polyatomic ions are typically more
separated in the solid, or even insulated from each other by the covalent moieties. The typical polyatomic ions, called oxyanions,
are species with a large number of oxygen atoms bonded to the central atom that could be a metal or nonmetal.
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06-1 The octet rule
Formation of a stable octet electron configuration can be achieved by ionic or covalent bonding,
but metals participate in metallic bonding due to an insufficient number of valence electrons
Through our exploration of the electron configuration of atoms, their ionization energies, and their electron affinities
we have learned that filled-shell noble gas configurations are especially stable, with their perfectly spherical electron
densities. It should not be at all surprising that achieving such a configuration is the driving force in formation of atomic
assemblies. A system composed of atoms may lower its energy if the atoms can achieve the electronic configuration of a
noble gas by reapportioning their electrons when they come together.
With the exception of helium (which has two valence electrons), a noble gas configuration means that there are eight
electrons in the valence shell of the atom. An atom will have to lose, gain, or share electrons to achieve this number. Any
such adjustments may only involve the valence shell electrons; they are energetically the easiest to remove or share,
and additional electrons can only be added to the outermost valence shell. We call this phenomenon the octet rule:
whenever possible, the electrons in a compound (or other bonded assembly) are distributed in such a way that eight
electrons surround each atom of the main group element. Hydrogen atoms are surrounded by only two electrons in their
structures to match the helium configuration.
The American chemist G. N. Lewis introduced a useful model to account for the valence electrons of atoms and their
apportionment among atoms participating in bonding. In this model, called a Lewis dot structure, the electrons are drawn
as dots surrounding atomic symbols (T06-1-1). The first four dots are placed separately around the four sides of the
atomic symbol, and any additional dots are then paired with those already present.
Figure F06-1-1. Lewis dot structures for the main group elements. All elements in a family (column) of the periodic table
have identical number of valence electrons.
The octet configuration can formally be achieved in two different ways, giving us two distinct forms of bonding: ionic,
and covalent. Metals that do not have sufficient number of valence electrons to form octets participate in the third type of
bonding (metallic bonding). All types of bonding are based on electrostatic attractions between positively charged nuclei
and negatively charged electrons; these interactions decrease the overall energy when atoms join together and result in a
more stable system.
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Ionic bonding is formed by ions of opposite charges. The lowering of the energy is a result of the electrostatic
attraction between anions and cations.
Covalent bonding is based on the sharing of electron pairs among directly bonded atoms. The shared electrons are
electrostatically attracted to the bonded nuclei, lowering the energy of the system.
In metallic bonding, because there are too few electrons to satisfy the octet rule, electron sharing is carried out to an
extreme. Valence electrons are shared among all metal atoms. In a way, metal cations are "bathed" in a sea of
electrons that move freely through the metallic solid.
Ionic bonds form when metals transfer electrons to nonmetals,
forming ions that attract each other electrostatically
Ionic bonding is found between atoms from opposite ends of the periodic table. As you may remember, metals have low
ionization energies (particularly group 1 alkali metals), and can easily lose electrons to form cations with noble gas
configurations. On the other hand, nonmetals, particularly halogens (group 17), have favorable negative electron
affinities, and easily gain electrons to form anions with noble gas configurations. These complementary characteristics of
metals and nonmetals are perfect for electron transfer between the bonding partners. For example, a sodium atom may
give up its valence electron to become a sodium cation (with a noble gas configuration), while a chlorine atom can accept
that electron, and be converted into a chloride ion, also with the noble gas configuration (F06-1-1). The resulting ions are
strongly attracted to each other by an electrostatic force, and form an extended crystalline solid NaCl (sodium chloride).
Figure F06-1-2. Ionic bonding in sodium chloride. An electron transfer from the sodium to the chlorine atom generates
ions of opposite charges that are attracted to each other by an electrostatic force. The Lewis structures of the atoms and
the ions formed by the electron transfer are also shown. Note that the ionic charges are shown as a superscript. Both Na+
and Cl−have noble gas configurations.
Covalent bonds form when nonmetals share electrons
Covalent bonds form mainly between nonmetal atoms. As you may recall, electrons in nonmetals are held quite tightly, as
is illustrated by their relatively high ionization energies. An octet configuration can be obtained by sharing electrons
rather than by giving them up (which would be energetically costly). For example, two hydrogen atoms may come together
to form an H2 molecule. In the H2 molecule the two electrons are attracted to both nuclei, formally giving both atoms the
[He] configuration. Similarly, by sharing two electrons, two fluorine atoms form an F2 molecule with an octet of electrons
around each. In the hydrogen fluoride molecule, a hydrogen atom and a fluorine atom combine, wherein the hydrogen has
the [He] configuration while the fluorine has an octet of electrons. The shared electron pairs are replaced by a line
connecting the atoms in the Lewis structure of covalently bonded atoms; in the standard chemical notation, this line
represents the bond.
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Figure F06-1-3. Covalent bonds in Lewis notation. The bond is a shared pair of electrons attracted to both nuclei (and
possibly screened by core electrons).
For nonmetals, simple arithmetic tells us how many covalent bonds the atom needs to form in order to satisfy the
octet rule. For example, referring back to Table 06-1-1, it is immediately obvious that carbon needs to form four bonds,
nitrogen requires three, and both fluorine and hydrogen require one. The number of desired covalent bonds must match
the number of unpaired electrons in the Lewis dot structure of the atom. You may recall that we already introduced the
concept of the valence of an atom as the number of covalent bonds an atom prefers to form; now we know that the octet
rule is behind that concept. The number of bonds needed by a neutral atom is 8 − N, where N is the number of valence
electrons.
The nuclei and core electrons in metallic bonding are immersed in an electron sea
composed of all valence electrons
Figure F06-1-4. Electron sea model of metals. The metal cations
(the nuclei and core electrons) are immersed in a "sea" of their
valence electrons which freely roam around the cations.
Metals, in general, do not have enough valence electrons to satisfy the octet rule by forming specific bonds between
atoms one electron pair at a time. To overcome this deficiency, the valence electrons are collectively shared by all metal
ions in the solid. A simple model that accounts for bonding in metals is called the electron-sea model. In this model, the
metal is an extended, tightly packed 3-dimensional network of metal cations (which consist of the nucleus plus core
electrons) immersed in a "sea" of mobile electrons. The electrons are confined to the solid by their electrostatic attraction
to the nuclei, but can easily "flow" via their thermal motions or if a voltage is applied; this explains why metals are good
conductors of heat and electricity. The fact that metal cations in the network may have up to 12 close neighbors also
accounts for metal malleability and ductility (ease of shape deformations), as ions can change their positions in space
relative to their partners easily, and electrons can adjust to this new spatial positioning of the nuclei without difficulty.
A full explanation of metallic properties requires refinement to the simple model described here. We will postpone
such in-depth discussion of metallic bonding until the second semester of general chemistry. In this lesson, we will
concentrate on probing ionic and covalent bonding.
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06-2 Ionic bonding
The formation of ions of opposite charges from neutral atoms is energetically unfavorable
in the gas phase
Let's return to our example of ionic bonding in sodium chloride. Since we know how much energy it takes to ionize
sodium in the gas phase (IE1 = 496 kJ/mol) and how much energy is released when chlorine accepts an electron (EA =
−349 kJ/mol), we can very easily evaluate the overall energetics of the formation of one mole of separated ions of each
kind in the gas phase.
+
Na(g) + Cl(g) ⟶ Na
(g) + Cl
−
(g)
Δ Hr xn = (496 − 349) kJ/mol = 147 kJ/mol
C06-2-1
This process is clearly endothermic, meaning that it will not occur spontaneously in the gas phase. Chlorine has the
most favorable electron affinity of all elements, but even if we match it with the easiest element to oxidize, (cesium,
with an IE1(Cs) = 376 kJ/mol) ion formation in the gas phase would still be unfavorable; the overall enthalpy change would
be ΔHrxn = 27 kJ/mol. In addition, we have not even taken into account the energy needed to produce gaseous sodium
and chlorine atoms from their most stable forms (solid Na, and gaseous Cl2). These energy expenditures are called
atomization energies and in this case are 108 kJ/mol to vaporize sodium metal, and 122 kJ/mole to break the Cl−Cl bonds
in Cl2(g).
We can contrast this endothermic process of making ions in the gas phase with the highly exothermic—indeed,
violent —reaction of sodium metal and chlorine gas, forming solid sodium chloride:
Na(s) +
1
2
Cl (g) ⟶ NaCl(s)
2
Δ Hr xn = −411 kJ/mol
C06-2-2
We know that solid NaCl is a stable and ubiquitous substance just from casual observation, so its formation must be
energetically favorable. Clearly, the cause of this stability is the powerful, energy-lowering electrostatic interaction between
the oppositely charged ions that is in effect as soon as they get within close distance in the crystalline solid. The
electrostatic energy can be evaluated with the help of the Coulomb equation (see E01-4-2):
Eel =
kQ1 Q2
E06-2-1
d
Here, k = 8.99 × 109 J·m/C2, and Q1 = −Q2 = −1.602 × 10−19C (one atomic unit of charge), while d is the sum of
the ionic radii for Na+ (116 pm) and Cl− (167 pm); this is the closest the ions can come to each other. The electrostatic
stabilization energy comes to a substantial −491 kJ per mole of ions—this is clearly enough to overcome the endothermic
ion formation in the gas phase (147 kJ/mol, see above). This is true even if atomization energies of 108 kJ/mol and 122
kJ/mol are included, giving a total of 377 kJ/mol of energy expenditure to make ions. But this is still not sufficient to explain
the highly exothermic formation of solid NaCl. Apparently, there is more to the electrostatic stabilization than what we have
accounted for so far.
Ionic solids form because of large stabilizing electrostatic interactions between ions
Let's examine the structure of crystalline NaCl a bit more closely (F06-2-1), concentrating on the sodium cation in the
middle. That sodium cation is interacting with six chloride ions (labeled A) in its direct vicinity, including the four that are
visible in the picture, and one more in each layer directly in front and behind the one shown. We have accounted for only
one interaction in our basic electrostatic calculation above (E06-2-1). But simply multiplying our number by six would not
be correct either, since we would neglect mutual electrostatic repulsion energies among the six Cl− ions surrounding our
selected Na+, as well as its own repulsions with 12 other Na+ ions (B) displaced "diagonally" in all three dimensions from
our ion of interest. Although the electrostatic interactions drop off as the distance separating the charges increases, more
refined calculations should include additional "spheres" of electrostatic influence (both attractive and repulsive) from the
further-removed ions. In general, if the crystal structures are known, such detailed electrostatic calculations are possible,
and indeed are often carried out to gain better understanding of ionic solids. These calculations are, however, outside of
the scope of our presentation. We will use a different approach to quantitate the strength of ionic bonding (see 06-3),
nevertheless, the electrostatic arguments presented here explain the stability of extended networks in ionic solids
satisfactorily.
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Figure F06-2-1. A model of a small fragment of crystalline NaCl. The grey Na+ cation in the center interacts with 6 Cl− ions
with which it is in direct contact; four are visible (labeled A), and additional two are in the crystal layers directly behind and
in front of the center grey cation. The center Na+ also has repulsive interactions with other Na+ ions (labeled B). Weaker
electrostatic interactions with further removed ions are also present (both attractions and repulsions). You may take a tour
of the surroundings of one of the sodium cations
The thermodynamic data presented here for NaCl also illustrate the limitations imposed by ionic bonding. Ionic
solids will be formed only between cations and anions. Usually this means metal cations (especially those on the left side
of the periodic table with relatively small ionization potentials) and nonmetal anions (especially those on the right side of
the periodic table with favorable electron affinities). For main group elements, ionic solids are limited to ions with a noble
gas configuration. Generating other ions by removing core electrons (as in Na2+, for example) or adding electrons beyond
the valence shell (as in, Cl2− for example), is energetically so costly that even the increased electrostatic stabilization does
not result in an overall favorable process. In general, electrostatic stabilization in solids cannot compensate for the cost of
the increasing ionization energies for each successive electron removed and the positive unfavorable electron affinities of
sequentially added electrons beyond the +3 or −3 charge. This ±3 charge is a practical upper limit for ions in ionic solids.
Similarly, transition metals (which may have more than three electrons beyond the noble gas core) will generally form ions
that do not have a noble gas configuration, demonstrating the limits of the octet rule. As a reminder, when forming ions
transition metals will first lose electrons from their valence ns2 subshell before losing any electrons from their (n−1)d
subshell.
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06-3 Lattice energy
Lattice energies that measure the strength of ionic bonding are calculated
using a thermodynamic cycle
As we have just discovered, the principal reason behind the stability of ionic compounds is the electrostatic attraction
between ions of opposite charge. The ions are drawn together, releasing energy while forming highly ordered 3dimensional crystalline lattices. The strength of such ionic bonding can be defined as the energy required to break up the
solid into individual ions in the gas phase. This energy is called the lattice energy. For example, it takes 788 kJ for one
mole of solid sodium chloride to completely separate into ions:
+
NaCl(s) ⟶ Na
(s) + Cl
−
(g)
C06-3-1
Δ Hlattice = +788 kJ/mol
Notice that the reverse process (the forming of an ionic solid from gaseous ions) is highly exothermic, signifying the
stability of the solid. In general, the higher the lattice energy, the more the ions are attracted to each other in the solid, and
the stronger the ionic bonding.
Lattice energies cannot be measured directly. An indirect approach is employed to obtain their values, based on a
thermodynamic cycle called the Born-Haber cycle. The thermodynamic principle behind this method is the fact that
enthalpy is a state function; the change in energy (ΔH) does not depend on the paths used in the preparation of the
compound, only on its structure and state. In fact, we employed similar thermodynamic principles in the previous section
(06-2) to learn about interactions between ions in NaCl.
The calculated difference in energy between the ionic solid and the gaseous ions gives us the lattice energy that we
were after. That energy difference is the measure of the net electrostatic attractions between ions in the solid; this is the
best gauge of the ionic bond strength. The lattice energy of several ionic solids obtained from analogous thermodynamic
cycles are collected in Table T06-3-1.
T06-3-1. Lattice energy of selected ionic solids
Compound
Lattice
energy
[± 1]
(kJ/mol)
Compound
Lattice
energy
[± 2]
(kJ/mol)
Compound
Lattice
energy
[± 3]
(kJ/mol)
LiF
1047
BeF2
3526
AlF3
6252
LiCl
860
MgF2
2978
CrCl3
5529
LiBr
820
MgCl2
2522
VF3
5890
LiI
761
MgBr2
2451
ScF3
5540
NaF
928
MgI2
2340
CrCl3
5529
Sc2O3
13708
NaCl
786
CaF2
2651
NaBr
751
CaCl2
2271
NaI
703
CaI2
2087
V2O3
14520
KF
826
SrF2
2513
Cr2O3
14957
KCl
717
BaF2
2373
Al2O3
15920
KBr
689
BaCl2
2069
Ti2O3
14149
KI
650
RbF
795
BeO
4443
ScN
7506
RbCl
695
MgO
3845
VN
8233
RbBr
668
CaO
3454
NbN
8022
RbI
632
SrO
3276
CsF
759
BaO
3054
CsCl
667
NiO
4010
CsBr
647
VO
3863
CsI
601
ZnO
3971
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The major factor controlling the magnitudes of lattice energies are the charges on ions,
with ion sizes being a secondary contributor
The major trends in lattice energy are accounted for by factors that control the electrostatic interactions, including the
magnitude of the charge on the ions, the ion sizes, and the specific crystal arrangements of the ions. Of these factors, the
magnitude of the ionic charge is by far the most dominant, as the electrostatic attraction is directly proportional to the
product of charges in Coulomb’s law. The doubling or tripling of that contribution upon going from a +1 ion to a +2 or +3
ion is clearly discernible in the data in Table T06-3-1.
The ion sizes vary much less from ion to ion, and so do their sums (values of d in E06-2-1). In general, the
differences in ion sizes contribute less to the observed magnitudes of lattice energy, but differing sizes explain the trends
among salts with the same ionic charges very well. Ions with a given set of charges often have identical or similar lattice
symmetries, and thus specific arrangements of atoms do not strongly influence the general trends, although they do affect
the magnitudes of specific lattice energies.
Figure F06-3-1. Examples of trends in lattice energies and melting points for simple ionic solids with common ions. The
values of d (the sum of the ionic radii, or the distance between the two ions) are shown below the ions in pm (100 pm = 1
Å). The lattice energies (and melting points) dramatically increase as ion charges increase and decrease as one of the
ions in the molecule increases in size (with the other one kept the same).
The data in Table T06-3-1 show that increasing ion size correlates with decreasing lattice energy. Direct comparison
of ionic solids where one of the ions is kept constant while the other ion increases in size illustrates this trend the best
(F06-3-1). The lattice energy steadily decreases with increase of d, as predicted by the inversely proportional relationship
between the electrostatic energy and the separation of charges (E06-2-1).
The comparison of the lattice energies of two ionic solids where the sum of the ionic radii (d) is almost the same
highlights both of the two most important trends. For example, SrO (d = 258 pm) has a lattice energy that is almost 4 times
larger than LiCl (d = 257 pm); as expected given the +2/−2 charges in the former (compared to +1/−1 charges in the later).
On the other hand, when the charges are the same, as in KCl (d = 319 pm) and NaI (d = 322 pm), the lattice energies
differ very little, even if the solids involve different ions of different sizes.
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Given the ionic bonding principles that we have just explored, we can understand the physical properties of ionic
solids. As we have seen, such solids are organized to maximize the electrostatic attractions between ions. This
organization is the basis of their crystalline, highly ordered periodic structure. Since there are no direct bonds between
ions (only electrostatic attractions) such crystals can easily be mechanically broken into smaller pieces by splitting along
crystallographic planes while preserving their crystalline order. We call such solids brittle. The large attractions between
ions as gauged by the high lattice energies indicate that individual ions are extremely hard to separate into the gas phase
from such solids, and even relatively minor dislocations of ions from their optimal positions within the lattice are
energetically costly. Therefore, ionic solids built of atomic ions have high melting points, since melting destroys the highly
ordered state (even if the close distance between the ions is retained). Indeed, as seen in Figure F06-3-1, higher melting
points correlate with higher lattice energies. You may, for example, predict with confidence that MgO will have a
significantly higher melting point than NaCl, or order the sodium halides according to their melting points even if you do
not know anything about their specific values. The ability to predict trends in physical or chemical properties of materials
based only on the understanding of their structures and some basic chemical principles demonstrates the power of the
scientific method. We will continue to draw connections between the atomic structure and macroscopic properties of
matter throughout our Lessons.
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06-4 Covalent bonding
Covalent bonds are made by sharing of electron density between atoms
The electrostatic forces at work in covalent bonding may be illustrated by analyzing H2, which is the simplest molecule
and is built of two protons and two electrons. When the two protons in the nuclei of the hydrogen atoms are brought close
together, there is electrostatic repulsion between them and the energy of the system increases. The shared electrons are
attracted to both nuclei, but their effectiveness in providing electrostatic attraction to counter the nuclear repulsions
depends on their position. An electron in positions close to the internuclear axis can attract both nuclei to itself and also
diminish their mutual repulsion by screening their charges. This stabilizing influence diminishes if the electron moves away
from the midpoint, and becomes destabilizing when it is in a position behind one of the nuclei; in such positions the
electron attracts the nearer nucleus more strongly than the one further away, and pulls the nuclei apart.
The region of space where electron-nuclei attractions provide stabilization may be "mapped out" by doing
electrostatic calculations for multiple points surrounding the nuclei (F06-4-1). The resulting bonding region shown in the
figure leads us to the conclusion that covalent bonding calls for increased electron density on and around the inter-nuclear
axis.
Figure F06-4-1. From left to right: (a) Electrostatic interactions in H2 or in any electron pair that is shared between two
nuclei. The orange vectors illustrate repulsive forces between like charges, and the blue vectors show attractive forces
between opposite charges. The particles are not drawn to scale. (b) The green area represents the bonding region (the
parabolic shape is darwn by considering electrostatic interactions of electrons). Higher of electron density in the marked
area contributes to bonding.
Sharing two electrons doubles the attractions, but also introduces electron-electron repulsion that destabilizes the
system somewhat (F06-4-1). The bond between the atoms maximizes electrostatic attractions by having the shared
electrons occupy the inter-nuclear bonding region (F06-4-1a), but also minimizes repulsions by having these electrons
avoid each other. Although the full elucidation of quantum effects is beyond the scope of our presentation, we should add
that the kinetic energy of electrons is lower in the inter-nuclear region, decreasing the overall energy of the system. In a
simplistic way you may visualize it as electrons being "trapped" in the bonding region and slowed down. The bottom line is
that the bonded molecule is more stable than two separate hydrogen atoms.
As you may remember, in quantum chemistry we deal with electron densities (which represent probabilities) rather
than specific electron location, and we use wavefunctions and orbitals to describe this probability. We will return to these
concepts to describe the details of how covalent bonds are formed in a later Lesson (Lesson 9-1). For now, we can
conclude that the attractive force between the positively charged nuclei and the negatively charged electron-pair cloud
between them holds atoms together.
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Lewis structures illustrate the apportionment of bonding and nonbonding electron pairs
to atoms in molecules
One pair of shared electrons (bonding electrons) is all that is available in the hydrogen molecule, and that is all that is
necessary to attain the noble gas configuration of [He]. More than one electron pair may need to be shared to reach the
octet configuration of a noble gas for atoms in other molecules. The number of shared electron pairs (bonds) is specified
by the valence of the atom. Let's consider some simple examples of molecules with bonds to hydrogen (F06-4-2):
Figure F06-4-2. The Lewis structures of hydrogen fluoride, water, ammonia, and methane. For nonmetals, the number of
bonds an atom forms with hydrogen to attain an octet follows the valence rule (8 − N), where N is the number of valence
electrons on that atom.
In our standard notation for a Lewis structure (F06-1-2), we show shared electrons as lines connecting the nuclei,
and unshared electron pairs (also known as lone pairs or nonbonding pairs) as pairs of dots. In all of these structures the
hydrogen atoms are surrounded by two bonding electrons ([He] configuration), and the other atoms are surrounded by a
full octet. The fluorine atom in the hydrogen fluoride molecule forms one bond to a hydrogen and has 3 nonbonding pairs.
The oxygen atom In a water molecule forms two bonds to hydrogen atoms and has two lone pairs. The nitrogen atom in
ammonia has three bonds to hydrogen atoms and one unshared electron pair. The carbon atom in methane forms four
bonds to hydrogen atoms and has no lone pairs. Each shared electron pair constitutes one covalent bond. If the two
atoms are joined by just one bond, that bond is referred to as a single bond.
The Lewis structures of F06-4-2 account for the apportionment of all valence electrons for all of the atoms
participating in bonding. They clearly show the number of bonds and lone electron pairs, and most importantly they show
the connectivity of the molecule, specifying which atoms are bonded to which atoms. However, these structures are
deficient in their lack of information about the three-dimensional positioning of the atoms in relation to each other. As we
will soon learn (Chapter 8), there are ways to use the Lewis structures to deduce information about the three-dimensional
positioning, but until then we will draw them in a way that keeps the bonding and lone pairs as far from each other as
possible in the plane of the drawing; we anticipate that electron pairs repel each other, and we need to maximize their
separation.
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06-5 Multiple bonds
Multiple bonds result when more than one electron pair is shared
between the same two atoms in molecules
Atoms in some molecules may share multiple electron pairs in order to achieve an octet configuration. If two electron pairs
are shared the bond is called a double bond, and if three electron pairs are shared we have a triple bond. Some transition
metals may form quadruple bonds, sharing four electron pairs, but these are outside of the scope of our Lesson. The
number of electron pairs shared is called the bond order. Examples of common simple molecules with single and multiple
bonds are shown in Table T06-5-1.
Table T06-5-1. Examples of single, double, and triple bonds and their bond lengths
Bond order of 1
hydrogen
peroxide
Bond order of 2
Bond order of 3
dioxygen
146 pm
hydrazine
121 pm
diazene
145 pm
ethane
dinitrogen
125 pm
ethene
154 pm
ethanol
ethyne
134 pm
142 pm
120 pm
carbon
monoxide
methanal
methylamine
110 pm
121 pm
hydrogen
cyanide
methyleneimine
147 pm
113 pm
127 pm
116 pm
Bond lengths depend on atomic bonding radii and bond order
Multiple bonds may form between atoms of the same element or between different elements, but they are usually limited
to nonmetals of the second row of the periodic table (we will address that limitation later in Lesson 9). Single bond lengths
can be predicted reasonably well by adding the values of the atomic bonding radii. For example, we know that atomic
radii shrink from left to right in a given row of the periodic table and would expect to see declining bond lengths in the
C−C, C−N, C−O series. This trend is indeed confirmed by the data in Table T06-5-1. The multiple bonds are shorter than
the single bonds between the same pair of atoms, and the bond lengths in general correlate with bond order; the higher
the bond order the shorter the bond. The bond lengths (single or multiple) between a given pair of atoms usually vary little
from compound to compound, and the average bond lengths (Table T06-5-2) describe the bonding distances with
adequate precision for most applications. Deviations, when observed, are readily explainable based on the details of the
electronic structure of the specific cases (we will discuss such cases later on, in Lesson 9).
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Table T06-5-2. Average bond lengths (in pm): Single Bonds
I
Br
Cl
S
P
Si
F
O
N
C
H
H
161
142
127
132
138
145
92
94
98
110
74
C
210
191
176
181
187
194
141
143
147
154
N
203
184
169
174
180
187
134
136
140
O
199
180
165
170
176
183
130
146
F
197
178
163
168
174
181
128
Si
250
231
216
221
227
234
P
243
224
209
214
220
S
237
218
203
208
Cl
232
213
200
Br
247
228
I
266
Multiple Bonds
Bond
Length
Bond
Length
N=N
124
C=C
134
N≡N
110
C≡C
116
C=N
127
C=O
123
C≡N
116
C≡O
113
N=O
122
O=O
121
Bonds are stronger if they are shorter or have higher bond order
As we have just learned, the strength of bonding in ionic solids is measured by the lattice energy, which expresses the
strength of the electrostatic interactions between collections of ions. In contrast, the strength of a covalent bond is specific
to the pair of atoms sharing the electrons, and is generally only weakly influenced by other atoms or bonds in the
molecule. The strength of a covalent bond is defined by the energy needed to break that exact bond while leaving other
parts of the molecule intact. This energy is called the bond dissociation energy (BDE), and the stronger the bond, the
higher its BDE. We will define the process involved in breaking the bonds later (Lesson 16). The bond strength of a given
covalent bond does not vary much from molecule to molecule, and average values are quite useful for qualitative
applications (Table T06-5-3). Deviations from the average values are found in some molecules, and the reasons for these
deviations are linked directly to the details of their structure.
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Table T06-5-3. BDEs, average single bond dissociation energies (in kJ/mol)
I
Br
Cl
S
P
Si
F
O
N
C
H
H
299
366
431
368
322
323
568
463
391
413
436
C
220
276
328
259
264
301
453
358
276
348
N
159
243
200
200
335
272
176
193
O
234
234
203
364
340
368
190
146
F
277
237
193
327
490
582
157
Si
234
310
464
226
P
184
264
319
218
253
242
S
Cl
208
218
Br
175
193
I
151
226
209
266
Multiple Bonds
Bond
BDE
Bond
BDE
N=N
418
C=C
620
N≡N
941
C≡C
815
C=N
615
C=O
745
C≡N
891
C≡O
1072
N=O
607
O=O
499
Even a casual scan of trends in bond lengths in Table T06-5-2 and bond strengths in Table T06-5-3 yields a general
and very useful correlation: the shorter the bond and the higher its bond order, the stronger it is. This trend holds for bonds
with different bond orders between the same pairs of atoms as well as for different bonds of the same order. We may also
notice that for most multiple bonds, the second and third bonds are not as strong as the first bond (the exceptions include
bonds between atoms with lone pairs such as oxygen and nitrogen). We will come to fully appreciate this difference when
we learn how these bonds are constructed (in Lesson 9).
Bond
C≡C
C=C
C−C
Length
113 pm
134 pm
154 pm
BDE
839 kJ/mol
614 kJ/mol
348 kJ/mol
Bond
C−F
C−Cl
C−Br
Length
134 pm
176 pm
193 pm
BDE
486 kJ/mol
327 kJ/mol
285 kJ/mol
Under standard conditions, compounds with covalent bonds are usually gases, liquids, or relatively soft solids with
low melting points. The molecules in such substances are held together by weak electrostatic attractions called
intermolecular forces. The molecules may move past each other or even separate with relative ease. In some covalent
compounds, however, if the atoms bond into extensive covalent networks, like in diamond for example, the solids may be
very hard and have extremely high melting points. In general. the bonds holding the atoms together within the molecules
(called intramolecular forces) are strong while the intermolecular forces are much weaker. We will learn much more about
these interactions and how they govern the properties of covalent compounds in future Lessons.
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06-6 Polyatomic ions
Atoms in polyatomic ions are held together by covalent bonds
So far we have studied compounds that were either ionic or covalent. We have learned how to identify ionic compounds
from the elements they contain, typically a metal and a nonmetal, and typically from opposite sides of the periodic
table. Metals have low ionization energies and form cations easily, and nonmetals have negative electron affinities and
form anions readily. The ions are held together by electrostatic interactions between positive and negative charges. The
ionic solids formed by the ions, called salts, can be recognized by their characteristic physical properties. They are brittle
solids at room temperature, and have high melting points.
Identification of covalent molecular compounds is also facilitated by identification of the type of elements in the
molecular formula. Covalent substances are composed of nonmetals or metalloids, which are the elements in the upper
right hand side of the periodic table. Atoms in molecular compounds are held together by strong covalent bonds, but the
individual molecules are held together by weak electrostatic attraction (intermolecular forces). Such compounds are often
gases or liquids at room temperature and pressure. In the solid state, molecular compounds are often soft and melt easily.
In general, they have low melting and boiling points.
However, there are also chemical species that are more complex. They have ionic character, but the ions involved
have internal covalent bonding. We have seen examples of such compounds when we discussed formulas and
composition. The simplest examples of such salts result from replacing one of the atomic ions with a polyatomic ion. For
example, in sodium hydroxide, NaOH, the hydroxide ion has a covalent O−H bond, and in ammonium chloride (NH4Cl) all
four hydrogens form covalent bonds to the nitrogen in the center of the ammonium cation.
The structural complexity of the polyatomic ions can progress quickly. You may be familiar with some of the common
polyatomic ions, such as carbonate (CO32−), nitrate (NO3−), peroxide (O22−), sulfate (SO42−) or phosphate (PO43−). You
might be surprised by permanganate, MnO4−, or dichromate, Cr2O72−, which still have internal covalent bonding even
though they are composed of metal/nonmetal combinations. You may have recognized some of the oxyanions listed
above since they are derived from familiar oxoacids including nitric acid, sulfuric acid, and phosphoric acid. What all of
these species have in common are multiple oxygen atoms covalently bonded to a central atom that could be a metal or
nonmetal. We will discuss bonding and structure of such species in forthcoming Lessons.
Table T06-6-1 Common polyatomic ions
Even though the polyatomic anions have only internal covalent
bonding, they combine with cations (atomic or polyatomic) to form
ionic solids. Usually such salts form precisely ordered crystals, but
typically with much lower melting points than purely ionic salts. The
electrostatic attractions between such ions are not as strong as in the
atomic ions; the covalently bonded fragments separate the charges
by a larger distance and even insulate the charges from each other.
Consider ammonium dichromate, (NH4)2Cr2O7, and its melting
point of 180 °C. Both the low melting point and a casual look at the
composition may lead one to identify this compound as covalent.
There is no metallic cation to start the formula, and the metal is
hidden in the part of the formula reserved for nonmetals. Yet, in the
signature process for salts, ammonium dichromate dissolves in water
and dissociates into the ammonium cations and the dichromate
anions. The compound is ionic, but its ions are constructed using
covalent bonds.
For now, we list the most common polyatomic ions that we will
encounter in our future Lessons (Table T06-6-1). We will discuss their
structures, and some aspects of their bonding later in the course.
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Formula
Name
Formula Name
NH 4+
H 3O+
ammonium
hydronium
SO 32−
SO 42−
sulfite
sulfate
HO −
O 22−
hydroxide
peroxide
NO 2−
NO 3−
nitrite
nitrate
CH 3COO− acetate
H 2PO4−
C 2O42−
oxalate
HPO 42−
CN −
SCN −
cyanide
PO 43−
thiocyanate
N 3−
azide
HCO 3−
CO 32−
ClO −
ClO 2−
ClO 3−
ClO 4−
dihydrogen
phosphate
hydrogen
phosphate
phosphate
hydrogen
carbonate
carbonate
hypochlorite
chlorite
MnO 4− permanganate
chlorate
CrO 42− chromate
perchlorate Cr 2O72− dichromate
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07 Molecular Structure
Once we have grasped the basic concepts of ionic and covalent bonding, we are ready to dive deeper and discover
that the two types of bonding are just extremes on a continuum of electron sharing schemes. The extent of bias in electron
density allocations to the bonding atoms and the resulting charge polarization are predetermined by the differences in
electronegativity between the atoms. The unequal sharing of electrons results in bond dipoles, and ultimately in the
polarity of an entire molecule, which in turn constitutes the critical component of intermolecular interactions, as we will see
later in the course.
Lewis structures are an important tool to account for atomic connectivity and for the distribution of all valence
electrons among bonded atoms. They allow us to begin to envision the 3-dimensional shapes of molecules and are basis
for the development of bonding theories. Mastering the skill to efficiently create Lewis structures and to interpret the
electron apportionments they represent is just the first step in our exploration of the electronic structure of molecules.
07-1 Electronegativity
Electronegativity is the tendency of an atom in a compound to attract electron density toward itself. It is usually expressed
on Pauling's scale, which is unit-less. Highly electronegative atoms behave as electron acceptors, while atoms with low
electronegativity act as electron donors. Electronegativity increases from the left to right and from the bottom to top in the periodic
table, with some deviations among the transition metals.
07-2 Bond dipoles
Bond dipoles for diatomic molecules provide a direct measure of the charge separation and the polarity of a species. The
analysis of bond dipoles through the prism of electronegativity differences between bonded atoms establishes boundaries for ionic
and covalent bonding, and allows us to evaluate the relative contributions of the two in the "gray" area of polar covalent bonds.
Exploring bond dipoles is the first step to understanding molecular polarity, and eventually the intermolecular interactions that
govern phase transitions.
07-3 Lewis structures
Lewis structures are symbolic representations of atom connectivity in a molecule or ion and account for the apportionment of
all valence electrons into bonding electrons and lone pairs. A systematic protocol for drawing Lewis structures facilitates their
generation; our procedure is guided by the octet rule. Determining the formal charges on each atom helps us to evaluate the
relative importance of alternative structures.
07-4 Resonance
Some molecules cannot be accurately represented by just one Lewis structure. The culprits are delocalized electron pairs. In
contrast to localized pairs, which are only shared between a pair of bonded atoms, three or more atoms share delocalized electron
pairs. Such increased sharing lowers the energy of the system. In order to adequately account for delocalized pairs, superposition
of multiple structures with alternative electron apportionments is needed; these are called resonance structures. The relative
importance of each resonance structure and their individual contributions are evaluated based on partial charge criteria.
07-5 Octet rule exceptions
There are three categories of compounds whose atoms do not always obey the octet rule. They include compounds with an
odd number of electrons (radicals), compounds with insufficient valence electrons to construct an octet, and hypervalent
compounds which have atoms with more than eight electrons. Hypervalent compounds can only occur when central atoms from
the 3rd row (or below) of the periodic table bond to small electronegative atoms.
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07-1 Electronegativity
Electrons in covalent bonds are not always shared equally
We recently learned about ionic bonds between ions, which are formed when electrons are transferred from metal atoms
to nonmetal atoms. We used NaCl as a prime example of ionic interactions wherein one atom (from the left side of the
periodic table) completely gives up its electron, while the other atom (from the right side of the periodic table) accepts it.
As we learned, the electrostatic attraction between the ions lowers the energy of the system.
We also explored covalent bonding, wherein the electrostatic attraction of one or more shared electron pairs to both
bonded nuclei lowers the system’s energy. The molecules H2 and Cl2 illustrate bonding where the electron sharing is
perfectly equal. However, with H−Cl (hydrogen chloride) we arrive at an interesting junction. Hydrogen chloride is a gas
under standard conditions, implying the presence of a covalent bond, even though hydrogen is from the same group as
sodium (group 1). So, why don't we get an ionic HCl with H+ and Cl− ions in this case?
You may recognize that hydrogen and chlorine are both nonmetals, whereas ionic solid formation requires the
combination of a metal and a nonmetal. If we take our analysis to the next level, we realize that hydrogen is a nonmetal
(despite being in group 1) because of its high ionization energy; IE1 for H is 1312 kJ/mol, while that of Na is only 496
kJ/mol. Thus, although chlorine is "willing" to accept the electron, hydrogen is "reluctant" to give it away. Does this mean
that the bonding electron pair is shared equally in H−Cl?
As we will learn shortly (Lesson 07-2), one can answer this question experimentally by measuring dipole moments.
For now, let's just jump to the results, which tell us that hydrogen has a slight positive charge (+0.18 au) in an H−Cl
molecule, while chlorine has a negative charge of equal magnitude (−0.18 au). This polarization indicates that chlorine is
pulling a larger share of the bonding electron pair toward itself. This tendency of an atom to attract electron density in the
bond toward itself is called electronegativity (χ).
Electronegativity is a measure of the tendency of an atom in a molecule
to attract shared electron density toward itself
American chemist Linus Pauling observed that the H−Cl bond (BDEH−Cl = 429 kJ/mol) was notably stronger than the
mean of the BDEs of H−H and Cl−Cl bonds (BDEH−H = 433 kJ/mole and BDECl−Cl = 240 kJ/mol). One would expect the
mean value to be closer to the observed value if electron sharing was equal in HCl. Pauling deduced that the difference is
due to the partially ionic nature of the bond (18%, based on the partial charges reported above), which adds stability. He
proposed to use the extra bond stability as a measure of the differences in electronegativity (Δχ) between bonded atoms
(H and Cl in this case). Using the available BDE values for various bonds (and an arbitrarily set value of electronegativity
for hydrogen) the Pauling electronegativity scale was developed. The accepted values shown in Figure F07-1-1 are based
on the most current BDEs (and use geometric means of BDEs as a reference).
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Figure F07-1-1. Pauling electronegativity scale. The least electronegative atoms are metals in the lower left corner of the
periodic table (light yellow) and the most electronegative atoms are in the upper right corner (dark orange).
Electronegativity increases going from the bottom-left to the upper-right of the periodic table, with some deviations
observed among transition metals. Click on the image to see a 3D view.
Pauling electronegativity is a unit-less number on an arbitrary scale, ranging from 0.79 for cesium to 3.98 for
fluorine. With the exception of some of the transition metals, electronegativity increases across the periodic table from left
to right and in a less pronounced fashion within the groups from the bottom to the top. The increase of electronegativity in
each row follows the trend in Zeff. Indeed, the increasing effective nuclear charge is responsible for increases in IE1 and
|EA|, and both of them contribute to electronegativity (see Mulliken electronegativity). Electronegativity is the best
expression of the tendency of an atom to donate or accept electron density within chemical bonds. Highly electronegative
atoms readily accept electron density and form negative ions. Atoms with low electronegativity, sometimes called
electropositive, readily donate electron density and form positive ions.
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07-2 Bond dipoles
Dipole moments are the experimental measure of the polarization
of electron density in molecules
When two electric point charges of opposite sign and equal magnitude Q are separated by a distance r, an electric dipole
forms. It is quantified by its dipole moment, μ:
E07-2-1
μ = Qr
The SI unit of the dipole moment is coulomb-meter (C·m), but it is typically expressed in debyes (D), a unit that
equals 3.34 × 10−30 C·m. For molecules, we usually use electron charge (1 au = 1.60 × 10−19 C) and separation distances
in picometers (pm) or angstroms (1 Å = 100 pm), matching units of bond lengths. For reference, the dipole moment of
(+1) and (−1) charges separated by 100 pm is:
μ = Qr = (1.60 × 10
−19
10
−12
m
C) (100 pm) (
1 D
)(
1 pm
3.34 × 10
−30
) = 4.79 D
E07-2-2
C⋅ m
The dipole moment provides us with a measure of charge separation, or the system polarity. By convention it is a
vector pointing from the center of positive charge to the center of negative charge, with the length representing the
magnitude of μ.
It turns out that the dipole moments of molecules can be experimentally measured by various methods. Such
measurements, when done on diatomic molecules in the gas phase (where the molecules are electrostatically
unperturbed by any neighbors), provide us with valuable information on the sharing of bonding electrons and the polarity
of molecules. We collect some pertinent data in Table T07-2-1.
Table T07-2-1. Experimental and calculated (fully ionic) dipoles of diatomic
molecules
μ
μ
calculated
Molecules
(observed)
(100%
ionic)
Bond
length
%ionic
D
D
pm
%
H−I
0.45
7.76
162
6
0.46
H−Br
0.83
6.85
143
12
0.76
H−Cl
1.11
6.13
128
18
0.96
H−F
1.83
4.45
93
41
1.78
Cs−F
7.88
11.25
235
70
3.19
K−Br
10.63
13.51
282
79
2.14
Na−Cl
9.00
11.30
236
80
2.23
K−Cl
10.27
12.79
267
80
2.34
Na−F
8.16
9.24
193
88
3.05
Δχ
(electronegativity
difference)
Let us first demonstrate how the dipole moments are calculated assuming a fully ionic bond. In the H−Cl molecule
the bond length is 128 pm. We can take advantage of the calculation in E07-2-2 where the separation distance was 100
pm, and just use the ratio of distances here. If the charges were fully separated (+1 on H and −1 on Cl), the dipole
moment would be (128/100) × 4.79 D = 6.13 D. Yet the measured dipole moment is only 1.11 D, or 0.18 (1.11/6.13) of
that expected for the ionic molecule. We conclude that H−Cl bond is 82% covalent and 18% ionic. We use the same
strategy to calculate the ionic dipole moments for other molecules in T07-2-1, using their bond lengths.
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If we return to our examples from the previous section (Lesson 07-1), we now understand the partial charge
separation that we used for the H−Cl molecule (δ = 0.18 of the full electron charge). Molecules of H2 and Cl2 have no
dipole moment (μ = 0), but molecules of H−Cl and Na−Cl do (Figure F07-2-1). Note that we are looking here at individual
Na−Cl molecules in the gas phase and not the ionic solid. Gaseous molecules of NaCl and other ionic substances shown
in the bottom half of Table T07-2-1 can be prepared by heating the ionic solid to a high temperature under vacuum. We
use them here for illustration purposes; they are not stable under standard conditions.
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Charge polarization is visualized by listing partial charges on atoms, drawing dipole vectors,
or displaying maps of electrostatic potential energy
μ=0D
μ=0D
μ = 1.11 D
μ = 9.00 D
Figure F07-2-1. Molecules of H2, Cl2, HCl and NaCl. The dipole moment, partial charge, and full charge are illustrated on
the Lewis structures. The partial charge of an HCl molecules is δ = 0.18. The dipole moment vector for NaCl is not shown,
but would be 8 times longer than the one shown for HCl, if presented on the same scale. On the electrostatic potential
maps (MEP, drawn here to reflect relative molecular sizes and on a common ±200 kJ potential energy scale), reds
represent areas of high electron density, blues represent low electron density, and greens are "neutral'.
Molecules of H2 and Cl2 have nonpolar covalent bonds; there is no charge separation (μ = 0). An H−Cl molecule has
a polar covalent bond. It has a dipole moment, μ = 1.11 D, with partial charge separation; chlorine has δ−= −0.18 au, and
hydrogen has δ+ = 0.18 au, as we calculated above. The dipole moment vector (F07-2-1) shows the direction of the
electron density shift. An Na−Cl molecule is 80% ionic. The partial covalent bond is shown in F07-2-1 as a broken line,
and charges are shown as full charges (which is an approximation). It has a large dipole moment, μ = 9.00 D; the dipole
moment vector would have to be about 8 times longer than one for H−Cl if drawn to scale.
The colored surfaces shown in Figure F07-2-1 are maps of electrostatic potential energy (MEP); essentially, they
show the electron density in a molecule as mapped on its surface. First, a boundary surface for the molecule is drawn that
contains 95% of electron density (as we did when we studied atomic orbitals). For all practical purposes, this surface is
the same as the van der Waals surface that we discussed previously, and corresponds to the sum of the non-bonding
radii. It is also what we call the space-filling model of a molecule, as it shows the volume of its electron cloud. Next, we
use a positive +1 point charge as a probe and move it on the surface, measuring its electrostatic energy. We finish by
painting the surface with colors representing that energy. Areas in red represent high electron density, while those in blue
represent low electron density (greens are in the middle-range). Such MEPs can only be generated by computers, but are
visually very informative. It’s like having a pair of "superman glasses" and being able to see and distinguish molecular
regions of varying electron densities.
The electronegativity differences between bonded atoms determine if the bonds are
covalent, polar covalent, or ionic
The general trends observable in Table T07-2-1 clearly indicate that there really is a continuum of the degree of
polarization of electron density involved in bonding. At one end is equal sharing (where there is no difference in
electronegativity between bonded atoms), in the center are polar covalent bonds (with a moderate contribution from ionic
bonding), and at the other end are highly polarized, essentially ionic bonds between atoms (where electronegativity
differences are large). In other words, purely ionic bonds and nonpolar covalent bonds merely represent the extremes of
bonding possibilities. The electronegativity scale (F07-1-1) provides us with a way to rapidly predict the kind of bonding
present and assess the polarity of the bond by looking at the difference in electronegativity. The difference in
electronegativity, Δχ, is detailed in the last column of table T07-2-1. Although it is impossible to draw clear dividing lines, it
is common to assume that for atoms with Δχ ≥ 1.8 there is full or almost full electron transfer, resulting in the formation of
ionic solids (F07-2-2) On the other hand, bonds between atoms with Δχ ≤ 0.4 are deemed nonpolar covalent bonds. In the
midway range bonds are polar covalent, with their polarity increasing as Δχ increases (Figure F07-2-2). The suggested
cutoff values are only given here to build up your chemical intuition. There are many complicating factors that make
these arbitrary dividing lines very fuzzy, indeed.
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Figure F07-2-2. The difference in electronegativity, Δχ, can be used to predict the bond polarity. Bonds with Δχ ≤ 0.4 are
deemed to be nonpolar covalent, bonds with Δχ between 0.4 and 1.8 are polar covalent, and bonds with Δχ > 1.8 are
ionic. All MEPs are at the common ±200 kJ scale.
The concept of polarity is not limited to just diatomic molecules. Any molecule in which the centers of positive and
negative charges do not coincide is a polar molecule. We will learn later (Lesson 08-4) how to determine polarity of
molecules by analyzing bond dipoles, but even now it is important to appreciate that polarity determines interactions
between molecules, called intermolecular forces (Lesson 11-2) which are responsible for the properties of gases, liquids,
and solids, as well as the transitions between these phases. Polar molecules interact with each other and other polar
molecules and ions through electrostatic forces. Dipole "ends" of a particular sign (+ or −) are attracted to ions or dipole
ends with the opposite sign, and are repelled by ions and dipole ends of the same sign. Although dipole-ion and dipoledipole interactions are much weaker than the ion-ion interactions of ionic bonding, they are comparable to the kinetic
energy of thermal motions around room temperature (25 °C). The relative magnitude of these energy terms influences the
phase behavior of substances around us. To illustrate, the polarity of water molecules is the reason that water remains a
liquid at temperatures at which most other small molecules exist as gases. Since liquid water is necessary to support life
as we know it, these intermolecular interactions are clearly worth extensive examination (Chapter 11).
To build our understanding of the interactions responsible for holding atoms together, we initially treated ionic and
covalent bonding as two separate concepts. We recognize that when covalent bonding is dominant, the compounds exist
as molecules in the form of gases, liquids, or soft solids (characterized by relatively low melting and boiling points), with
the exception of extended covalent networks such as those in diamond or silicon. We learned that both the polarity of
covalent bonds and the overall polarity of molecules are very important in determining the properties of covalent
substances. On the other hand, when ionic bonding is dominant the compounds are brittle, high-melting solids, with
extended lattice structures that dissociate into aqueous ions if soluble in water. This new understanding of
electronegativity gives us the ability to quickly recognize the predominant bonding interactions in a substance and
understand its properties, even if we were initially unsettled by the continuity of the bonding models from covalent
nonpolar, through covalent polar, to ionic. We can now add more sophistication to our approach and move beyond simple
metal-nonmetal (or ionic) and nonmetal-nonmetal (or covalent) paradigms.
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07-3 Lewis structures
Lewis structures show atom connectivity and apportionment
of all bonding and nonbonding electrons to individual atoms in molecules
As we mentioned before, Lewis structures are in essence an accounting scheme for apportioning valence electrons
among atoms in molecules. They show connectivity between atoms, and they show the number of bonding and lone
electron pairs on each atom. The octet rule and the valence rules derived from it govern the possible arrangements of
electron pairs in most compounds. Lewis structures are also the starting point to understand the 3-dimensional shapes
of molecules, and to eventually understand theories of bonding. They constitute the symbolic language of chemistry.
Fluency in both reading and writing this language is a critical skill every student of chemistry must master. With practice,
reading Lewis structures should become equivalent to reading a novel; one does not see individual letters or words, but
assimilates the meaning of whole sentences. Similarly, a practiced chemist does not see the lines or electron dots, but
grasps the molecular shape, with its intricate bonding arrangements and regions of high and low electron density, as well
as their effects on molecular properties or reactivity.
Let us begin with the basic steps:
1. Sum the valence electrons for all atoms. Based on the molecular formula, add together the number of valence
electrons, using the periodic table as a guide. If the species is an ion, adjust the result by adding one electron for each
negative charge, and subtract one for each positive charge.
2. Establish the connectivity of the atoms. Draw a single line to represent a bond between all directly bonded
atoms. Connectivity information may be provided by the name of the compound; a specific name always corresponds to a
defined atom connectivity. Connectivity may also be suggested by the way the formula is written (atoms are often listed in
the order they are connected). Frequently, the central atom is listed first, followed by the other atoms attached to it. If no
information is provided, decide which atom is the central atom; this usually should be the least electronegative element.
Distribute the atoms and bonds evenly in the plane of the drawing in order to provide clarity to the drawing. The number
of single bonds drawn to an atom must not exceed one for H, four for any atom of the second row, and six for any other
atom.
3. Distribute the remaining electrons. Take the total number of valence electrons and subtract the number of
electrons used to form single bonds (2 electrons per bond). Place the remaining electrons pairwise as lone pairs on nonhydrogen atoms, starting with the most electronegative atoms, completing an octet on each. If there are too few electrons
to complete octets, go on to step 4. If there are too many electrons after octets have been completed, go on to step 5.
4. Make multiple bonds if needed. If the central atom ends up with less than octet, use lone pairs on atoms
bonded to it to form double or triple bonds.
5. Place any leftover electrons on the central atom. If there are any valence electrons not assigned to specific
atoms affix them to the central atom even if it results in exceeding the octet on that atom
We can illustrate this process by drawing the Lewis structure of formaldehyde (methanal), CH2O.
1. Valence electrons
NV (C) = 4
NV (O) = 6
NV (H) = 2 × 1
Total = 12
2. Connectivity
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The valence electrons of all atoms are added together. There
are 12 valence electrons to be apportioned to carbon and
oxygen. Hydrogens form one bond each and do not accept any
additional electrons.
The name unambiguously defines the species. If it was not
given, the atom order suggests that carbon is the central atom,
which is consistent with it being the least electronegative atom
(hydrogens cannot serve as central atoms, they may only form
one bond).
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3. Distribute electrons
Six electrons were used to make three single bonds. The
remaining 6 are allocated, pairwise, to the oxygen atom.
Carbon does not have an octet.
4. Multiple bonds
Since all valence electrons have been apportioned, one of the
oxygen's lone pairs has to be reassigned to be shared with
carbon, forming a double bond. Both carbon and oxygen have
octet, and both hydrogens have two valence electrons. The
Lewis structure satisfactorily accounts for bonding in
formaldehyde.
5. Extra electrons
This step is not needed since we were able to draw a Lewis
structure with complete octets using all of the valence
electrons.
The Lewis structures that best describe electron distribution in molecules have
the smallest formal charges possible
Often, some additional insight can be gained about the structure by assigning formal charges. Formal charges are
essential when identifying the charge-bearing atoms in molecules or in polyatomic ions (see below). Also, if multiple
Lewis structures can be drawn with the same atom connectivity but different electron apportionment to individual atoms,
the structure that best fits the actual molecule can be selected by choosing the structure with the smallest formal charges.
Formal charges (FC) are assigned following a simple "electron-bookkeeping" procedure. The number of electrons on
a given atom in the Lewis structure, NLS, is the sum of all unshared electrons on that atom plus half of the electrons
shared with bonding partners. The formal charge on each atom is then calculated by subtracting NLS from the number of
valence electrons that atom has based on its electronic configuration (NV).
E07-3-1
F C = NV − NLS
Let's practice the whole procedure on thiocyanate ion, NCS−.
1. Valence electrons
NV (N) = 5
NV (C) = 4
NV (S) = 6
−1 charge = 1
Total = 16
The valence electrons of all atoms are added together. The
negative charge on the anion indicates the presence of one
extra electron, which is added to the total.
2. Connectivity
The name unambiguously defines the species. If it was not
given, the atom order suggests that carbon is the central atom,
which is consistent with it being the least electronegative atom.
If no information were given, other options to consider would be
C−N−S, C−S−N, or a cyclic structure.
3. Distribute electrons
Four electrons were used to make two single bonds,
represented by lines. The remaining 12 electrons are
distributed pairwise, first by giving three pairs to nitrogen, then
by giving three pairs to sulfur. Electrons should be distributed to
the most electronegative atoms first.
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4. Multiple bonds
Since all electrons have been apportioned, and the central atom does not yet have an
octet, multiple bonds need to be formed. There are three options to form these bonds:
using both lone pairs on nitrogen, using both lone pairs on sulfur, or using one lone pair
from each. In each structure all atoms have a complete octet. We indicate that the
species is an ion by enclosing the structure in brackets and specifying its charge as a
superscript outside of the bracket.
5. Formal charges
Since we have more than one possible structure, we must calculate the formal charge
on each atom to decide which is the "best". For each structure NLS is subtracted from
NV, to yield the formal charge on each atom. The individual formal charges must always
add up to the total charge on the species (−1 in this case), and are written above the
corresponding atom. The formal charges are often shown as part of the Lewis
structure; zero formal charges may be omitted, as can "1"s following the "+" or "−"
signs. If formal charges are assigned to atoms, it is not necessary to draw the bracket
with the overall charge.
All three Lewis structures drawn in our exercise describe the thiocyanate anion. In all structures N, C, and S have
octets; they differ only in the way electrons are apportioned to the atoms. We say that the structures contribute to a
description of the true electron distribution in this ion. Using formal charges, we can decide which of the structures
contributes the most, i.e., which is the dominant ("best") structure. The following guidelines are used in selecting the
dominant structure:
the dominant structure has the smallest charge separation, preferably with zero formal charges or the smallest
possible formal charges
the positioning of the formal charges in the dominant structure is consistent with the relative electronegativities of
atoms; the dominant structure should have negative formal charges on the most electronegative atoms, and positive
charges on the least electronegative atoms
Applying these guidelines, we can easily determine that the middle structure in the example above is going to
contribute less than the other two, as it has the largest charge separation (−2 and +1). The remaining structures have the
smallest charges (−1) possible for the anion. We can select the dominant structure as being the one with −1 formal charge
on nitrogen (above, on the right), as nitrogen is more electronegative than sulfur.
We have arrived at a peculiar junction: we have gone from only being able to draw Lewis structures for the simplest
molecules, to being forced to select the dominant structure from an abundance of possibilities. Why do we have multiple
Lewis structures? Are there different thiocyanate ions? We will address these questions in the next section.
It is important to realize that formal charges are only an accounting tool: they do not represent true charges (see 072), but they give us some guidance on how charges are distributed among atoms in molecules or polyatomic ions. We
have just explored (see above) how to use formal charges to determine the relative contributions of various Lewis
structures to the actual structure, but even if only one Lewis structure can be drawn, we often show formal charges on
specific atoms. For example, we can use square brackets to show the overall charge on the hydroxide (HO−) or
hydronium ions (H3O+), or we can specifically assign the formal charges to the oxygen atoms in the two ions. In the overall
neutral form of the simple amino acid glycine (below right), there appears to be no need for formal charges, yet when
calculated, they give us some insight into the structure of the molecule, showing its zwitterionic nature.
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We may also note that atoms with full octets, but with non-zero formal charges, do not follow our simple valence
rules; they form fewer bonds than their valence number if negatively charged, or more bonds if positively charged. These
changes reflect converting bonding pairs into lone pairs, or vice versa. We will later return to these issues to refine our
arguments and improve our understanding of Lewis structures, but it is important that you practice drawing and reading
Lewis structures until it becomes second nature.
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07-4 Resonance
Molecules with delocalized electrons require more than one Lewis structure
to describe them adequately
We previously mentioned that ozone, O3, is a less stable allotrope of oxygen than O2. If we draw its Lewis structure by
following the procedure outlined in the previous section, we arrive at structure I in step 3, as shown in Figure F07-4-1. This
structure has serious shortcomings: it does not have an octet on the central oxygen, and it has large formal charge
separation. In step 4 of our procedure we add double bonds to rectify these shortcomings. We can form a double bond
with the blue oxygen on the left (II), or with the green oxygen on the right (III). Both II and III are satisfactory Lewis
structures. They are equivalent, i.e., they present the same bonding pattern, but they are not the same as the terminal
oxygens are distinct atoms (in structure II, blue oxygen is doubly-bonded, but it is singly-bonded in structure III).
Figure F07-4-1. Resonance structures for ozone. Structure I does not have an octet on the central oxygen and does not
contribute significantly to the description of the molecule. II and III are equivalent and contribute equally to the molecule.
That is, the real structure of ozone is a superposition (blending) of structures II and III. Formal charges are shown in
orange.
However, there is a big problem with either of these structures: in an ozone molecule both O−O bonds have been
measured to be identical in length, specifically 128 pm (1.28 Å). These bonds are shorter than a typical single O−O bond
(146 pm), but longer than a typical double bond (121 pm). The experimental data tells us that our structures are wrong!
Interestingly, even if both are incorrect, the average, or superposition, of II and III gives a correct description of
ozone. Just as two defective pictures of the same object, each with missing pixels, generate a sharp image when
combined, superposition of two (or more) Lewis structures gives us a more realistic picture of the molecule. Such a
superposition is called resonance, and I, II and III are resonance structures or resonance forms of each other, as marked
by the two-headed arrows. Structure I does not contribute significantly because it has high formal charges and its central
oxygen does not have an octet, but II and III contribute equally.
The term resonance might be misleading, implying oscillations between structures. In reality, there is no oscillation of
any kind. The individual resonance structures do not exist as molecules, and the molecule that exists cannot be
adequately described by any one Lewis structure. To fix the notation problem (the "broken" picture) we have to use
multiple Lewis structures (multiple defective pictures).
To trace the origins of this difficulty in notation, let's look at structure I and deconstruct how II and III are made from
it. We observe that of the lone pairs on terminal oxygen atoms, two pairs on each atom stay put, while the third is shared
(forming the second bond to the central oxygen). Thus we have a pair of electrons on each terminal oxygen atom that
either forms the second bond or plays the role of the lone pair depending on the resonance structure. In the real
molecule, all three oxygen atoms share these electron pairs. So far we have dealt only with localized electron pairs, which
are shared between two directly bonded atoms. Resonance describes pairs of delocalized electrons that are shared by
more than two atoms. In Lewis structures we do not have a way to designate electrons shared by three (or more) atoms,
so we are left with resonance notation. In an effort to show such sharing, one may draw broken-line multiple bonds (IV),
which are meant to show that the O−O bond orders are 1.5 for both bonds (midway between a bond order of 1 and 2).
Such structures are not easily readable, and are not used frequently.
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Figure F07-4-2. Resonance structures for ozone.Structures II and III are combined to give IV, in which the electron
sharing between three atoms is shown with a broken line. Structures such as IV are difficult to read (as they have
fractional bonds, see below) and are not used frequently.
Since more sharing strengthens the bonding, the ability to draw more resonance structures indicates that the
molecule has higher stability. The term "resonance-stabilized,” is commonly used to highlight the presence of delocalized
bonds, despite the fact that resonance is merely a notation issue and not a physical phenomenon.
For molecules represented by equivalent resonance structures the bond orders and formal
charges are averages of the values found in these structures
For an additional example of resonance, let’s consider the nitrate ion NO3−, shown below with one oxygen in green and
one in orange (F07-4-3). In this case we can draw three resonance structures with all atoms satisfying the octet rule. The
structures are equivalent, as a rotation around N by 120° converts one into another, reproducing bonding arrangements,
but they are not actually the same; after the rotation the atom colors do not match. The green oxygen atom cannot be
turned into an orange oxygen atom; they are atoms with distinct identities. This distinction is important, because the
result is three equivalent structures that contribute to the description of the real anion. If they were the same, we would not
have resonance!
Figure F07-4-3. Three equivalent resonance structures for a nitrate ion. The average of the three structures (right)
yields N−O bond order of 1⅓ and formal charges of −2/3 on each oxygen.
Let's concentrate on the green oxygen. It forms a single bond (bond order of 1) to nitrogen in each of the first two
structures and a double bond (bond order of 2) in the third structure. Since the real ion is the superposition of equal parts
of all resonance structures, we can expect that O−N bond order to be an average of 1⅓ (which is equal to (1+1+2)/3).
We also expect the bond length to be shorter than a single bond, but longer than a double bond. The same is true for the
other two N−O bonds to the orange and black oxygen atoms. This expectation is borne out: the N−O bonds are all equal
in length at 126 pm, which is in-between the lengths of a single bond (average N−O is 136 pm) and a double bond
(average N=O is 122 pm).
Similarly, the formal charge on the green oxygen is −1 in the first two structures, and 0 on the third, for an average of
−2/3. All of the oxygens repeat the pattern. The outcome of this analysis is that the real nitrate ion is perfectly symmetrical,
with three identical bonds and the formal charge distributed equally among the three oxygens. None of the individual
resonance structures are able to show it, but the superposition of all three does the job, as shown by a unique structure
with fractional bond orders (1⅓) and fractional formal charges (−2/3); this is shown on the right in F07-4-3.
For molecules represented by non-equivalent resonance structures the dominant contributor
is the structure with the lowest formal charges
In many instances, the resonance structures are not equivalent, as we saw in the case of the thiocyanate ion. We can
select the dominant contributor (left) and the second best contributor (middle) in F07-4-4 by using the formal charge
criteria presented previously (F07-4-4) Such estimates of the relative contributions of various structures are qualitative.
We are not able to assign specific weights to individual structures, or to calculated bond orders or partial charges.
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Figure F07-4-4. Resonance structures of a thiocyanate anion.The structure on the left contributes the most to the
description of the ion because it has the smallest formal charge (−1) on the most electronegative atom. The structure on
the right contributes the least because its formal charges deviates the most from 0. The relative contributions of each
structure cannot be quantified.
Cyclic delocalization of bonding electrons may lead to increased stability
Resonance is very important in describing many organic molecules with delocalized bonding. We will demonstrate just
one case here, and present a few more later on in the course. The well-known molecule benzene is a cyclic hydrocarbon
with apparent alternating single and double bonds. It is often drawn as a skeletal structure where each vertex (corner)
represents a carbon atom and the hydrogen atoms are not shown explicitly; each carbon is assumed to be bonded to a
sufficient number of hydrogen atoms to make a total of four bonds. The two equivalent resonance structures shown in
F07-4-5 signal that all C−C bonds should be equal in length and strength. They are indeed equal, as the bond length of
140 pm is intermediate between a C−C single bond of 154 pm and a C=C double bond of 134 pm. This is consistent with
a bond order of 1.5. This cyclic resonance is often shown in a shorthand notation with a circle in the middle of a hexagon.
This replaces two resonance structures with one, and also stresses that the cyclic delocalization of bonding electrons is
providing special stability, even beyond that of normal non-cyclic resonance. Indeed, benzene rings are found in many
compounds in nature and in commercial products because of this dramatically increased stability.
Figure F07-4-5. Resonance in benzene. For organic molecules, skeletal Lewis structures are used, in which each vertex
represents a carbon atom. The hydrogens are assumed to be bonded to the carbons according to octet rule and are not
drawn. All C−C bonds in benzene are equal (with a bond order of 1.5), and all carbon atoms share 6 of the bonding
electrons equally. The resulting delocalized electrons are represented by the circle, as shown in the structure on the right.
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07-5 Octet rule exceptions
So far we have relied on the octet rule so much in our discussion that it feels unseemly to consider cases where its
simplicity and usefulness are violated. Yet there are many instances where we need to extend our horizons beyond
counting to eight. Broadly speaking, there are three categories of exceptions:
1. molecules or polyatomic ions with an odd number of electrons
2. molecules or polyatomic ions with atoms that have fewer than an octet of valence electrons
3. molecules or polyatomic ions in which atoms have more than an octet of valence electrons
Radicals, molecules with unpaired electrons are generally very reactive
The first group is the least numerous. The members are often called free radicals, or just radicals. By their nature they
cannot have all their electrons paired, and cannot reach an octet. The presence of an unpaired electron makes them very
reactive. As we will see later, the pairing of electrons is part of the driving force leading to bond formation. Radicals often
exist only as short-lived intermediates in chemical reactions, or in the gas phase under very low pressure, as long as they
are separated from other radicals or molecules with which they would otherwise react. Stable species with an odd number
of electrons are quite rare.
Molecules with atoms that have an insufficient number of valence electrons
tend to combine with other molecules to increase the number of shared electrons
The second group of exceptions occurs if there are fewer than eight electrons around an atom in the molecule. Such
exceptions are also rare. For some molecules there are simply not enough valence electrons to formulate an octet. For
example, beryllium hydride (BeH2) has only 4 valence electrons, and borane (BH3) has only 6. These molecules may exist
in the gas phase under low pressure, where they cannot combine with other molecules to supplement their contingent of
valence electrons. Such electron deficient molecules or ions are also encountered as fleeting reaction intermediates on
their way to rapidly quench their thirst for electrons by sharing electrons with other molecules.
One interesting example is boron trifluoride, BF3, a chemical cousin of BH3. Boron can borrow electrons from
fluorine in the Lewis structure of BF3, forming a double bond (F07-5-1). Following step 5 of our Lewis structure procedure,
we end up with three resonance structures. How do we decide which resonance structure is the most important and
contributes the most? On the one hand, we have a resonance structure with only three single bonds and no octet on
boron, but with zero formal charges. On the other hand, we have three equivalent resonance structures that contain a
double bond; each has an octet on boron, a formal positive charge on fluorine, the most electronegative atom, and a
formal negative charge on boron, an atom with low electronegativity. Here, experimental results come to the rescue, and
we find that the B−F bond length in BF3 is 131 pm, which is shorter than a standard B–F single bond (137 pm). This
suggests that resonance structures with double bonds must contribute significantly. However, BF3 reacts with compounds
that have lone pairs (such as ammonia) to form stable complexes, indicating that boron's electron deficiency is not
sufficiently quenched by sharing electrons with fluorine, and implying that the non-octet structure is also important. We are
not able to declare a winner in the best structure competition, but at least we know that the octet formation is going to be
dominant in all other cases. Even the most electronegative atom in the periodic table can give up some electron density to
help a non-octet atom.
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Figure F07-5-1. Resonance structures in BF3. In the first structure (left), boron has no octet, but the formal charges on all
atoms are zero. In the three equivalent structure (right), boron has a full octet, but the formal charges on fluorine and
boron are against the electronegativity trends. BF3 reacts with ammonia, forming a stable complex with full octets on all
non-hydrogen atoms.
Atoms of the 3rd period and below can expand their valence to more than eight electrons
by bonding to small electronegative atoms
The third group of exceptions to the octet rule is by far the largest. There are numerous molecules and polyatomic ions
that have atoms with more than eight electrons around them. Such species are called hypervalent, as they exceed the
valence rules that we explored before. The central atoms of hypervalent molecules or ions must satisfy several conditions.
They have to be from the 3rd period of the periodic table or below, and be large. In most cases, they have to form bonds
with small, electronegative atoms (such as F, Cl, or O). Second row atoms do not form hypervalent compounds or ions.
Some examples of hypervalent species are shown in Figure F07-5-2.
Figure F07-5-2. Examples of hypervalent molecules and ions: phosphorus pentachloride, PCl5, sulfur hexafluoride SF6,
hexafluoroantimonate anion, SbF6−, thionyl tetrafluoride, SOF4, and xenon tetrafluoride, XeF4. In each case, the central
atom exceeds octet.
The main reasons for the restrictions listed above are the relative sizes of the atoms involved. Second row atoms
are just too small to accommodate more than four bonded neighbors. Bigger atoms (from the 3rd row or below) have room
for up to six neighbors around them, especially if the neighbors are relatively small. For example, nitrogen is too small to
form bonds to five fluorines, but phosphorus can bond to five fluorines in PF5 or even five chlorines in PCl5. Bromine and
iodine are too large to fit comfortably around a central phosphorus atom; PBr5 is very unstable, and PI5 does not exist.
Hypervalent resonance forms with minimized formal charges contribute
to a more precise presentation of electronic structure
For atoms in the 3rd row (or below) we may encounter difficulties when selecting the most important Lewis structure, which
is the one that contributes the most to the description of the actual molecule or ion. Since the octet of electrons is no
longer a formal limit, we may have some controversial decisions to make, a bit similar to the BF3 case presented above.
Specifically, our decision may be between a structure that has an octet on the central atom, but also has many formal
charges, and a structure with a hypervalent central atom, but diminished formal charges. Consider the phosphate trianion
(F07-5-3), for which we can write a Lewis structure with an octet on the phosphorus atom with the overall charge of the
species shown outside of the bracket, or with formal charges assigned to individual atoms (+1 on P and −1 on each
oxygen). Since the phosphorus atom can form hypervalent compounds (F07-5-3), we can also write four equivalent
resonance structures with −1 charges appearing on three oxygen atoms, i.e., the minimum possible for a trianion.
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Figure F07-5-3. Resonance structures for a hypervalent atom. For the phosphate ion, PO43−, one can write a resonance
structure with an octet on phosphorus and multiple formal charges (left), or four equivalent resonance structures (right) in
which phosphorus is hypervalent (5 bonds = 10 electrons), but with fewer formal charges.
Experimental measurements show that the negative charge is distributed equally in the species; all oxygen atoms
are equivalent. Also, all P-O bonds are of equal length. Both structural notations, the octet structure (on the left in F07-54), and the hypervalent resonance structures (on the right in F07-5-3) match these experimental observations. However,
the P-O bond length in phosphate trianion (153 pm) is notably shorter than an average single P-O bond (165 pm),
suggesting that resonance structures with the double bond contribute significantly to the electronic description of the ion.
The double-bonds that are formally drawn for hypervalent atoms are distinct from the double bonds formed by octetsatisfying atoms. Their constructions would require participation of d orbitals, but as mentioned above, these orbitals do
not participate in bonding to any significant extent in the compounds under consideration. To account for the increased
bonding (shorter, stronger bonds) a different theoretical model is applied (molecular orbital theory), which we will cover in
the second semester of general chemistry. For now, for simplicity, we can rationalize it as a strong electrostatic attraction
between the central atom (with a positive formal charge) and the negatively-charged atoms bonded to it.
In our course, if a Lewis structure can be drawn where there are complete octets on all non-hydrogen atoms, it is not
necessary to draw additional resonance structures, even if structures with lower formal charges result. The Lewis
structure with complete octets can be used to determine the three-dimensional shape of the molecule and most of its
properties of interest, such as polarity and applicable intermolecular forces, topics which we will discuss in future lessons
(see Chapter 8 and Chapter 11). However, chemists routinely draw double bonds from hypervalent atoms, including
those in resonance structures. Such structures should remind us about increased bond strengths (and shorter lengths),
and in the case of resonance structure, about increased stability of the whole molecule. In Figure F07-5-4, we show some
examples of such notation for a couple of hypervalent ions.
and 4 additional (equivalent) resonance
structures
and 1 additional (equivalent) resonance
structure
Figure F07-5-4. Examples of Lewis structures for hypervalent atoms with and without double bonds. The sulfate ion
(SO42−) is drawn with an octet on sulfur (top left) and in hypervalent representation (top right) with two double bonds and a
zero formal charge on sulfur. The S-O bond in sulfate is 149 pm in length (an average S-O single bond is 170 pm, while
an average S=O double bond is 143 pm). The chlorate ion (ClO3−) is shown with an octet on chlorine (bottom left) and
with a hypervalent chlorine atom (bottom right) that has double bonds, but a zero formal charge. The Cl-O bond in
chlorate is 149 pm (an average Cl-O single bond is 171 pm).
In summary, if multiple Lewis structures can be drawn with the same atom connectivity, but different electron
apportionment to individual atoms, our rules for evaluating the relative contribution of these resonance forms to the
description of the actual electronic structure of the molecule are as follows:
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For 2nd period atoms, structures where all non-hydrogen atoms have an octet contribute more than structures where
atoms have incomplete octets. Hypervalent structures with more than eight electrons around period 2 atoms are not
possible. If all atoms have octets, then the structure with lowest formal charges is most dominant. Among structures
with the same formal charges, the one with formal charges that match the electronegativity trends is a more
important contributor.
For 3rd period atoms (and below), hypervalent structures with more than eight valence electrons per atom are
possible. We will consider non-hypervalent structures to be sufficient to describe most properties of a species, even if
a hypervalent structure with smaller formal charges can be drawn. However, such hypervalent constructs add to our
understanding of the structures, bond lengths, and properties of these species.
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08 Molecules in 3D
Lewis structures display the connectivity of atoms and apportionment of all valence electrons into bonding and lone
pairs. However, they do not explicitly depict one of the most important aspects of molecules: the 3-dimensional disposition
of their bonded atoms. Information about the shapes and sizes of molecules, or about the overall polarity resulting from
uneven distributions of electron density, is central to understanding the properties of the molecules and substances
around us. Though this information is not explicitly displayed in Lewis structures, we can extract it from them, using what
we know about geometrical and electrostatic patterns in molecules.
08-1 VSEPR
Both bonding and nonbonding electron pairs on an atom repel each other electrostatically. The energy of a molecule is
lowered if such repulsions are minimized for all atoms in the molecule. This can be achieved by placing any atom’s electron pairs
as far as possible from each other in three dimensions. The geometrical model we use to understand this placement is called
Valence Shell Electron Pair Repulsion or VSEPR. All electron pairs around a given central atom are divided into bonding and
nonbonding electron domains. The number of bonding domains is equal to the number of atoms bonded to the central atom. If
bonding involves a double or a triple bond, all shared electrons in that bond count as one domain. Nonbonding pairs (lone pair
electrons), if any exist, constitute additional electron domains. The total number of domains is called the steric number of an atom.
The steric number determines the canonical shape of the molecule, which is called the electron-domain geometry. Possible
electron-domain geometries include linear (for a steric number of 2), trigonal planar (for 3), tetrahedral (for 4), trigonal bipyramidal
(for 5), and octahedral (for 6).
08-2 Canonical geometries
Molecular geometry is defined by the distances between nuclei (bond lengths) and the angles between their inter-nuclear
axes. Lone pairs, though critical to determining geometry, do not have well-defined axes, and therefore, angles to them cannot be
explicitly specified. Therefore, molecular geometry only describes the positions of atoms and bonds, not those of the lone pairs.
For each of the canonical molecular shapes determined by electron-domain geometry there are a number of molecular geometries
available, each pre-determined by the number of nonbonding domains present and each described by its own AXE formula. In
AXE notation the number of bonded partners, X, and nonbonding domains, E, are listed as subscripts.
08-3 Molecular shapes
We can further refine our understanding of molecular geometry by acknowledging that the mutual repulsions of the domains
are not equal. Lone pair domains repel most strongly, multiple-bond domains are next, and single-bond domains repel the
least.This gradation in repulsive interactions allows us to make qualitative predictions about how bond angles deviate from their
canonical values.
08-4 Molecular polarity
Maps of electrostatic potential. The 3-dimensional structure of molecules can be used to predict molecular polarity. The
presence of polar bonds can be deduced from electronegativity differences between the bonded atoms. Utilizing the structuredependent spatial orientation of bonds, the bond dipole vectors can then be added together to decide whether or not the molecule
has a net dipole moment, i.e. whether it is polar, and to qualitatively predict its magnitude and direction.
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08-1 VSEPR
Electron domains are distributed around central atoms to minimize electron-pair repulsion
The underlying principle governing the formation of covalent bonds is the sharing of valence electron pairs. While an
atom may form bonds to several bonding partners, some electrons may remain unshared (which we call lone pairs), or
even, on rare occasions, unpaired (which happens in radicals). As we have learned, Lewis structures provide a
convenient accounting technique to keep track of how valence electrons are apportioned to bonding partners, telling us
how many partners each atom has, how many electron pairs it shares with each of them to form single, double, or triple
bonds, and how many lone pairs (or single electrons) it has. However, with the exception of trivial cases such as diatomic
molecules, Lewis structures do not provide any direct information on molecular shapes. We can easily determine the
number of bonding partners and lone pairs on any atom of interest in a molecule, but the 3-dimensional disposition of
these bonding or lone electron pairs around the atom requires further analysis.
So far, we have drawn Lewis structures with all the atoms in one plane, with angles between bonds and disposition
of nonbonding pairs around atoms arbitrarily chosen for clarity of presentation. To fully describe molecular shape, we need
realistic bond lengths and true bond angles. We already know about bond lengths, and how to estimate them from
atomic sizes, or to approximate them with average values. It turns out we can also get surprisingly good values for the
bond angles from simple electrostatic and geometric arguments.
Since electrons repel each other electrostatically, electron pairs on a given atom will arrange themselves in space to
be as far from each other as possible. This mutual evasion minimizes the energy of the molecule by minimizing the
electron repulsion. This model is appropriately called Valence Shell Electron Pair Repulsion (VSEPR).
Let's start with the simplest set of cases: a central atom (A) with no lone electron pairs, singly bonded to an
increasing number of identical other atoms (X). With just one bonding partner (A−X), the situation is trivial: we do not
have any bond angles to analyze, and therefore, we can omit it from our examination. If there are only two bonding
partners, X−A−X, the two bonds arrange themselves pointing in opposite directions, with a 180° angle between them. The
molecule adopts a linear shape (Table T08-1-1). When three bonding partners are present, the three bonds in AX3 can
minimize their repulsion if they point to the vertices (corners) of an equilateral triangle that has the central atom in its
center, resulting in trigonal planar geometry, with 120° angles between bonds. Likewise, the optimal arrangement of the
four bonds in AX4 is tetrahedral, with the central atom in the middle of the tetrahedron and the four bonding partners at the
vertices. All bond angles are 109.5° (it requires some trigonometry to calculate this). With five bonding partners, the five
bonds in AX5 minimize their repulsion by pointing to the corners of a trigonal bipyramid that has the central atom in the
middle of the polyhedron. Here, the bond angles are 120° between equatorial bonds (those laying in the trigonal base
common to both pyramids) and 90° between any pair of equatorial and axial bonds (axial bonds are those pointing toward
the apexes of the pyramids). The axial bonds are 180° apart. Similarly, the six bonds in AX6 form an octahedron with the
central atom in the middle and bond angles of 90° or 180°. There are only a few special cases where a very large central
atom can bond to more than 6 other small atoms (AX7, AX8, or AX9). They are outside the scope of our Lesson, but even
these cases mostly follow the logic of electron-pair repulsion outlined here.
Canonical structures represent the optimal possible spatial arrangement of atoms
bonded to the central atom
The regular polyhedra we have described serve as canonical structures, and set standards to which all further
modifications will refer. They represent arrangements of atoms bonded to a central atom that maximizes both distances
between these atoms and the separation between the bonding electron pairs. Such ideal structures can only exist when
all atoms bonded to the central atom are identical, as shown by examples in T08-1-1. Any perturbations, such as the
presence of lone pairs, multiple bonds, or even atoms of varying electronegativity, will cause small deviations from the
canonical shapes.
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Table T08-1-1. The canonical molecular shapes
Lewis structure
Geometry
Polyhedron
Name
Example
AX 2
linear
BeH 2
AX 3
trigonal
planar
BH 3
AX 4
tetrahedral
CH 4
AX 5
trigonal
bipyramidal
PF 5
AX 6
octahedral
SF 6
We already know that single bonds are actually clouds of shared electrons that occupy internuclear space. Lone
electron pairs similarly occupy certain volumes of space that point in a particular direction (even if there is no bonded atom
at the "end of the cloud"). We call either of these directional electron density volumes electron domains. All electrons
shared by the central atom and any given bonding partner must belong to the same electron domain as they are all found
along the same internuclear axis. Therefore a multiple bond in a molecule constitutes just one domain, as do all single
bonds and lone electron pairs, or any unpaired electrons if present. The number of electron domains around any central
atom is called its steric number.
linear
trigonal planar
tetrahedral
trigonal bipyramidal
octahedral
Figure F08-1-1. The arrangements adopted by two, three, four, five, and six connected balloons (from left to right).
Electron domains are similar to these balloons and avoid each other in space by minimizing electron pair repulsions.
The electron domains protruding from the central atom behave like balloons tied together, trying to share the available
space, and pushing each other away (Figure F08-1-1). This analogy describes the electron pair repulsions that we used
to arrive at our canonical shapes (shown in Table T08-1-1) quite well.
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08-2 Canonical geometries
Electron-domain geometries result in specific molecular geometries
when the presence of lone electron pairs is considered
We can extend the balloon analogy to construct shapes for molecules with lone pairs and multiple bonds. A central
atom’s number of electron domains (its steric number) is read directly from the Lewis structure. The "AXE” notation is
commonly used to systematize this approach. As we already learned in the previous section, the “A” in AXE notation
designates the central atom, while “X” represents the bonding domains and “E” represents the nonbonding pairs (or single
electrons if present). Table T08-2-1 lists the possibilities.
Table T08-2-1. Molecular geometries classified in the AXE notations
Steric number
Zero lone pairs
One lone pair
Two lone pairs Three lone pairs
2
AX2E0
linear
3
AX3E0
trigonal planar
AX2E1
bent
AX4E0
tetrahedral
AX3E1
trigonal pyramidal
AX2E2
bent
AX5E0
trigonal bipyramidal
AX4E1
seesaw
AX3E2
T-shape
AX6E0
octahedral
AX5E1
square pyramidal
AX4E2
square planar
4
5
AX2E3
linear
6
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To determine the molecular structure of a molecule or ion we follow a series of well-defined steps:
1. We start by drawing the Lewis structure for the molecule or ion. We count the number of bonding domains
(including single and multiple bonds), and the number of nonbonding domains (lone pairs) on the central atom. The
number of bonding domains equals the number of atoms bonded to the central atom. In rare cases a single
(unpaired) electron will constitute its own nonbonding domain when we deal with radicals.
2. Next, we write the AXE formula for the central atom, with subscripts indicating the number of domains of each type.
The total number of domains equals the steric number of the atom. If the structure has multiple central atoms, we
analyze each one in turn.
3. The steric number tells us the electron-domain geometry (ED geometry). We have only five options in this category:
linear, trigonal planar, tetrahedral, trigonal bipyramidal, or octahedral; these are shown in Table T08-1-1, and
repeated in the first column of Table T08-2-1.
4. Depending on the number of nonbonding pairs, we then arrive at the molecular geometry. The molecular geometry
only describes the relative position of atoms (i.e., their nuclei) neglecting lone pairs. As you may remember from our
quantum introduction, electrons behave as density clouds that reflect the probability of finding electrons in a given
region of space. Therefore we cannot pinpoint their exact locations (or any bond angles involving them). On the other
hand, the positions of the nuclei are well-defined, and the separation between inter-nuclear axes unambiguously
defines the bond angles between atoms.
5. Finally, we adjust the ideal bond angles from the canonical structures to correct for unequal repulsions between the
various types of electron domains. We will learn how to carry out this final refinement in the next Lesson.
Molecules with a steric number of 3 can have trigonal planar or bent molecular geometries,
whereas a steric number of 4 yields tetrahedral, trigonal pyramidal, or bent molecular
geometries
For now, let's examine these geometries in more detail. A molecule with a steric number of 3 has trigonal planar ED
geometry, giving us two possible molecular geometries: trigonal planar (with no lone pairs), or bent (with one electron pair)
as shown in the second row of T08-2-1. With a steric number of 4, the ED geometry is tetrahedral. We now have three
options, with zero, one, and two electron pairs. The resulting molecular geometries are tetrahedral, trigonal pyramidal,
and bent, respectively. In Figure F08-2-1, we maintain the tetrahedron outline for AX3E1 and AX2E2 to show how the
geometries are related.
⇒
AX4E0
tetrahedral
⇒
AX3E1
trigonal pyramidal
AX2E2
bent
Figure F08-2-1. The geometries of molecules with a steric number of 4. The lone pairs point in the direction of the vacant
vertices.
Molecules with a steric number of 5 can have trigonal bipyramidal, seesaw, T-shaped, or linear
molecular geometries
With a steric number of 5, the electron-domain geometry is trigonal bipyramidal. Depending on the number of lone pairs,
we have four possible molecular shapes in this case: trigonal bipyramidal (no lone pairs), seesaw also called a sawhorse
(one lone pair), T-shaped (two lone pairs), and linear (three lone pairs). The relationship between these shapes is shown
in F08-2-2.
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⇒
⇒
AX5E0
trigonal bipyramidal
AX4E1
seesaw
⇒
AX3E2
T-shaped
AX2E3
linear
Figure F08-2-2. The geometries of molecules with a steric number of 5. The lone pairs point in the direction of the vacant
vertices.
For AX4E1 we have an interesting choice to make: should the lone pair be in the equatorial position, as shown in
Table T08-2-1, or in the axial position (F08-2-3). As we will see in the next section, the nonbonding domains are more
repulsive than bonding domains. The lone pair in the equatorial position has two repulsive interactions with axial bonding
pairs 90° apart, while the lone pair in the axial position has three such interactions with the equatorial bonding pairs. Since
repulsions are much greater when the domains are situated 90° apart rather than when they are 120°, the equatorial lone
pair results in lower energy. By the same logic, both lone pairs in AX3E2 are equatorial (see F08-2-3), which keeps them
120° apart from each other.
Figure F08-2-3. Two possible placements of the lone pair in AX4E1. The lone pair experiences less repulsion in the
equatorial position.
Molecules with a steric number of 6 can have octahedral, square pyramidal or square planar
molecular geometries
With a steric number of 6, the electron-domain geometry becomes octahedral (F08-2-4). The possible molecular
geometries include octahedral (no lone pairs), square pyramidal (one lone pair), and square planar (two lone pairs). Since
all positions in the octahedron are equivalent, the positioning of the first lone pair is immaterial, but the second one is best
placed 180° to the first, to minimize the repulsions between them.
⇒
AX6E0
octahedral
⇒
AX5E1
square pyramidal
AX4E2
square planar
Figure F08-2-4. The geometries of molecules with a steric number of 6. The lone pairs point in the direction of the vacant
vertices.
We have just taken the first step in learning how to translate collections of lines (bonds) and dots (electron pairs) into
reasonably good approximations of molecular shapes. It takes some practice to be able to look at Lewis structures and
visualize how the atoms are positioned in 3 dimensions. So far we have explored how to interpret such structures for
ideal, abstract molecules. Next, we will analyze some real examples, adding more sophistication to our model.
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08-3 Molecular shapes
The repulsion caused by lone pair electron domains is greater than the repulsion caused by
multiple-bond domains, which in turn is greater than that caused by single-bond domains
As we have already alluded, the canonical shapes and bond angles in Table T08-1-1 result only when all of the X atoms
bonded to the central atom are the same. Even minor perturbations in the size or electronegativity of X causes small
deviations from the perfect polyhedra; the bond lengths are no longer all equal and the bond angles depend on the
relative strengths of repulsions between different electrons domains. In general, bonding electron domains are less
repulsive than non-bonding electron domains. The bonding electrons are attracted to the nuclei of both bonding partners,
so they are farther away from the central atom on average than lone-pair electrons, which gain stabilization only through
attraction to the central atom’s nucleus. You can imagine this phenomenon as if the nonbonding electrons pushing their
way closer to the nucleus electrostatically shoved the other domains closer to each other. Therefore, the repulsion caused
by the lone pairs (lp) is greater than the repulsion caused by the bonding pairs (bp). As a result, all else being equal, lp-lp
repulsion is stronger than lp-bp repulsion, and in turn, lp-bp repulsion is stronger than bp-bp repulsion. Similarly, bonding
domains comprised of multiple bonds (which have higher electron density) are somewhat more repulsive than single-bond
domains, although less so than lone pairs. This gradation in the energetic penalties resulting from repulsive interactions
among the various electron domains has qualitative predictive value with regard to bond angles. We illustrate these
repulsive interactions and their influence on the fine details of molecular shapes for different AXE categories below.
Molecules with a steric number of 3 have bond angles close to the canonical angle of 120°
The trigonal planar canonical structure has 120° bond angles, as found in BF3 (Figure F08-3-1). Even if we cannot decide
which resonance structure is dominant or what the bond order is, the three B−F bonds are equivalent and they repel each
other equally, preserving the symmetry of the structure. In carbonyl dichloride (phosgene), the double bond domain repels
the single bond domains more strongly than they repel each other. The O−C−Cl angles open up from the canonical 120°
to 124°, while the Cl−C−Cl bond angle gets smaller. A similar situation is found in ethene (ethylene) regarding its double
bond. In ozone, the lone-pair domain repels both bonding domains more forcefully than they repel each other. This is true
even though the bond order is 1.5 because of resonance, and the bonds have higher electron density than would single
bond domains. Similarly, the repulsion between the double-bond and the single-bond domains in trans-diazene is weaker
than the repulsion that they are both getting from the lone pair domains. As a result, the H−N−N bond angle decreases
significantly from the canonical 120°. This is one of the largest departures from standard bond angles that we will
encounter. Typically, the deviations are just a few degrees, only occasionally exceeding 10°.
trigonal planar
trigonal planar
trigonal planar
bent
bent
AX3E0
AX3E0
AX3E0
AX2E1
AX2E1
boron trifluoride
carbonyl dichloride
ethene
ozone
trans-diazene
Figure F08-3-1. Examples of bond angles in structures with trigonal planar electron-domain geometry. The central atoms
used for analysis are shown in red. For clarity, only lone pairs on the central atoms are shown. The imaginary axes for
lone pairs are added to help relate the structures to their canonical equivalents. In trans-diazene, the trans- prefix
indicates that the hydrogens are on the opposite sides of the double bond (we will discuss such geometrical isomers
later in the course). Notice that for planar geometries the angles around the central atom must add up to 360° (within
rounding error).
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Molecules with a steric number of 4 have bond angles close to the canonical angle of 109.5°
Perfect tetrahedral bond angles of 109.5° are observed only when the four domains are identical, as illustrated by
methane in Figure F08-3-2. If one of the domains is a lone-pair domain, the angles between the bonding domains shrink
noticeably, as seen in ammonia. This angle contraction is not halted even if the bonding domains have higher electron
density because of their increased bond order (due to resonance, as happens in the sulfite ion (SO32−). When there are
two lone-pair domains, as in water molecules, the effect is even more pronounced, as the lone-pair domains strongly repel
not only the bonding domains, but each other as well. Deviations from the canonical structures are even caused by what
might be considered minor perturbations. In methyl chloride (four bonding domains), the electrons in the C−Cl bond are
polarized away from carbon and toward chlorine due to the electronegativity difference. This shift diminishes the
repulsion between the C−Cl and the C−H bonding electrons, which are much closer to the central carbon because of the
reversed electronegativity difference and the short length of the C−H bonds. As the result, Cl−C−H bond angles close up a
bit, while the H−C−H bond angles open up.
tetrahedral
trigonal pyramidal
bent
trigonal pyramidal
tetrahedral
AX4E0
AX3E1
AX2E2
AX3E1
AX4E0
methane
ammonia
water
sulfite ion
methyl chloride
Figure F08-3-2. Examples of bond angles in structures with tetrahedral electron-domain geometry. The central atoms
used for analysis are shown in red. For clarity, only lone pairs on the central atoms are shown. The imaginary axes for
lone pairs are added to help relate the structures to their canonical equivalents.
Molecules with a steric number of 5 have bond angles close to the canonical angles of 90° and
120°
With a steric number of 5, the canonical electron-domain geometry is trigonal bipyramidal. Phosphorus pentafluoride (F083-3) is one example of such a structure. In this geometry we have two groups of angles to analyze: the angles between
the equatorial domains, and the angles between the equatorial and axial domains. As we discussed previously, the
placement of the lone pair in the equatorial position leads to less repulsion and a lower energy of the system. Sulfur
tetrafluoride illustrates such a situation. The equatorial lone pair repels the other equatorial domains (S−F bonds) more
strongly than they repel each other, squeezing them together to form a 102° F−S−F angle. Similarly, the axial S−F bonds
are pushed back away from the lone pair (the standard inter-axial bond angle would have been 180°). When a second
lone pair is present the lowest energy structure is also obtained when it is in the equatorial position, as in chlorine
trifluoride. Again, the axial Cl−F bonds are pushed back away from the lone pairs. When a third lone pair is present, as in
xenon difluoride, it is positioned in the equatorial plane and returns the structure to the higher symmetry. The repulsions of
all equatorial domains are now balanced, resulting in a perfectly linear structure. Thionyl tetrafluoride shows the effect of a
double bond domain, and has a structure very similar to sulfur tetrafluoride; the domain with the strongest repulsion is that
of the equatorial S=O double-bond instead of the lone pair.
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trigonal bipyramidal
seesaw
T-shaped
linear
AX4E1
AX3E2
AX2E3
AX5E0
sulfur tetrafluoride
chlorine trifluoride
xenon difluoride
thionyl tetrafluoride
AX5E0
phosphorus
pentafluoride
trigonal bipyramidal
Figure F08-3-3. Examples of bond angles in structures with trigonal bipyramidal electron-domain geometry. The central
atoms used for analysis are shown in red. For clarity, only lone pairs on the central atoms are shown. The imaginary axes
for lone pairs are added to help relate the structures to their canonical equivalents. Notice that angles between the
equatorial domains must add up to 360° as these domains lie in one plane.
Molecules with a steric number of 6 have bond angles near the canonical angle of 90°
In the canonical octahedral electron-domain geometry all positions are equivalent. Even if we label two separate angles in
SF6 (F08-3-4), there are no equatorial and axial positions that can be distinguished, as any four of the identical atoms
(fluorines in this case) can formally constitute the base of what could be called a "square bipyramid". Replacing one of the
bonding domains with a lone pair breaks that symmetry. As shown in iodine pentafluoride, the lone pair repels the bonding
domains away from itself, squeezing the remaining bond angles to below 90°. A second lone pair should then be placed
180° from the first, to minimize their mutual repulsions. Xenon tetrafluoride illustrates such a situation where the repulsive
action of both lone pairs on the remaining bonding domains balances out.
octahedral
square pyramidal
AX6E0
AX5E1
square
AX4E2
sulfur hexafluoride
iodine pentafluoride
xenon tetrafluoride
Figure F08-3-4. Examples of bond angles in structures with octahedral electron-domain geometry. The central atoms
used for analysis are shown in red. For clarity, only lone pairs on the central atoms are shown. The imaginary axes for
lone pairs are added to help relate the structures to their canonical equivalents.
As we were focusing on the geometry of our examples, and not their makeup, you may not have noticed that we
named two xenon compounds among them. Yes, the noble gas xenon can form compounds! It turns out that the largest
noble gases may form stable hypervalent compounds. Examples are limited to combination with the most electronegative
element, fluorine; XeF2, XeF4, and XeF6 form reasonably stable crystalline substances. There are also a few unstable
xenon oxyfluorides and oxides, and a few very transient compounds of krypton (KrF2) and even argon (HArF). No
compounds of helium or neon have ever been prepared, and radon is not studied due to its radioactivity.
If a molecule has multiple central atoms, they all have to be analyzed in turn. For example, glycine, the simplest
amino acid, has 4 atoms that are bonded to at least two other atoms. The electron-domain and molecular geometries of
these atoms are listed in Table T08-3-1. The predicted values for several bond angles are also included (experimental
values confirm our expectations). However, one issue remains unaddressed: the relative orientation of all "fragments"
around the central atoms. We can identify the geometry around each central atom, but how are these geometries stitched
together in the molecule of glycine? To answer this question we must use even more sophisticated models of electron
repulsion in a process called conformational analysis. Although that topic is beyond the scope of general chemistry, we will
address some of its simplest concepts in later Lessons.
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Table T08-3-1. Structural data for glycine
Glycine
Central
Steric
atom
number
O
4
AX2E2
C
3
AX3E0 trigonal planar
C
4
AX4E0
N
4
AX3E1
AXE
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ED
geometry
Molecular geometry
Bond angles
predicted / observed
bent
α
<109.5°
106°
trigonal planar
β
<120°
117°
tetrahedral
tetrahedral
γ
109.5°
109.5°
tetrahedral
trigonal pyramidal
δ
<109.5°
107°
tetrahedral
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08-4 Molecular polarity
Molecules are polar if they have polar bonds and a geometry such that
their bond dipoles do not cancel each other out
We previously explored the concept of molecular polarity, starting with diatomic molecules where bond dipole moments
gave us information about uneven electron density distributions among bonding partners. Larger molecules can also be
polar overall, if the centers of the positive and the negative charges do not coincide. Two conditions have to be satisfied
for a molecule to show a net dipole moment:
1. The molecule has to have polar bonds, i.e. bonds between atoms of differing electronegativity. The larger the
electronegativity difference, the bigger the magnitude of the resulting bond dipoles.
2. The bond dipoles must not cancel. The bond dipole moments are vectors that have magnitude and direction. The
magnitude of the vectors is dictated by the electronegativity differences between the bonded atoms. The relative
spatial positions of the vectors and their orientation (the direction they point in) is determined by the 3-dimensional
shape of the molecule.
We need to perform vector addition of all bond dipoles to qualitatively establish whether a given molecule has a
dipole moment (and to determine the magnitude and direction of that dipole). Let's look at a couple of examples to
familiarize ourselves with the procedures involved. In a water molecule, we have two polar O−H bonds (χ(O) = 3.44, χ(H)
= 2.20). Even if we do not know the specific magnitudes of the bond dipoles, we know they must be substantial because
of the relatively large electronegativity difference between O and H (Δχ = 1.24). The water molecule is bent, with an
H−O−H bond angle of 104°. In the Figure F08-4-1 we draw the bond dipoles along the O−H bonds on an arbitrary scale,
with arrows pointing toward the more electronegative atom. The two bond-dipole vectors have the same magnitude, but
point in different directions. To add the vectors graphically, we use the "head-to-tail" method; we translate one vector
(green) to the end of the other vector (blue) without changing its direction or magnitude. The vector sum shown in
orange is obtained by connecting the tail of the first vector with the head of the other. The vector sum indicates that there
is a net dipole moment for the whole molecule, showing an overall shift of electron density toward the oxygen. That
polarization may be alternatively illustrated by the MEP of water. The experimentally determined dipole moment of water is
1.85 D.
Figure F08-4-1. The bond dipoles of water molecule are added graphically by the "head-to-tail" method, where the line
starts with a cross at the atom with positive partial charge and ends with an arrow at the atom with a negative partial
charge. In a water molecule, H (white) has positive charge and O (red) has negative charge. Negative charge on oxygen
means that electron density shifts toward it and electrons spend more time near oxygen. The polarization of electron cloud
is apparent in the MEP image (full scale of potential energy).
Figure F08-4-2. The bond dipoles of
carbon dioxide cancel out. The molecule
has no dipole moment, even though it
has polar bonds. The MEP illustrates
the symmetrical charge polarization in
this molecule.
On the other hand, a carbon dioxide molecule also has polar bonds (χ(O) = 3.44, χ(C) = 2.55), but the bond dipole
moments cancel out because the molecule has linear geometry. The bond dipole vectors are equal in magnitude, but point
in opposite directions, and the vector sum is zero. The CO2 molecule is non-polar (μ = 0).
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Symmetrical molecules with identical atoms bonded to the central atom
have no net dipole moments, even if they have polar bonds
Keeping the results for the CO2 molecule in mind, we can intuitively conclude that certain symmetrical molecules where all
atoms bonded to the central atom are identical will have no dipole moments, even if they have polar bonds. This is
obvious for linear and octahedral molecules (AX2E0 and AX6E0) as for each bond dipole there is another of the same
magnitude but pointing in the opposite direction (Figure F08-4-3). The same is true for the linear AX2E3 and square planar
AX4E2. It is less obvious for the trigonal planar or the tetrahedral molecules, but vector addition (F08-4-3) demonstrates
that the bond dipoles cancel out in these cases as well. The trigonal bipyramid is just a sum of the linear (axial bonds) and
trigonal planar geometries (equatorial bonds) which both have cancelling bond-dipole vectors.
Figure F08-4-3. Symmetrical molecules where the bond dipoles cancel and the net dipole of the molecules is zero. In the
first four structures from the left, the canceling of dipole vectors is directly visible. In AX3E0 the vector sum of the two
"upper" bond dipoles is equal but opposite to the dipole moment of the "lower" bond. In AX4E0 the vector sums of the two
bond dipoles on one side of tetrahedron cancel out with those on the other side. AX5E0 is a combination of AX2E0 (axial
bonds) and AX3E0 (equatorial bonds). Click on the image to see the dipole vectors.
Strictly speaking, we should also consider the contributions of lone pair dipoles, which have dipole moment vectors
pointing away from the nucleus along the lone pair axis. However, the nonbonding electrons are non-directional (spacially
quite diffuse) and held closely to the nuclei, so these contributions are usually small and may be neglected in our
qualitative reasoning. For symmetrical molecules (see above) the lone pair dipoles cancel if the bond dipoles do.
Molecular dipole moments are vector sums of bond dipole
Let's consider some instructive examples (F08-4-4). In methyl chloride the C−Cl bond dipole is enhanced by the sum of
three C−H bond dipoles, as carbon is less electronegative than chlorine but more electronegative than hydrogen. The
molecule is polar and the measured dipole is 1.87 D. In ammonia, the N−H bond dipoles (Δχ = 0.84) are augmented a bit
by the lone pair dipole, while in nitrogen trifluoride the bond dipoles are in the opposite direction (Δχ = 0.94), and are
partially canceled by the lone pair dipole. As a result the molecular dipole of ammonia is larger and in opposite direction
than that of NF3. As we noted above, CO2 has no dipole moment, but SO2 does due to its bent geometry. Similarly, CF4,
being perfectly tetrahedral, does not have a dipole moment, but SF4 does due to the presence of the lone pair that
"breaks" molecular symmetry. These pairs of contrasting cases stress the need for careful evaluation of individual
molecular structure; do not be fooled by the seemingly fortuitous similarity of some formulas. The number of lone pairs
and the 3-dimensional structure should be carefully considered before assessing overall polarity.
μ = 2.1 D
μ = 1.7 D
μ = 0.3 D
μ = 2.0 D
μ = 1.7 D
Figure F08-4-4. The bond dipoles and molecular dipoles (μ) of several molecules. The numbers listed are
electronegativity values. Click on an image to explore bond dipoles (orange), molecular dipoles (red) and MEPs (on 100
kJ scale) of the molecules.
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The logic of predicting a dipole (qualitatively) from the electronegativity difference and molecular structures may
occasionally be reversed. Sometimes structures may be distinguished from each other based on the experimental polarity
criterion. For example, three isomeric dichloroethylenes are shown in Figure F08-4-5. Their dipole moments have been
experimentally measured as 0 D, 1.47 D, and 1.96 D. Which isomer has no dipole moment? Which is the most polar? A
simple examination of bond dipoles clearly shows that I has no dipole, as its bond dipoles cancel exactly. The other two
require some simple head-to-tail vector addition before we can conclude that III is the most polar. The bond angles and
the bond-dipole vector angles decide the outcome, since all contributing bond dipoles (C−Cl and C−H, respectively) are
the same.
Figure F08-4-5. The bond dipoles for C−H and C−Cl bonds in isomeric dichloroethylenes. The C−H (smaller arrows) and
C−Cl bond (larger arrows) dipole vectors are added "head-to tail" in each molecule. In I, the bond dipoles cancel out
and in II and III, the sum of the vectors results in a green vector (II) and an orange vector (III). The side-by-side
comparison of green and orange vectors clearly shows that III has the largest vector sum.
Now we can see the true nature of Lewis structures. They are not just a bunch of lines and dots, they are highly
sophisticated ideograms. They can communicate extensive information on molecular shape and polarity if supplemented
with electronegativity data. Even when drawn with awkward geometries and bond angles, experienced viewers can
visualize the 3-dimensional molecular structures in exquisite detail. We will need all of this information when we start
exploring molecular interactions in future Lessons.
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09 Valence Bond Theory
At the beginning of our course we studied quantum wavefunctions and the resulting atomic orbitals as
representations of electron density around individual atoms. When atoms combine to form molecules these orbitals
interact to form new receptacles for the redistributed valence electrons. Valence bond theory is one of the bonding models
that describe the bond formation process in terms of interpenetrating atomic orbitals. Even if there are other theories of
bonding, such as molecular orbital theory, our approach is to use the simplest model that accounts for the phenomenon of
interest.
09-1 Orbital overlap
When two atoms approach each other their orbitals overlap. The wavefunctions of two electrons, one on each atom,
interpenetrate and add together. This results in a wavefunction with increased amplitude and higher electron density (which is the
square of the wavefunction). The increased electron density along the internuclear axis is called a σ bond. The length of the bond
corresponds to the distance at which the electrostatic energy is minimized. The electrons constituting the bonding pair must have
opposite spins in order to obey the Pauli exclusion principle. The making of bonds by spin-pairing electrons in overlapping orbitals
is the essence of valence bond theory.
09-2 Hybridization
Atoms can be considered to pre-mix their valence orbitals to form hybrid orbitals before bonding. Depending on the number
of orbitals participating in mixing, there are three hybridization types; sp3 forms four hybrids, sp2 forms three hybrids and leaves
one p orbital unhybridized, and sp forms two hybrids and leaves two unchanged p orbitals. Either the atomic or the hybrid orbitals
can overlap head-on producing a σ bond. For atoms following the octet rule, the number of electron domains on the atom
determines the hybridization type.
09-3 π bonds
The p orbitals left out of hybridization may overlap sideways forming π bonds. The π bonds have electron density above and
below the internuclear axis, but not along the axis. They are weaker than σ bonds, because of diminished overlap in the sideways
approach. If an atom participates in formation of a double or triple bond, only one of the bonds is the σ bond; the remaining bonds
are π bonds. The molecules that require resonance structures for adequate representation have a delocalized π system built of
more than two overlapping p orbitals on adjacent atoms. Such a π system holds delocalized π electrons that are shared by more
than two atoms.
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09-1 Orbital overlap
The energy of two interacting hydrogen atoms has a minimum when the atoms are 74 pm apart,
which is the H-H bond length in an H2 molecule
Over the last few Lessons we have learned a lot about molecules. We know how to write Lewis structures showing the
connectivity between atoms and any nonbonding pairs. We can translate these structures into molecular shapes and
reasonably estimate bond lengths and angles. We can analyze bond dipoles and molecular polarity. However, we have
not yet explained how the bonds are made. To address that fundamental question we need to return to the quantum
chemistry concepts that describe electron density distribution in atoms, which we introduced at the beginning of our
course. After all, atoms are our building blocks and we need to figure out how they join together to form bonds.
We begin with construction of the simplest molecule, H2, from two hydrogen atoms. As you recall, each hydrogen
atom has one electron in a 1s orbital in its ground state. Two hydrogen atoms at a large distance from each other do not
interact at all. The electrostatic energy of the system is zero as the electrostatic forces are infinitesimal (Figure F09-1-1).
At closer distances the 2 electrons are attracted by both hydrogen nuclei and the energy of the system drops. The
decrease is small at first, when the two atoms are at the van der Waals distance, and then greater when they are within
bonding distance. At even closer separation, the nuclear-nuclear repulsion becomes dominant, and the energy rapidly
increases as the nuclei get very close to each other.
Figure F09-1-1. The energy of two hydrogen atoms as the function of internuclear separation. At large distances the
atoms do not interact electrostatically and there is no stabilization energy. At closer distances (for example at the van der
Waals distance) the energy gets lower, and it reaches a minimum at the bonding distance of 74 pm. At shorter distances,
the nuclear-nuclear repulsions increase the energy of the system.
The energy of the H2 molecule reaches a minimum at a separation distance of 74 pm; that distance is what we call
the bond length. It corresponds to the lowest energy compromise between the attractive and repulsive interactions, and
the energy lowering at this point equals the negative of the bond dissociation energy (BDE).
The σ bond in a H2 molecule is formed by the overlap of the two 1s orbitals of the hydrogen
atoms
As the two hydrogen atoms get closer their 1s orbitals overlap and interpenetrate, creating a space for the shared
electrons. The overlap of orbitals is equivalent to constructive interference of the two 1s wavefunctions. When two
water-wave crests interfere a bigger wave is created. Similarly, when two wavefunctions with the same algebraic sign
overlap they add together, creating a new wave with a larger amplitude in the internuclear space (F09-1-2).
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Figure F09-1-2. Constructive interference (addition —) of two 1s wavefunctions (----). The amplitude of the wave increases
in the space between the nuclei. The new volume with increased electron density is called the σ bond. It is shown as a
90% boundary surface.
Since the electron density is the square of the wavefunction, the probability of finding the electron in the space
along the internuclear axis also increases. As you may remember, that space between nuclei was identified as the
"bonding region" in our electrostatic analysis done previously. Thus, the overlap of two atomic orbitals generates a new
space where the electron density is shared effectively. That new electron "cloud" (the new orbital) is called the σ bond. As
we have done previously for atomic orbitals, we represent it with a boundary surface encompassing 90% of bonding
electron density. A characteristic of a σ bond is that it is cylindrically symmetric about the internuclear axis. Note that it
encloses both bonded nuclei.
The σ bond, like any orbital, can hold up to two electrons, but the two electrons must have opposite spins due to the
Pauli exclusion principle. The spin-pairing requirement is responsible for the fact that all bonding and nonbonding
electrons that we have considered in Lewis structures are always present as pairs. The radicals are a rare exception,
and as we have alluded previously, the driving force to pair the odd electrons is responsible for the high reactivity of such
species.
In Valence Bond theory σ bonds form by the head-on overlapping of orbitals
The overlap of atomic orbitals is not limited to 1s orbitals. We can easily extend that paradigm to the overlap of any
valence orbitals, for example a 1s orbital on hydrogen overlaps with a 2p atomic orbital on fluorine in the formation of a
σ bond in an H−F molecule (Figure F09-1-3). In a Cl2 molecule two 3p orbitals overlap to form a Cl−Cl σ bond. In each
case, the single valence electron from each atom participating in bonding is spin-paired in the bond. Notice that the
overlap of orbital lobes of the same algebraic sign (in-phase) is needed for constructive interference of wavefunctions.
Figure F09-1-3. Overlap of 1s and 2p orbitals in an HF molecule, and two 3p orbitals in Cl2. The overlapping orbitals have
matching phases (algebraic signs of their wavefunctions).
The bonding theory we have just described is called Valence Bond theory (VB). Within its framework, bonds form by
sharing spin-paired electrons between bonding partners. That sharing manifests itself in increased electron density along
the internuclear axis in the space generated by the overlap of atomic orbitals, and is called a σ bond. The shared
electrons are localized between the bonded atoms. We have previously described single bonds with electron domains in
the VSEPR model, or lines connecting atoms in a Lewis structure; both are representations of the σ bond of VB theory.
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Atomic orbital overlap does not adequately explain bonding in molecules
Unfortunately, we quickly encounter difficulties when analyzing other simple molecules. Consider methane, CH4, with its
tetrahedral shape and bond angles of 109.5°. Carbon has one 2s and three 2p valence orbitals in its ground state
configuration; 2 paired electrons reside in the 2s orbital, and two unpaired electrons reside in the 2p orbitals. (Figure
F09-1-4). Following our VB recipe, one would predict that carbon should form a CH2 molecule by overlap of the two
unpaired electrons in 2p orbitals with two 1s orbitals on the hydrogens. The resulting two C−H bonds would be at 90° to
each other, as that is the angle between any two p orbitals. In such a compound carbon would not have an octet. We
know from experimental measurements that methane consists of a tetrahedral carbon atom σ-bonded to four hydrogen
atoms. To match the true methane structure we need to prepare carbon for bonding by first making four equivalent orbitals
from the four available valence orbitals (one 2s and three 2p). That process is called hybridization.
Figure F09-1-4. The valence electrons of the ground state configuration of carbon and the formation of σ bonds by overlap
of singly occupied 2p orbitals (two loops) with 1s hydrogen orbitals (green circles). The filled 2s orbital is omitted for clarity.
The CH2 molecule is unstable because carbon atom does not have an octet.
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09-2 Hybridization
Hybrid orbitals better suited for bonding are created by the mixing of atomic orbitals
To prepare four orbitals for bonding with four hydrogen atoms, the carbon atom needs four unpaired electrons. That
requires promotion of one of the 2s electrons to an empty p orbital. The excited state that is produced has high energy
(402 kJ/mole above the ground state); it still has three degenerate p orbitals at 90° to each other, along with the lower
energy 2s orbital. These orbitals would not form four equivalent bonds at 109.5°. To make the orbitals equal we need to
mix them all together (combine their wavefunctions mathematically) and form 4 new orbitals. Each new orbital is
constructed from 1/4 of 2s orbital (it has 25% s character) and the rest of it (75%) is mixture of all three 2p orbitals. This
process is called hybridization and the mixed orbitals are called hybrid orbitals.
In a way the process is equivalent to making mixed drinks. Imagine that we have 3 glasses of pineapple juice (the
2p orbitals) and one glass of coconut cream (the 2s orbital), and we have to make four identical mixed drinks ("Coco
Coladas"). First we split the glass of coconut cream among 4 new glasses, 1/4 per glass, and then we mix all the
pineapple juice in a pitcher and fill the remaining room in each of the four new glasses to the rim. Notice that the number
of glasses of mixed drinks (and orbitals as well) must equal the number of glasses of original components before mixing.
Additionally, we report the "strength" of the mixture as the ratio of the two components, 1:3 in our case, or sp3 in the case
of orbitals; this indicates that the hybrid orbital contains 25% of the s orbital (one part in four).
The mixing of one s and three p orbitals creates four sp3 hybrid orbitals that are 109.5° apart,
with large front lobes ready to form bonds or hold lone pairs
Returning to the orbitals, the mixing does not require any additional investment of energy; the resulting sp3 hybrids have
lower energy than the 2p orbitals, but higher energy than the 2s orbital. On average the energy is the same. Each of the
four hybrids then overlaps with a 1s orbital on a hydrogen atom, forming the four equivalent σ bonds of methane. The
energy released by bond formation more than compensates for the initial investment that was needed to form the excited
state (this energy is equivalent to the BDE of the C−H bonds in methane, or 410 kJ/mol for each bond).
Figure F09-2-1. Hybridization scheme for the generation of sp3 hybrids. The promotion of one of the 2s electrons to the
2p orbital generates an excited state with four unpaired electrons, followed by the mixing of orbitals to make four
equivalent hybrid orbitals.
The hybridization affects the energy of the orbitals, and also their shape and orientation. Generally, the mixing of s
and p orbitals can be understood in the context of the interference of their wavefunctions. On the side of the p orbital
where its sign matches the sign of the s orbital, the waves add, giving a "swollen" large lobe, while on the side where the
wave signs are mismatched, the waves subtract, giving a shrunken lobe (F09-2-2). The hybrids will be able to use their
large front lobe (large "nose") to better overlap with orbitals on the bonding partners. The small back lobe (small "tail") will
point away from the bonding region, and will often be omitted from the pictures for clarity. The hybrids with larger s
percentages have bigger noses (they are more spherical) and smaller tails.
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Figure F09-2-2. Mixing of s and p orbitals. The lobes of the same sign add, while those of opposite sign subtract, yielding
hybrid orbitals of uneven lobes; the shape depends on the amount of s orbital contribution. The relative sizes of lobes can
be explored in 3D by selecting the %-s character or by clicking on specific hybrids.
The angles between orbitals also change upon mixing. Although it requires some significant trigonometry (which we
will skip), it can be shown that the sp3 hybrids are 109.5° apart. The new hybrids can now make σ bonds with hydrogen
atoms and form methane with the correct shape.
→
=
Figure F09-2-3. The four atomic orbitals (s and three p) mix to form four hybrid orbitals on carbon (top). The four sp3
hybrids point to the corners of a tetrahedron, with angles between them of 109.5o. When superimposed, the tail lobes are
often omitted for clarity (middle right). The hybrids' large lobes overlap with s orbitals of four hydrogen atoms to form four
σ bonds (bottom), each holding two electrons (bottom).
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The procedure laid out above does not describe the actual reaction that makes methane, but only serves as a model on
how the structure of methane can be explained in terms of the valence atomic orbitals that are available on carbon and
hydrogen atoms. The geometry of methane is an observable fact; the hybridization procedure is a conceptual model
accounting for that geometry.
The sp3 hybridization scheme applies to all atoms with 4 electron domains that obey the octet rule. Any such atom
with AX4E0, AX3E1, or AX2E2 molecular geometry is indeed sp3 hybridized, even if it does not form 4 bonds. For example,
in ammonia the nitrogen atom is sp3 hybridized and its lone pair resides in a sp3 hybrid orbital. In water, oxygen is sp3
hybridized, and both of its lone pairs are held in sp3 hybrid orbitals. (Figure F09-2-4)
Figure F09-2-4. Hybridization of nitrogen in ammonia (left) and oxygen in water (right). All lone pairs reside in sp3
hybrid orbitals.
The mixing of one s and two p orbitals creates three sp2 hybrid orbitals that are 120° apart,
leaving one p orbital unhybridized
The BH3 molecule has 3 bonds, 3 electron domains, and its hybridization scheme has to be adjusted accordingly. Since
boron's ground state electronic configuration has only one of its three electrons unpaired (Figure F09-2-5), we again have
to first promote one electron from the 2s orbital to a 2p orbital to produce an excited state with three unpaired electrons. In
this case we need to produce three equivalent hybrids. This can be achieved by mixing the 2s orbital with the two
occupied 2p orbitals, leaving one empty 2p orbital behind. This hybridization produces different hybrids than before, with
1:2 ratio of s to p. These are sp2 hybrids, each containing 33% s orbital character. As expected for orbitals with a larger s
fraction, they will have bigger front lobes and lower energy than the sp3 hybrids. The overlap of three hybrids with three 1s
orbitals on hydrogen atoms generates the three σ bonds of a BH3 molecule.
Figure F09-2-5. Hybridization scheme for the generation of sp2 hybrids. The promotion of one of the 2s electrons to the
2p orbital generates an excited state with three unpaired electrons, followed by the mixing of orbitals to make three
equivalent hybrids.
As before, the hybridization also changes the angle between orbitals. For sp2 hybrids that angle is 120°, matching
the bond angles in the trigonal planar molecular geometry of BH3. The new feature in this hybridization type is the
presence of the unhybridized atomic p orbital that is perpendicular to the plane containing the three sp2 hybrids. In the
BH3 molecule, that p orbital is empty. As we have discussed before, this molecule is unstable, boron has an incomplete
octet, and it will react with other molecules that can provide electrons to fill the vacancy.
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Figure F09-2-6. The three sp2 hybrids of BH3, and the unhybridized atomic p orbital that is perpendicular to the plane
containing the hybrids.
The mixing of one p and one s orbital creates two sp hybrid orbitals that are 180° apart,
leaving two perpendicular p orbitals unhybridized
In a BeH2 molecule, the hybridization is different again, and involves only two orbitals. The ground state Be has two paired
2s electrons (F09-2-6). In the excited state one of them occupies the 2p orbital, and the mixing of 2s and 2p singly
occupied orbitals then leads to two sp hybrids. Two empty 2p orbitals do not participate in the hybridization. The s to p
ratio in the hybrids is 1:1, resulting in even larger front lobes and lower energy hybrid orbitals (the energy is the average of
the energy of the original 2s and 2p orbitals). This is as expected for an orbital with a large s contribution (50%).
Figure F09-2-7. Hybridization scheme for the generation of sp hybrids. The promotion of one of the 2s electrons to the 2p
orbital generates an excited state with two unpaired electrons, followed by mixing of these orbitals to make two equivalent
hybrids. Two unhybridized p orbitals remain empty.
The two sp hybrids are 180° apart, and when they form bonds by overlapping with the two 1s hydrogen orbitals the
resulting molecular geometry is linear. The two unhybridized p orbitals are perpendicular to the axis of the two hybrids,
and to each other (F09-2-8).
Figure F09-2-8. The two sp hybrids and the two empty unhybridized atomic p orbitals, perpendicular to the axis
containing the sp hybrids.
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The number of hybrid orbitals for the three hybridization types matches
the number of electron domains for atoms with steric numbers 2, 3, and 4
The three hybridization types (sp, sp2, and sp3) match the linear, trigonal planar, and tetrahedral electron-domain
geometries of atoms following the octet rule. This immediately tells us how to decide the hybridization type of an atom in
a molecule. It is sufficient to count electron domains around that atom in the Lewis structure and match that number to the
number of hybrid orbitals. The steric number determines the number of hybrids needed to form the σ bonds and hold lone
pairs; sp hybridization provides two hybrids, sp2 provides three hybrids, and sp3 provides four hybrids.
As we have learned, there are also hypervalent atoms with trigonal bipyramidal or octahedral electron pair
geometry. The trigonal bipyramidal ED geometry requires 5 hybrid orbitals, and the octahedral geometry calls for 6 hybrid
orbitals. We know that only atoms from the third row or below form hypervalent compounds. It would be tempting to
include d orbitals to produce the correct number of hybrids, sp3d and sp3d2, respectively. It turns out, however, that d
orbitals do not mix well with the s and p orbitals of the same shell (n = 3 or higher) because they have significantly higher
energy; that is why there are no 3rd row elements with d electrons in the periodic table.
The explanation of bonding in such hypervalent atoms requires a more sophisticated valence bond theory (which is
beyond the scope of our presentation), or a model called molecular orbital theory (MO theory); this is briefly covered in the
second semester of general chemistry. However, this limitation of VB theory illustrates clearly that models in science are
not reality, but only approximations. We use the simplest model that correctly accounts for the structures and properties of
interest, but we must be aware that a more sophisticated theory may need to be used, or even invented, when the theory
does not explain an experimental observation.
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09-3 π bonds
π bonds form by sideways overlap of p orbitals on adjacent atoms
We have just learned how to identify the hybridization of an atom based on the Lewis structure. Consider for example,
ethene (ethylene) in Figure F09-3-1. Both carbons have three electron domains, including two single bond domains and
one double bond domain. The steric number of 3 requires an sp2 hybridization type for both carbons. The only difference
between carbon and the boron atom that we have discussed previously (F09-2-5) is that carbon has four valence
electrons instead of three. That one extra electron resides in the unhybridized p orbital.
Figure F09-3-1. Hybridization of carbons in ethene (ethylene) The two carbon atoms have three electron domains each,
implying sp2 hybridization.
Both sp2 hybridized carbons form two σ bonds to hydrogens by overlapping their sp2 hybrids with a 1s orbital on a
hydrogen atom. The third σ bond is the C−C bond, and it is formed by overlapping two sp2 hybrids from the two carbons.
(Figure F09-3-2). The two unhybridized p orbitals on the carbon atoms are singly occupied, and can overlap sideways to
share their unpaired electrons. This type of overlap generates another type of bond called a π bond.
Figure F09-3-2. Formation of σ and π bonds in ethylene. The p orbitals perpendicular to the σ network overlap sideways,
forming a π bond.
In contrast to σ bonds, the π bond has increased electron density above and below the internuclear axis, but not
along the axis itself. The sideways overlap is not as efficient as head-on overlap, and π bonds are generally weaker than
σ bonds and in fact do not exist alone (without a σ bond)
In a molecule of ethyne (acetylene) each carbon has two electron domains, and therefore both carbons are sp
hybridized (Figure F09-3-3). As in BeH2, the sp hybrids on each carbon form σ bonds to hydrogens, and the other two sp
hybrids form the C−C σ bond. The two unhybridized p orbitals on each carbon hold one electron each (they were empty in
BeH2). As in ethene, the p orbitals can overlap sideways forming π bonds. Since the two sets of p orbitals are 90° to each
other (and to the internuclear axis), the two π bonds that are formed lie in two perpendicular planes.
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Figure F09-3-3. Hybridization of carbons in ethyne (acetylene). The two carbon-hydrogen bonds and the C−C bond are
made using sp hybrids on carbons. The two π bonds are perpendicular to each other and to the C−C axis.
σ bonds formed by head-on overlap of atomic or hybrid orbitals exhibit free rotation, whereas
π bonds formed by sideways overlap of p orbitals cannot rotate around the internuclear axis
We can now generalize our observations about σ and π bonds. Only one of the multiple bonds formed by an atom that
follows the octet rule can be σ; the other bonds formed in a double or triple bond must be π bonds. Although all bonds
look equivalent in a Lewis structure, we now know that that the multiple lines represent different types of bonds.
The σ bonds are made by head-on overlap of various combinations of atomic or hybrid orbitals (F09-3-4), producing
increased electron density along the internuclear axis. The electron density in σ bonds has cylindrical symmetry (any
perpendicular cross-section is a circle), and rotation around σ bonds does not change the overlap. In general, such
rotations are facile, requiring only very small amounts of energy. This so called free-rotation is responsible for the relative
orientation of individual central atoms with all their electron domains, and therefore, for the overall shape of multi-central
atom molecules. The different shapes are called conformations, and their relative energies depend on repulsions
between bonding and nonbonding electron domains on adjacent atoms. The analysis of such interactions is called
conformational analysis and it is outside of the scope of our course.
Figure F09-3-4. Various orbitals overlapping head-on to make σ bonds.
Only the sideways overlap of p orbitals can make π bonds. By contrast, rotation around
double bonds does not occur freely at all; it requires breaking of the π bond, which is
energetically costly. As we have mentioned, a π bond by itself is weaker than a σ bond, but it
still takes a lot of energy to break (above 250 kJ/mol for an average π bond). Double bonds
occur frequently with C, N, and O, but not so with larger atoms; σ bonds to larger atoms are
longer, making sideways overlap ineffective.
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Delocalized electrons in molecules with resonance reside in networks of overlapping π orbitals
As an additional exercise, let's see if we can deduce the hybridization of all atoms in the nitrate ion. As you may
remember, we can draw three equivalent resonance structures for this anion. (F09-3-5). The green oxygen has four
electron domains in the first two structures (I and II), but only three in the third (III), which would require that the green
oxygen is sp3 hybridized in I and II, but sp2 hybridized in III. That accounting is against the resonance rules, however. The
resonance forms may differ in electron apportionment to different atoms, but since they represent one real molecule, any
atom (an oxygen in our case) must have the same electron-domain geometry in all resonance structures. Otherwise, we
would have oscillating structures, which is not experimentally observed. Since all oxygens are equivalent, the only way to
satisfy the resonance requirement is to have all oxygen atoms be sp2 hybridized. You see that requirement manifest itself
in our "average" Lewis structure on the right. A general lesson for us is that hybridization of an atom that holds
electrons participating in resonance has to be read from the resonance form wherein that atom has the smallest steric
number (smallest number of electron domains).
Figure F09-3-5. Resonance structures for the nitrate ion. The average structure with delocalized bonds is marked on the
right.
If all oxygens and the nitrogen atom in the middle of the nitrate ion are sp2 hybridized, then there is an unhybridized
p orbital on each atom of this anion. The p orbitals overlap sideways with each other, resulting in an extended π network
through which π electrons can freely move. That π network explains what delocalized electrons are and how are they
shared between more than two atoms. Any time a molecule needs resonance structures to adequately represent its
electron distribution (i.e. it has delocalized electrons), it has to have delocalized π bonding (a π system built of more than
two p orbitals). Any atom contributing a p orbital to that network must be sp2 or sp hybridized.
Figure F09-3-6. Each atom in the nitrate ion contributes one p orbital to the π system, which accounts for delocalized
electrons. Click on the images for 3D displays.
Similar delocalization takes place through the π system of benzene. All carbons are sp2 hybridized, and each
provides one p orbital and one π electron to the cyclic π system. That cyclic electron delocalization provides extra stability
to the benzene molecule called aromaticity.
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Figure F09-3-7. Delocalized π electrons in benzene. Each carbon atom in benzene is sp2 hybridized and contributes one
p orbital to the cyclic π system. The cyclic delocalization of π electrons is responsible for the extra stability found in
benzene and related molecules. Click on the images for 3D displays.
Delocalized electrons in delocalized π systems are modeled better with molecular orbital theory, which is covered in
the second semester of general chemistry.
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10 Organic Molecules
Compounds of carbon constitute the largest group of molecular structures; millions of such compounds are known,
some appear in nature, while others are synthesized in industry or in laboratories. The vast array of carbon compounds of
various sizes exists due to carbon’s ability to easily form bonds not only to other carbons, but also to any element in the
periodic table. Carbon even forms π bonds to nonmetals in the 2nd and 3rd rows of the periodic table. The versatility of
carbon manifests itself in the fact that living organisms are made up of carbon compounds, and function by performing
chemistry on such compounds. Life itself is organic chemistry.
10-1 Alkanes
Alkanes have a general formula of CnH2n+2. They constitute one class of hydrocarbons, compounds built only from hydrogen
and carbon. All carbon atoms in alkanes are sp3 hybridized and all bonds are single bonds. Because of the free rotation around
the σ bonds, alkanes are flexible and can easily change shape (conformation) in order to minimize repulsions between bonds on
adjacent atoms. Alkanes with more than 3 carbons can exist as multiple variants of the same formula with different atom
connectivity, called constitutional isomers. As the number of carbons in the formula grows, the number of possible isomers
increases rapidly.
10-2 Other hydrocarbons
Unsaturated hydrocarbons have on average fewer hydrogen atoms per carbon atom. They are divided into four groups:
alkenes, alkynes, aromatics, and cycloalkanes. Alkenes have one or more double bonds and sp2 hybridized carbons. When
alkenes contain more than 3 carbons, they can exist as constitutional and stereochemical isomers. The stereochemical isomers,
called in this case geometrical isomers, have the same connectivity but different disposition of atoms around the double bond
marked as trans and cis. Alkynes have one or more triple bonds and sp hybridized carbons; they have four fewer hydrogens per
each triple bond than their saturated analogues. Aromatic hydrocarbons have cyclic π systems built from the p orbitals of sp2
hybridized carbons. The cycloalkanes are examples of unsaturation due to ring formation, which requires the formal removal of
two hydrogens. With the exception of cyclopropane (C3H6), all cycloalkane rings are puckered.
10-3 Functional groups
Functional groups are individual atoms or groups of atoms of a preset structure that are responsible for the characteristic
properties and chemical reactions of a molecule. Members of the same functional group will undergo the same types of reactions
with little influence from the hydrocarbon residue to which they are bonded. The common functional groups include alcohols,
amines, aldehydes, ketones, and acids and their derivatives (esters and amides). The ability to recognize functional groups allows
one to predict the chemical behavior of the compound containing them.
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10-1 Alkanes
Carbon is special because its atoms can bond to each other and to atoms of other elements
to a practically unlimited degree
Compounds of carbon have a special place in chemistry. Carbon has 4 valence electrons and forms four bonds in the vast
majority of its compounds. Because it is small, it forms short strong bonds both to other carbons and to most other
elements, nonmetal and metal alike. It forms double and triple bonds with other carbons and with nonmetals of the second
and the third row of the periodic table. There is no other element with such a rich repertoire of bonding arrangements; it is
not surprising that nature recruited carbon as the element of life. The main constituents of living matter are carbon
compounds of varying complexity, ranging from relatively simple compounds with just a few atoms, such as glycine or
glucose (which we discussed earlier), to large and exquisitely intricate compounds containing millions of atoms, such as
molecules of DNA (deoxyribonucleic acid), RNA (ribonucleic acid), or proteins.
Table T10-1-1. The first eight normal (linear) alkanes.
Formula
Name
CnH2n+2
CH 4
Lewis structure
(condensed structure)
methane
3D structure
(skeletal structures)
No.
of
isomers
1
CH 4
C 2H6
ethane
1
CH 3CH3
C 3H8
propane
1
CH 3CH2CH3
C 4H10
butane
2
CH 3CH2CH2CH3
C 5H12 pentane
3
CH 3CH2CH2CH2CH3
C 6H14
hexane
5
CH 3CH2CH2CH2CH2CH3
C 7H16 heptane
9
CH 3CH2CH2CH2CH2CH2CH3
C 8H18
octane
18
CH 3CH2CH2CH2CH2CH2CH2CH3
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The chemistry of compounds of carbon, called organic chemistry, is a vast sub-discipline of chemistry encompassing
petroleum products, polymers (plastics and fibers), pharmaceuticals, pesticides, paints, and a multitude of advanced
materials. There are millions of organic compounds known, either found in nature or prepared in industry or the laboratory.
In alkanes, the simplest hydrocarbons, carbon and hydrogen atoms are bonded only through
single bonds
Our introduction to organic chemistry starts with hydrocarbons, which are compounds containing only carbon and
hydrogen. They are the simplest, in terms of composition, yet even with just these two elements, there is a rich variety of
possible classes and structures. Hydrocarbons are divided into four subgroups including alkanes (compounds containing
only single bonds with names ending in "ane"), alkenes (compounds with one or more double bonds with names ending
with "ene"), alkynes (compounds with one or more triple bonds with names ending in "yne"), and aromatics (compounds
with cyclic π systems).
Alkanes have a general formula of CnH2n+2, which can be appreciated by looking at the Lewis structures of
compounds with straight chains (normal alkanes) forged from –CH2– units with only the terminal carbons requiring a third
hydrogen to close the octet. Table T10-1-1 lists such linear alkanes containing between 1 and 8 carbons. In addition to
straight chain alkanes, branched structures are also possible (see below) for compounds with more than 3 carbon atoms.
It is often convenient to represent organic molecules in a simplified structure called a skeletal formula (T10-1-1),
representing carbon backbone of a compound. The tetravalent nature of carbon allows us to simplify the Lewis structure
notation by omitting the redundant hydrogen atoms. In such skeletal formulas each vertex represents a carbon atom, and
the number of implied hydrogens on each carbon is such as to assure that it has four bonds. Skeletal formulas are
strongly preferred for their clarity.
Differently branched alkanes of the same composition are constitutional isomers
Branched structures can arise when an alkane has 4 or more carbons in the formula. For C4H10 (butane) it is possible to a
have either a straight chain of four carbon atoms, or a chain of 3 carbon atoms with a fourth carbon bonded to the central
carbon atom (Table 10-1-2). Such structures are referred to as branched hydrocarbons, since new chains of carbons
branch off of the main chain. In both versions of the butane molecule we still need 10 hydrogen atoms to complete the
octets on the carbon atoms, so both will have the same molecular formula but different connectivity between atoms.
Compounds with the same formula but different structures are called isomers. Specifically, isomers with different atom
connectivity are called constitutional isomers. Five-carbon alkanes can exist in 3 isomeric forms, and for C6H14 the
number of possible isomers is 5. The number of possible isomers rapidly increases with the number of carbons in the
formula (Table 10-1-1); for an alkane with 20 carbons the number of possible isomers is over 366,000!
Different isomers have their own systematic names, but some have common names that have been in use for a long
time and are deeply entrenched. For systematic names, the longest carbon chain is counted and this number is used as
the parent root of the name. Substituents (which are shorter hydrocarbon chains that are called alkyl groups) are indicated
by their chain length and position on the main chain, with the counting starting at the end of the chain closest to the
branching point.
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Table T10-1-2. Isomers of butane, pentane, and hexane (common names are in
parentheses)
Straight chain
(normal)
C4H10
butane
Branched chains
2-methylpropane
(isobutane)
C5H12
pentane
2-methylbutane
(isopentane)
2,2dimethylpropane
(neopentane)
C6H14
hexane
2-methylpentane
(isohexane)
3-methylpentane
2,3dimethylbutane
2,2dimethylbutane
(neohexane)
All carbon atoms in alkanes have four electron domains (four bonds) and are tetrahedral, with bond angles very
close to the ideal tetrahedral angle of 109.5°. All carbons are sp3 hybridized and all C–C bonds are formed by overlap of
sp3 hybrids on adjacent carbons, while all C–H bonds are formed by overlapping sp3 hybrids on carbon with the 1s orbital
on hydrogen. The C–C bond lengths are about 154 pm, and the C–H bond lengths are about 110 pm. When drawing
skeletal structures, it is customary to show bonds to any carbon linked to four other carbons as solid and broken wedges,
accurately illustrating their disposition above and below the plane of the drawing (Table T10-1-2).
Free rotation around C‒C bonds affects the overall shape of alkane molecules
Since the σ bonds have cylindrical symmetry, there is "free rotation" around all C–C and C–H bonds. While the latter is
inconsequential (hydrogens have spherical symmetry), the former affects the overall shape of the molecule, reorienting
the individual tetrahedral carbon domains relative to each other in space. Because the energy costs of such rotation are
minimal, the different shapes, called conformations, constantly and rapidly interconvert. In general, multiple conformations
may be present at room temperature. For example, 77% of hexane molecules exist in one of the conformations shown
below (F10-1-1). This point illustrates the flexibility of molecular shapes, as well as the fact that the skeletal structures can
be drawn in multiple ways; some of these drawings may not represent the actual shape of the molecule, but may instead
favor clarity, if the structures are branched.
Figure F10-1-1. Different skeletal drawings of hexane, all representing the same molecule. Usually no conformational
information is implied in such drawings (unless the hydrogens are explicitly drawn with solid and broken wedges).
Conformations that may approximate the drawings can be viewed in 3D by clicking on the skeletal structures. All these
conformations (and all others not pictured) interconvert rapidly at room temperature.
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10-2 Other hydrocarbons
Hydrocarbons other than alkanes have varying degrees of unsaturation
All alkanes have the maximum possible number of hydrogens that a hydrocarbon can have, which is two per each carbon,
plus two extra to cap the ends of the chain. Alkanes are called completely saturated. Other classes of hydrocarbons can
have fewer hydrogens per carbon, and are formally unsaturated. For example, if two "missing" hydrogens are replaced
with a double bond we obtain a hydrocarbon known as an alkene. Alkenes are a class of hydrocarbons containing one or
more C–C double bonds.
Alkenes have at least one C‒C double bond and may have geometrical isomers in addition
to constitutional isomers due to restricted rotation around double bonds
Alkenes containing one double bond have a formula of CnH2n, indicating one degree of unsaturation; i.e., two hydrogens
less than the corresponding alkane. Some simple examples of alkenes are collected in Figure F10-2-1. In ethylene (C2H4)
both carbons have three electron domains and are sp2 hybridized. The un-hybridized p orbitals form a π bond. The bonds
are slightly distorted from the canonical values as predicted by VSEPR model. The C–C double bond length of 134 pm is
shorter than a C–C single bond (154 pm), which is in agreement with the bond order of 2. Similar geometry is found
around all C–C double bonds.
Figure F10-2-1.Simple examples of alkenes. The drawing of ethylene (C2H4) illustrates typical bond angles and bond
lengths. For butene (C4H8), there are 3 constitutional isomers possible, one of which exists as a pair of geometrical
isomers called cis and trans. In the cis isomer the methyl groups marked by the blue arrows are on the same side of the
double bond. In the trans isomer the marked methyl substituents are on opposite sides of the double bond.
When an alkene contains four carbons, constitutional isomers are possible (F10-2-1). The double bond can be at
position 1, as in 1-butene (a terminal alkene) or at position 2 as in two isomeric 2-butenes (internal alkenes). There is also
an isomer with a branched chain, 2-methylpropene. The two isomers of 2-butene are of a different type than the
constitutional isomers. They have the same atom connectivity, but differ in the spatial (3-dimensional) orientation of the
atoms. Such isomers are stereoisomers—specifically, isomers with a differing disposition of substituents on the double
bond are called geometrical isomers. In the trans isomer the two methyl substituents on the "ends" of the double bond are
on the opposite side, while in the cis isomer they are on the same side (F10-2-1). With a larger number of carbons, or
multiple double bonds, the number of possible constitutional and stereochemical isomers both increase rapidly.
Alkynes have at least one C‒C triple bond
Alkynes are the class of hydrocarbons that have one or more C–C triple bonds. Alkynes containing one triple bond have
the formula of CnH2n-2 and are formally referred to as having two degrees of unsaturation (four hydrogens are missing).
Some simple examples of alkynes are shown in Figure F10-2-2. In ethyne (C2H2), both carbons are sp hybridized, and as
expected with two electron domains, the molecule is linear (180° bond angles). The two unhybridized sets of p orbitals
form two π bonds in mutually perpendicular planes. The triple bond so formed (one σ bond and two π) is very short (120
pm). The C–H bonds are also a bit shorter than those in alkanes or alkenes due to improved overlap between the sp
hybrids on carbon and the s orbitals on hydrogens.
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Figure F10-2-2. Simple examples of alkynes. The drawing of ethyne illustrates typical bond angles and bond lengths.
There are two constitutional isomers possible for butyne (C4H6): one has a terminal triple bond (1-butyne) and the other
has an internal triple bond (2-butyne).
For C4H6 there are two constitutional isomers possible (F10-2-2). With a larger number of carbons, or more than one
triple bond, the number of possible constitutional isomers increases.
Aromatic compounds contain benzene rings
Aromatic compounds constitute another group of unsaturated hydrocarbons; benzene is one that we have already
mentioned on several occasions. Its formula of C6H6 indicates that it has four degrees of unsaturation. It would take 8
extra hydrogens to reach the saturated formula of an alkane with the same number of carbons (C6H14). We can easily
conclude that the three double bonds account for the three degrees of unsaturation. The ring itself explains the fourth
degree of unsaturation; in formally joining the ends of hydrocarbon to form a ring, we lose two hydrogen atoms. Some
examples of benzene derivatives are displayed in Figure F10-2-3. In benzene all carbons are sp2 hybridized with trigonal
planar geometry. The 6 unhybridized p orbitals (one per carbon) form a delocalized π network, granting extra stability. All
C–C bonds are of equal length (140 pm), which is shorter than a single bond (154 pm), but longer than the double bonds
(134 pm) in other hydrocarbons.
Figure F10-2-3. Aromatic hydrocarbons: benzene and its derivatives. Benzene illustrates the typical C–C and C–H bond
lengths and angles found in aromatic hydrocarbons.
Cycloalkanes have only single bonds, but are unsaturated because their carbon atoms form
rings
Having realized that cyclic hydrocarbons are also formally unsaturated, we can take a brief look at a few other possibilities
(F10-2-4). Cyclopropane (C3H6), the smallest cyclic hydrocarbon, has apparent C–C bond angles of 60°. The sp3
hybridization of the carbons (containing four single bonds) seems to be inconsistent with such bond angles. A closer
analysis (which is beyond the scope of our course) indicates that in this molecule the C–C σ bonds are "bent", i.e. the
bonding electron density does not lie along the internuclear axis. The bonding is clearly not conventional in this strained
molecule. On the other hand, cyclopentane (C5H10) and cyclohexane (C6H12) have conventional tetrahedral bond angles
about carbon. The carbon rings are not flat (with all carbons in one plane) as skeletal structures would indicate, but have
puckered conformations, called envelope for cyclopentane and chair for cyclohexane.
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Figure F10-2-4. Examples of simple cycloalkanes. Cyclopropane has unusual bent σ bonds between carbons, which
explains the apparent 60° angles. Cyclopentane and cyclohexane have puckered shapes. Click on the pictures for 3D
models.
A summary of hydrocarbons is presented in Table T10-2-1. Click on the name of the example molecule to see and
manipulate a 3-D structure. The examples are limited to six-carbon representatives from each group.
Table T10-2-1. Hydrocarbon categories and six-carbon examples
Hydrocarbon Bonding
Example
Formula
Alkanes
hexane
single bonds
Degree of
unsaturation
C nH2n+2
0
Cycloalkanes ring
cyclohexane
C nH2n
1
Alkenes
double bond
hexene
C nH2n
1
Alkynes
triple bond
hexyne
C nH2n−2
2
Aromatic
conjugated double bonds in a ring benzene
C 6H6
4
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10-3 Functional groups
Functional groups have similar chemical properties
that are only weakly influenced by other parts of the molecule
The hydrogen atoms in hydrocarbons can be substituted with other atoms or groups of atoms called functional groups. A
functional group is composed of a discrete atom or a group of atoms with a set structure, and gives an organic molecule
its distinctive chemical properties. A given functional group chemically behaves the same way in every molecule, and is
only slightly affected by the rest of the molecule (its carbon backbone).
Table T10-3-1. Common functional groups
Functional group
General formula Example
Halides
ethyl bromide
Alcohols
ethanol
Ethers
diethyl ether
Amines
diethylamine
Aldehydes
acetaldehyde
Ketones
acetone
Carboxylic acids
acetic acid
Esters
ethyl acetate
Amides
acetamide
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This consistency in chemical properties allows chemists to organize a massive amount of information into
manageable patterns. Rather than having to deal with millions of individual and potentially different compounds, we need
only to understand and characterize a couple dozens of different functional groups. We will limit our presentation to the
listing of the most basic functional groups and some of their characteristics (Table T10-3-1). In our general formulas, R
stands for the different hydrocarbon "residues" to which the functional group may be attached. Notice that in skeletal
structures hydrogen atoms are always explicitly shown when bonded to non-carbon atoms (so-called heteroatoms)
Functional groups typically contain heteroatoms such as halogen, oxygen or nitrogen atoms
The groups of molecules in Table T10-3-1 are obtained by bonding a functional group to a hydrocarbon residue R. In
halides, a hydrogen atom on the residue is replaced by one of the halogens (F, Cl, Br, or I). Halides are widely
commercially used as solvents, flame retardants and refrigerants, in polymer production, and as pharmaceuticals.
Alcohols are related to water, where a hydrocarbon residue replaces one of the hydrogens. They can serve as solvents,
fuels, or preservatives. Ethanol is "the component" of alcoholic beverages. In an ether, two hydrocarbon residues are
bonded to the same oxygen atom. Ethers are commonly used as solvents, and diethyl ether was one of the first
anesthetics ever used during surgery. Amines consist of derivatives of ammonia, in which one, two, or three hydrogens
are substituted with hydrocarbon residues. They have wide applications in the dye and the pharmaceutical industries and
often have unpleasant smells. Amines are weak bases, a property that will become important when we talk about
reactions (see C10-3-1). In acid-base reactions, the nitrogen lone pair of an amine can accept a proton, to form an
organic ammonium cation.
C10-3-1
The remaining functional groups in Table T10-3-1 contain an important subunit called the carbonyl group, which is a
carbon atom doubly bonded to an oxygen atom (C=O). In an aldehyde the carbonyl group is bonded to a hydrogen on one
side, and an R group on the other. In a ketone, the carbonyl group is bonded to two hydrocarbon residues R which can be
the same or different. The last three groups on our list are carboxylic acids (with a carbonyl group bonded to an O–H
group on one side and an R group on the other), esters (with a carbonyl bonded to an O–R group on one side and an R
group on the other) and amides (with the carbonyl bonded to an NH group on one side and an R group on the other).
Formally speaking, the last two are derivatives of carboxylic acids and alcohols or amines, respectively.
Carbonyl- containing compounds have interesting characteristics. Formaldehyde (which is the smallest aldehyde) is
used in the production of resins; other aldehydes are used in the production of perfumes and flavors. Ketones are very
common solvents, and are precursors for pharmaceuticals. Carboxylic acids are important ingredients in the production of
polymers and pharmaceuticals and other common substances. In fact, a dilute solution of the carboxylic acid acetic acid
(CH3COOH) is commonly known as vinegar. Carboxylic acids are weak acids that partially dissociate in water (C10-4-2).
We will learn more about this reaction in future Lessons.
C10-3-2
Esters are commonly used as solvents, odorants and fragrances in perfumery and in food. The amide functional
group plays an especially important role in biological and polymer chemistry. This functional group is formed when a
carboxylic acid and an amine react, linking the residues connected to them. It is utilized in many polymers; nylon and
kevlar are among the most well known. In biology this functional group is found in proteins and is referred to as a peptide
bond.
The specific chemical properties and reactions of the functional groups are a wide-ranging subject covered in
separate chemistry courses. Our goal here is much more limited: we want to familiarize you with the concept of functional
groups and enable you to recognize the most common ones listed above. To that effect, we provide below three examples
of ubiquitous pharmaceuticals and call your attention to the functional groups present in them (F10-3-1).
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Figure F10-3-1. Familiar over-the-counter medications with their functional groups identified. The hydrocarbon residues
(alkyl or aromatic) are not marked for clarity.
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11 Intermolecular Forces
In earlier Lessons we learned about atoms as the building blocks of chemistry. It was quickly evident that there are
very few substances composed of individual atoms. Most of the time atoms combine to form molecules held together by
chemical bonds. In subsequent Lessons we learned about the rules of molecular bonding and structure. Now that we
understand the 3-dimensional shapes of individual molecules and can describe their polarity, it is time to broaden the
scope of our investigations once more. Except for gases, molecules typically do not exist alone at ambient conditions (300
K and 1 atm), but aggregate to form liquids and solids. The interactions that bring them together are much weaker than
the intramolecular bonds responsible for their structural integrity. Still, these forces play a crucial role in determining the
properties of bulk materials, which are the properties that we observe in our macroscopic world. In this Lesson we will
explore the connection between the molecular structure of a material and the properties it displays for the first time.
11-1 States of matter
The vast majority of substances around us exist in one of the three basic states, gas, liquid, or solid. Molecules may
transition between these states or phases without any changes to their structure. Liquid and solid phases are called condensed
phases, where molecules are held together by various electrostatic intermolecular forces (IMFs). Their mutual attraction is
countered by the kinetic energy of their thermal motions, which is proportional to temperature. As the temperature increases, the
molecules move more rapidly, and may overcome their mutual attractions and transition from solid to liquid once a certain
temperature–called the melting point–is reached. With further increases in temperature all remaining intermolecular forces can be
overcome and the molecules transition from the liquid into the gas phase. When that transition happens throughout the liquid the
boiling point has been reached. The values of the melting and boiling points serve as indicators of the strength of the
intermolecular interactions in solids and liquids, respectively.
11-2 Electrostatic interactions
All substances have some type of intermolecular forces (IMFs) acting between neighboring particles. These IMFs are
electrostatic by nature, and can occur between particles that have permanent charges, have permanent dipole moments, or have
temporary dipole moments induced in them by other particles. The net electrostatic attraction between the particles is the sum of
all these forces. The strength of the net electrostatic attraction is determined by the magnitude of interacting charges (full, partial,
or induced), and is inversely proportional to the distance between the charges. The magnitude of the attractive interaction is
determined by the specific type of IMF, and ranges from 1/d to 1/d6 where d is the distance between the particles.
11-3 Ions and dipoles
The strongest interactions involve permanently charged particles, such as the ion-dipole interactions commonly encountered
when ions dissolve in polar liquids such as water. Dipole-dipole interactions are another type of intermolecular force, and are
common between polar molecules in liquids and solids.
11-4 Dispersion forces
Nonpolar substances also have electrostatic intermolecular forces acting between neighboring particles. In this case the
source of the interactions are temporary dipoles, which may be induced in nonpolar molecules through interaction with an ion, a
dipole, or even another nonpolar molecule in close proximity. These induced dipoles interact with other particles in a manner
similar to permanent dipoles. The generation of an instantaneous dipole depends on the susceptibility of a molecule to have its
electron cloud distorted; this property is called polarizability. Polarizability increases with the total number of electrons present in
the molecule, and also correlates with its size. The intermolecular forces created by such instantaneous dipoles are called London
dispersion forces (LDF), and, together with dipole-dipole interactions, are referred to as van der Waals forces. The total sum of all
intermolecular forces present dictates the properties of substances in an additive manner. For molecules of similar shapes and
sizes, dipole interactions are primarily responsible for any difference in the melting or boiling points. For molecules of varying
sizes, dispersion forces usually dominate over the dipole-dipole interactions in determining the properties of a substance.
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11-5 Hydrogen bonding
Hydrogen bonds are a special type of dipole-dipole interaction; they are stronger than London dispersion forces or other
dipole-dipole interactions. Hydrogen bonds form if there are N–H, O–H, or F–H bonds present in the molecules. These three types
of bonds are highly polar, with a significant partial positive charge on the almost bare hydrogen nucleus. Because of its small size
this hydrogen may approach the lone pairs on atoms of other molecules closely, and interact quite strongly to form a hydrogen
bond. Hydrogen bonds are directional, like covalent bonds (although much weaker). They are responsible for the many unique
properties of water and are very important in biology and biochemistry.
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11-1 States of matter
The balance of intermolecular electrostatic attraction and the thermal energy
of molecular motions determines the state of matter
In previous Lessons we learned a lot about atoms, their electronic structures, and how they bond together to form
molecules. These bonds within molecules, called intramolecular forces, hold the molecule together. We have also
explored molecular shapes and the polarity of individual structures. Now, we are ready to translate our knowledge of
atoms and molecules into an understanding of the directly observable, macroscopic behavior of substances that surround
us. We are entering the second stage of our chemical education: finding connections between microscopic (nanoscopic)
structures and the physical and chemical properties of "bulk" matter.
All particles – ions, molecules and atoms – are susceptible to electrostatic forces. The particles may have
permanent charges, dipole moments, or they may acquire instantaneous dipole moments. When they are in close vicinity,
opposite charges attract each other. These electrostatic interactions between neighboring molecules or atoms are called
intermolecular forces (IMFs); they lower the energy of the system, keeping particles in close contact and aggregating them
into bulk materials.
Even though the particles in bulk materials are in close contact, the particles are in constant motion. They vibrate or
move and collide with each other and with the walls that contain them. In these collisions they transfer and redistribute
their kinetic energy. This incessant movement constitutes the thermal energy of the sample. The average kinetic energy of
all particles in the sample is directly proportional to its absolute temperature. Only at the absolute zero (expressed in
Kelvin, 0 K or –273.15 °C) does all motion and thermal energy vanish. We will develop these ideas more fully in the next
Chapter, when we discuss the kinetic molecular theory of gases.
Intermolecular attractions and the kinetic energy of thermal motions are "antagonistic," in the sense that the first is
keeping the particles in close contact, while the second makes them fly apart. The relative magnitudes of these energies
influence the state (or phase) of the sample.
In gases thermal motions dominate, and in solids the intermolecular forces prevail, whereas
in liquids the thermal motions and intermolecular forces are comparable
We are all familiar with the three basic states of matter, gas, liquid, and solid, illustrated in Figure F11-1-1.
Figure F11-1-1. The three basic states of matter: gas, liquid and solid. The phase transitions are associated with changes
in the thermal energy of the particles.
Each of the phases of matter has its own characteristics. In gases, kinetic energy overpowers the energy of the
intermolecular interactions. The atoms or molecules have barely any contact with each other (except for very brief
collisions). Freed from interactions, the particles move in all directions, going past each other and through any available
small openings. Gases flow readily and rapidly and fully fill any container they occupy. They have no shape or volume of
their own. The gas particles are widely separated. Since the volume occupied by the molecules is small compared to the
total volume of the sample, gases are readily compressible through the application of external pressure.
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In liquids, the kinetic energy of thermal motions is comparable in magnitude to the energy of the intermolecular
interactions. The molecules constantly interact and electrostatically attract each other; they are closely packed in space.
Consequently, liquids are much denser than gases, as the empty space between particles has been largely eliminated.
Application of pressure to a liquid sample does not significantly change the volume, (liquids are incompressible). Yet even
with the restriction of motion caused by the dense packing, the molecules can move past each other (albeit covering much
smaller distances between collisions than gas molecules). As a result, liquids can flow and will assume the shape of their
container, but only the portion that they occupy – they do not expand in volume to fill the container.
In solids, the intermolecular interactions overpower the kinetic energy of the particles. The particles are "glued"
together and held tightly in place by the IMFs. They vibrate in their positions, but move past each other only very slowly
and with great difficulty. Solids have their own shapes and volumes and are essentially incompressible. Often, the
particles in solids are well organized, forming crystalline patterns or lattices to maximize their intermolecular interactions.
Liquids and solids are called condensed phases, as the molecules are tightly packed, with various opportunities for
interaction with multiple neighboring molecules.
Atoms or molecules transition between phases
in response to temperature and pressure changes
It is an amazing fact that molecules can transition, unchanged, between the three states. There is no change in the
chemical formula of a molecule or the types and numbers of chemical bonds as it undergoes a phase transition. Many
substances can exist as any of the three phases within relatively narrow ranges of pressure and temperature. For
example, at atmospheric pressure (1 atm), water exists as a solid up to 273 K, as a liquid at higher temperatures (273 373 K) and as a gas above 373 K. Additionally, all three phases can coexist in equilibrium at the so-called triple point
(near 273 K and 0.006 atm). Although intermolecular attractions are much weaker than the bonds between atoms that
hold molecules intact, they are strong enough to compete with thermal motions and hold molecules together as liquids or
solids.
Figure F11-1-2. The three states of water. In the solid, the molecules are organized in a repetitive pattern, in the liquid
they are in close contact and can move past each other freely, and in the gas they are separated from each other, with
very little molecular interactions between them.
To illustrate the competition between intermolecular forces and thermal energy, let us describe the changes that
would take place on a microscopic level in the heating of a solid. The solid is built of molecules initially at room
temperature (around 300 K). If we increase the thermal energy by increasing the temperature, the molecules, on average,
will vibrate faster and with larger amplitudes. Eventually, the molecules will gain sufficient energy to break some of the
intermolecular attractions to their neighbors, and will break out of the organized pattern of the solid; they will start moving
past each other and the solid will turn into a liquid. The temperature at which this happens is called the melting point.
Qualitatively, the numerical value of the melting point is a measure of the strength of the molecular interactions in the
solid; the stronger the interactions (IMFs), the higher the melting point. Providing more energy to the molecules (now in
the liquid phase) increases their kinetic energy further. Some molecules may have enough speed to break free of the
remaining intermolecular interactions and escape the liquid to enter the gas or vapor phase. At a certain temperature,
called the boiling point, bubbles of the vapor form in the liquid, and all molecules may enter the gas phase (provided there
is sufficient thermal energy). The boiling point is the measure of the strength of intermolecular interactions in a liquid: the
stronger the interactions, the higher the boiling point. In the gas phase, the molecules are so separated that their
intermolecular interactions are no longer significant relative to their kinetic energy. Increasing the temperature further only
causes the molecules to move faster and experience more energetic collisions.
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Let us imagine reversing the order of phase transitions by starting with a substance that is a gas at room
temperature. In addition to lowering the temperature, we can also increase the external pressure to bring molecules closer
together and increase their intermolecular interactions. If we lower the temperature sufficiently (i.e. remove kinetic energy
from the sample) we can liquefy the gas. However, all gases have a so-called critical point (temperature/pressure point),
above which the gas cannot be liquefied. In other words, bringing molecules closer by increasing pressure above the
critical point temperature is ineffective in making them to aggregate. With further cooling, we can solidify the liquid. This
example illustrates that pressure also affects the distance between molecules, and thus affects the strength of their
interactions (the closer they are the stronger the IMF).
Phase diagrams illustrate phases and phase transitions
as a function of temperature and pressure
Chemists often use phase diagrams to illustrate how the temperature and pressure affect the state of a sample. A typical
phase diagram is shown in Figure F11-1-3. The phase diagrams of different substances have a similar overall "shape,”
differing only in some details and in the pressure/temperature ranges of the phases, which is dependent on the strength of
the IMFs.
Figure F11-1-3. A typical phase diagram. The boundaries between phases correspond to different sets of temperature and
pressure at which the two phases are in equilibrium. Every point along these boundaries is a phase transition point, where
either melting, boiling, or sublimation take place. At the triple point all three phases are in equilibrium. Above the critical
point, the substance is a supercritical fluid.
In the plots, solids occupy the region where temperature is low (low kinetic energy) and pressure is high (forcing the
molecules closer to each other, which strengthens their interactions), while gases dominate where temperature is high
(and they have a lot of kinetic energy) and pressure is low. At low pressures, solids may directly convert into gases (a
process called sublimation). Notice that the melting points and boiling points (and sublimation points as well) lie along the
boundary lines between phases, and change with pressure. We usually refer to melting and boiling points at 1 atm of
pressure as normal. We will discuss the phase diagrams in more detail later in the course.
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11-2 Electrostatic interactions
Intermolecular interactions are caused by electrostatic forces between ions, dipoles,
and induced dipoles
All intramolecular and intermolecular forces arise because matter is composed of charged subatomic particles. These
interactions are electrostatic in nature, but we must differentiate between the two types of attractions because they differ in
magnitude and properties. The bonds between atoms are the intramolecular forces responsible for maintaining the
integrity of discrete molecules or extended networks. They are strong, directional (i.e. point from an atom to its bonding
partners) and act at short range (the nuclei are short distances from each other). On the other hand, intermolecular forces
occur between neighboring molecules; they are generally weaker than chemical bonds, are less directional, and operate
at longer range (the nuclei of the interacting particles do not get very close).
In fact, we have already learned about the extreme examples of such interactions: ionic bonds. Ion-ion forces are
long-ranged and are not directional; ions interact equally strongly with all neighboring ions, as we discovered previously
when we examined lattice energies. However, ionic bonds have strengths in the several hundred to several thousand
kJ/mol range, and are comparable in strength to covalent or metallic bonds which also have bond strengths on the order
of several hundred to a thousand kJ/mol. Ionic bonds are strong because ions with full charges (or even multiple charges)
attract each other with energy that is proportional to the charge (Q) and inversely proportional to the distance between
them (1/d).
Electrostatic interactions also involve non-ionic particles that do not carry full charges. Polar particles have only
partial charges associated with their molecular dipole moments (μ). The nonpolar particles depend on induced dipoles
for their intermolecular interaction. An induced dipole is an instantaneous temporary dipole moment that results when the
electron cloud of a particle is distorted. The ease with which the electron density cloud is distorted is called the
polarizability (α). Attractive forces produced by these temporary dipole moments are known as dispersion forces (also
called London dispersion forces, or LDFs).
The energy of the electrostatic interactions of polar and nonpolar molecules will be much smaller than ion-ion
interactions, as the charges involved are of a much smaller magnitude. Even more importantly, this interaction energy is
more sensitive to the distance between the interacting partial charges; it is inversely related to the distance raised to a
higher power (between 1/d2 and 1/d6, depending on the type of interaction). Typically, the energies of such interactions
are, at most, a few dozen kJ/mol, and are often no more than 15% of the strength of intramolecular bonding interactions.
The electrostatic intermolecular interactions are summarized in Table T11-2-1. Interactions between electricallyneutral molecules (non-ionic particles) include dispersion forces and dipole-dipole attractions (collectively called van der
Waals forces or vdWF), as well as hydrogen bonds, which are a special case of directional dipole-dipole interactions. We
will discuss all these forces in detail below.
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Table T11-2-1. Types of intermolecular interactions
Interactions
Energy
ion-dipole
∝
Qμ
d
dipole-dipole
μμ
∝
d
ion-induced
dipole
∝
dipole-induced
dipole
∝
dispersion
∝
3
Qα
d
4
μα
d
6
αα
d
hydrogen
bonding
2
6
μμ
∝
d
3
Situations
Examples
Typical
strength
(kJ/mol)
ions in polar
liquids
Na+/H2O
50 - 200
polar liquids or
solids
CH3CN(ℓ)
or (s)
5 - 25
ions in nonpolar
liquids
Na+/Ar
3 - 15
nonpolar
molecules
in polar solvents
I2/CH2Cl2
2 - 10
nonpolar liquids
or solids
N2(ℓ), I2(s)
1 - 40
H bonded to N, O,
F
H2O(ℓ)
10 - 40
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11-3 Ions and dipoles
Ions in solution are stabilized by strong interactions with the dipoles of polar solvent molecules
The strongest intermolecular interactions occur when ions interact with polar molecules. The most typical examples of
such ion-dipole forces are found in solutions of salts in water. As we will learn in more detail later, ionic solids such as
NaCl dissociate into individual ions when they dissolve in water. The positive ions attract the negative ends of the water
molecule dipoles, and repel the positive ends. Thus, a sphere of water molecules encloses the cation, with oxygen
atoms interacting with the cation (Figure F11-3-1). The negative chloride ions, conversely, attract the positive ends of the
water dipoles. The anion is surrounded by H–O bonds of water molecules, forming hydrogen bonds (see Lesson 11-5).
Figure F11-3-1. Ion-dipole interactions for Na+ and Cl– ions in water. The cation (left) attracts the negative end of the water
dipole (oxygen atom), while the anion (right) is hydrogen-bonded (see below) to H–OH groups. Click on the images to
view 3D models.
The strength of dipole-dipole interactions depends on the orientation of interacting molecules,
and diminishes strongly with increasing intermolecular distances
Dipole-dipole interactions are important between polar molecules in both liquids and in solids. These forces depend not
only on the magnitude of the molecular dipoles μ, but also on the relative orientations of the dipoles. Figure F11-3-2
illustrates two possible orientations for the dipole-dipole interaction between polar molecules of HCl. The total dipole
interaction is averaged over many orientations that result from the constant motion of the particles. The average strength
of dipole interactions is on the order of 0.2 kJ/mol, which is very weak when compared to the much larger H–Cl bond
dissociation energy of 431 kJ/mol. Since the energy of the dipole-dipole interactions falls off with 1/d3, it is very distancesensitive. For ion-ion interactions, if the distance between ions increases by the factor of two, the interaction energy also
decreases by a factor of two. For dipole-dipole interactions, the same change in distance between dipoles leads to an 8fold decrease in the interaction energy.
Figure F11-3-2. Dipole-dipole interactions in HCl. Only two of the many possible relative orientations of the dipoles are
presented. The bond strength in HCl is 431 kJ/mol. All dipole-dipole interactions, including those shown, are about 0.2
kJ/mol. All dipole-dipole interactions account for only 10% of the total IMFs in liquid HCl. The other 90% are dispersion
forces (see the next Lesson). MEP are shown at the full range of electrostatic energy.
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This drop-off in energy with distance has even more significance when comparing interactions between polar
molecules in the solid and liquid phases. For example, consider how molecules of acetonitrile (CH3CN) fit together in a
crystalline solid and in a liquid sample (Figure F11-3-3). In the solid phase the molecules are neatly arranged, with the
positively-charged ends of their dipoles (the –CH3 end) close to the negatively-charged ends of the dipoles on nitrogen
atoms. This arrangement minimizes positive-positive and negative-negative repulsive interactions. In the liquid phase, the
molecules move constantly, rearranging their positions and orientations. There are, on average, more attractive than
repulsive interactions, but the molecules are much more disordered and separated (the density of the solid is higher than
the density of the liquid). Both of these factors are responsible for weaker intermolecular interactions in the liquid phase.
Figure F11-3-3. Acetonitrile (H3C-C≡N:) in the solid and liquid phases. In the solid (left), the molecular dipoles are
organized to maximize the attractions between the positive dipole ends (light-blue –CH3) and the negative ends (red N).
In solution (right), the thermal motion randomizes the orientations, and although the attractive forces dominate (since they
lower the energy), there are also repulsive interactions present. MEP are shown at the full range of electrostatic energy.
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11-4 Dispersion forces
Nonpolar molecules develop temporary dipoles when approached by ions, dipoles,
or even other nonpolar molecules
Intermolecular interactions are not limited to charged or polar molecules. Molecules or atoms without a permanent dipole
moment can have a temporary dipole moment induced. Intermolecular forces that depend on these induced dipole
moments are called London dispersion forces (LDFs) or simply dispersion forces. An induced dipole moment is caused by
a temporary shift in the electron cloud density. Even though the electrons are distributed symmetrically in non polar
molecules or atoms, the electron cloud symmetry can be perturbed by the presence of an electrical charge in its proximity,
such as an approaching ion or a molecule with a permanent dipole.
Figure F11-4-1. Induced dipole in a nonpolar species. The approaching Na+ ion polarizes the electron cloud of argon,
which induces a temporary dipole in the argon that is attracted to the sodium ion and other cations in the vicinity.
When an ion approaches and interacts with a nonpolar species the interaction is due to ion-induced dipole forces.
For example, an argon atom has a perfectly spherical electron cloud. If a sodium cation approaches closely, the electrons
on argon will be attracted toward it, causing deformation of the electron density (Figure F11-4-1). There will be a slight
excess of electron density on the side of the approaching Na+, and a small deficiency on the opposite side of the argon
nucleus. A temporary dipole will form on the argon atom, and there will be a net attraction between the two species. The
magnitude of this induced dipole will be proportional to the polarizability (α). These attractions will be similar to those
previously described between a sodium cation and water, but weaker, because of the smaller magnitudes of such
induced dipoles and the inverse dependence of their interaction energy on a higher power of separation distance (1/d4).
Both positive and negative charges on ions, as well as dipoles of polar molecules, can polarize nonpolar atoms or
molecules generating a temporary dipole. In general, the induced dipoles will track the motion of the polarizing (dipoleinducing) particle, and will disappear when the interacting molecules separate. The individual interactions are weak, and
effective only at short range, but are very general and prevalent.
Furthermore, no polarizing charge or dipole is necessary to induce an instantaneous dipole moment. Let's consider
two helium atoms in close proximity (Figure F11-4-2). On average, they have perfectly spherical electron density, but
electrons are in constant motion which can result in both electrons being on the same side of the He nucleus at any one
moment in time, yielding an instantaneous, if temporary, asymmetry of the electron distribution. This instantaneous dipole
moment can induce a temporary dipole moment in a second He atom, which happens to be in close proximity, and other
neighboring atoms will be similarly affected. These transient, fluctuating dipoles attract one another in the same way that
permanent dipoles do (see above). The interactions are weaker because the dipoles are smaller, (proportional to the
polarizability of the molecule) and the energy dependence on distance is 1/d6. Nevertheless, this type of interaction is
sufficient to liquefy helium (which is not very polarizable) if thermal energy is withdrawn by cooling the gas to a low
temperature of 4 K.
Figure F11-4-2. Dispersion forces in helium. The momentary asymmetry of electrons around neutral helium (green)
creates a dipole that polarizes helium atoms in close proximity (rainbow color). The induced dipoles attract each other
and polarize more helium atoms.
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The attraction forces due to such "self-polarization" are called London dispersion forces (LDF), or just dispersion
forces. Their strength depends on the polarizability of atoms or molecules, which can be thought of as the squashiness of
the electron cloud; the more polarizable the electron density, the stronger the interactions. In general, the polarizability
increases with the number of electrons; heavier and bigger atoms or molecules have stronger dispersion forces than do
lighter particles. They have more electrons, and their outermost electrons are farther from the nucleus, and thus are held
less tightly.
Boiling points are a good measure of the strength of London dispersion forces
for nonpolar molecules
As we mentioned before, boiling points are a measure of the strength of the intermolecular interactions in liquids. When
we compare boiling points of non-polar noble gases and diatomic halogen molecules (Figure F11-4-3), we can easily
observe the LDF trends presented above. The boiling points for noble gases increase steadily from helium (4 K) to Xe
(166 K), in agreement with the increasing number of electrons (and size), and thus, the increased polarizability. A similar
trend is observed among the halogens. The smallest halogen (F2) has the lowest boiling point (85 K) and the largest (I2)
has the highest boiling point (458 K). It may be tempting to invoke the high electronegativity of fluorine and expect some
"polar" contributions (and an increase in its boiling point), but such logic does not hold; the F2 molecule has no permanent
dipole moment (nor does any other dihalogen), and dispersion forces are the only intermolecular interactions present.
Figure F11-4-3. As the sizes of noble gas and halogen atoms increase the dispersion forces among the atoms increase,
causing the boiling points to increase as well.
Cross-group comparison of boiling points provides some additional insight. For example, F2 (18 electrons, MW = 38
g/mol) and Ar (18 electrons, AW = 40 g/mol) have the same number of electrons and very similar masses. Their boiling
points are also very similar (85 K vs 87 K). However, Cl2 (34 electrons, MW = 71 g/mol) and Kr (36 electrons, AW = 83
g/mol) have very different boiling points (239 K vs. 121 K). The source of the stronger LDFs in Cl2 is the cylindrical shape
of the molecule, which has a greater contact surface than a spherical Kr atom. The chlorine molecules can interact with
each other ("touch") over a larger area than Kr atoms can (for comparison, the ratio of the long molecular axis to the
atomic sphere diameter is 1.1 for F2/Ar and 1.4 for Cl2/Kr).
The same phenomenon is observed for more complex molecules. The boiling points of three isomeric
pentanes are 309 K, 301 K, and 283 K for pentane, isopentane, and neopentane, respectively (F11-4-4). The straightchain alkanes are cylindrical in shape and have more potential contact area with their neighbors than the more spherical
neopentane; the linear alkanes have stronger LDFs and higher boiling points. The irregularly-shaped isopentane
occupies the middle ground.
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pentane
bp = 309 K
isopentane
bp = 301 K
neopentane
bp = 282 K
Figure F11-4-4. Dispersion forces in pentanes. The boiling points for nonpolar molecules of the same mass depend on the
total area of interaction over which the dipoles may be induced. Cylindrical alkanes such as pentane have a large contact
areas, while more compact isomers have limited contact area which reduces to just a "point contact" for nearly spherical
neopentane.
The same dispersion forces account for the steady increase in the boiling points and melting points of unbranched
alkanes (Figure F11-4-5). Hydrocarbons are considered nonpolar, so no significant dipole-dipole interactions are present.
However, each added CH2 group adds more electrons than can be polarized and more contact points along the chain that
may develop instantaneous dipole moments.
Figure F11-4-5. Dispersion forces in linear (unbranched) alkanes. Increasing chain length of the alkanes causes both
melting and boiling points to increase.
It is important to note that dispersion forces are always attractive, and are found in all substances, charged or
uncharged, polar or nonpolar. LDF’s depend on a very shallow penetration of molecular electron clouds. Since molecules
have stable and filled valence shells (due to the octet rule), any further interpenetration leads to repulsive forces. These
weak attractions are in contrast to bonding interactions (in which octets are completed), which require very substantial
overlap and deep interpenetration of atomic or hybrid orbitals. The intermolecular repulsions set the size limits on
nonbonding radii that were introduced earlier. The nonbonding distances are known as the van der Waals radii, and
represent the distances at which the dispersion and dipole-dipole interactions stop being attractive; this is also the
minimum possible distance between molecules in the condensed phase. The space-filling models we have used
throughout our lessons represent such molecular sizes.
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Dispersion forces always contribute to the intermolecular interactions of uncharged molecules
When several kinds of intermolecular forces are present in a substance the effect is additive, and it can be difficult to
untangle the relative importance of each one, especially when polar molecules are involved. If the substances to be
compared have similar molecular weights and shapes, then the dispersion forces will contribute about equally, and any
differences in their total IMFs would be driven by differences in polarity and would manifest themselves by variations in
melting points or boiling points. For example, in a series of molecules of very similar shape and almost the same
molecular weight, the magnitudes of the dipole moments will dictate the trend of the overall strengths of IMF in action; the
more polar molecules will have higher boiling points (F11-4-6).
carbon dioxide
MW = 44 amu
μ=0D
bp* = 216 K
propane
MW = 44 amu
μ = 0.1 D
bp = 231 K
dimethyl ether
MW = 46 amu
μ = 1.30 D
bp = 248 K
oxirane
MW = 44 amu
μ = 1.9 D
bp = 284 K
acetaldehyde
MW = 44 amu
μ = 2.7 D
bp = 294 K
acetonitrile
MW = 41 amu
μ = 3.9 D
bp = 355 K
Figure F11-4-6. Comparison of intermolecular forces (IMFs) for molecules with similar dispersion forces but different
polarities. All molecules have essentially the same molecular weights and very similar shapes, causing their dispersion
forces to be similar. The polarity differences set the trend in their boiling points, with liquids comprised of more polar
molecules boiling at higher temperatures (All MEPs are on a common ±200 kJ scale).
When molecules differ substantially in size (i.e. molecular weights), then LDF usually dominate the interactions, and
the larger molecules will have higher melting or boiling points, even if they are less polar. For example, in the hydrogen
halides series shown in Figure F11-4-7, polarity and molecular mass follow opposite trends: the mass increases from H–F
to H–I, while the dipole moments drop in the same order, as dictated by differences in electronegativity. Dispersion forces
(dependent on size) take precedence, as evidenced by the boiling point ordering except for HF, which has a boiling point
that is far too high to simply dismiss as a rare outlier. The reasons for the exceptionally strong IMFs in HF will be
discussed in depth in Lesson 11-5.
H−F
MW = 20 amu
μ = 1.83 D
bp = 292 K
H−Cl
MW = 36.5 amu
μ = 1.11 D
bp = 188 K
H−Br
MW = 81 amu
μ = 0.83 D
bp = 207 K
H−I
MW = 128 amu
μ = 0.45 D
bp = 239 K
Figure F11-4-7. Comparison of the intermolecular forces (IMFs) for molecules with differing polarities and molecular
weights. With the exception of HF (see Lesson 11-5), the London dispersion forces dominate the intermolecular
interactions. For HCl, HBr, and HI, dispersion forces increase with molecular weight, which goes against the dipole
moment trends. (All MEPs are on a common ±100 kJ scale).
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11-5 Hydrogen bonding
Hydrogen bonds are directional dipole-dipole interactions
We have just observed that H–F molecules have intermolecular interactions larger than we would expect given the
dispersion forces and dipole-dipole interactions of similar molecules. We can identify more "exceptions" quite readily just
by looking at other simple compounds of nonmetals with hydrogen (F11-5-1). The 14th group (4A), hydrides, (compounds
of the carbon family with hydrogen) follow the expected trend: their boiling points increase with size, i.e., with increasing
dispersion forces. Due to the symmetry of the molecules, polarity does not play any role. The hydrides for groups 15, 16
and 17 (5A, 6A and 7A) with the exception of the first member of each series, also follow this trend. But, water (H2O),
ammonia (NH3), and hydrogen fluoride (HF) all show strong deviations; they all have substantially higher boiling points
than expected.
Figure F11-5-1. Comparison of intermolecular forces (IMFs) in molecules with differing polarities and molecular
weights. The boiling points of the hydrides of groups 14, 15, 16, and 17, (4A, 5A, 6A and 7A) increase along with their
molecular weights, indicating that London dispersion forces dominate. The exceptional behavior of NH3, H2O and HF is
due to hydrogen bonding.
The extraordinary deviations from the boiling-point trends observed for NH3, H2O and HF are due to hydrogen
bonding, which is a special case of directional dipole-dipole interactions. Hydrogen bonding occurs when a hydrogen atom
is bonded directly to one of the highly electronegative atoms N, O, and F. Due to their small size and high
electronegativity, their bonds to hydrogen are quite polar. Since hydrogen has no core electrons, there is a significant
partial positive charge on the exposed hydrogen nucleus. This positive charge is attracted to the areas with higher
electron density, such as lone pairs, on the electronegative atoms in other molecules. Since hydrogen is so small, it can
approach the lone pair of the electronegative atom on another molecule quite closely, which increases the strength of the
interaction, especially if the approach is along the axis of the lone-pair.
Hydrogen bonds are formed when hydrogen atoms bonded to F, O, and N interact
with electron pairs on electronegative elements, especially F, O, and N
Only electron pairs on the most electronegative neutral atoms (F, O, N) participate in hydrogen bonding, accepting the
positively-polarized hydrogen. The capability to form stronger hydrogen bonds increases with the increased electron
density on the atom accepting the hydrogen. Thus, the negatively-charged F, O, or N ions form even stronger hydrogen
bonds, and anions of other less-electronegative elements (such as halides or sulfides) are also able to accept the
hydrogen bonds.
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The name of the hydrogen bond denotes both the increased strength of the interactions and its directionality, similar
to covalent bonds. Hydrogen bonds are stronger than other dipole-dipole and dispersion forces, (but are weaker than
covalent bonds). Typically, their strength is on the order of 5-25 kJ/mol, but in special cases may reach 100 kJ/mol. In the
strongest hydrogen bonds, the angle between hydrogen and the two atoms it interacts with is 180°; i.e., the atom holding
the hydrogen (the donor), the hydrogen itself, and the atom with the lone pair to which the hydrogen bond is made (the
acceptor) are collinear.
Figure F11-5-2. Hydrogen bonds and their relative strengths. Hydrogen bonds may form between molecules of one
substance (NH3, HF or H2O, top), or between different substances (bottom). The relative strengths depend on the bond
dipole (H–N < H–O < H–F), which depends on the electronegativity difference between the atom providing the lone pair
(hydrogen-bond acceptor) and the atom accepting the electrons (electron acceptor, in this case hydrogen). More
electronegative atoms are more reluctant to share their electrons and produce a larger dipole.
Hydrogen bonding dominates the intermolecular interactions of neutral molecules
and plays an important role in biological structures and processes
Hydrogen bonds are generally stronger than dipole-dipole forces or London dispersion forces, and if present in the
substance, they usually provide the largest contribution to intermolecular interaction, resulting in higher melting and boiling
points relative to similar compounds lacking hydrogen bonds. Like other intermolecular forces, they are additive;
molecules capable of forming multiple hydrogen bonds have dramatically increased intermolecular interactions. Indeed,
many of the unique properties of water are directly related to the ability of water molecules to form multiple hydrogen
bonds. We previously noted the unusually high melting and boiling points for a substance comprised of molecules of such
small size; Solid water (ice) is less dense than the liquid phase, thanks to the highly organized network of hydrogen bonds
(four per water molecule) that support a structure with large voids (Figure F11-5-3), which collapse on melting. As a
consequence, water expands when it freezes, and ice floats on top of water. Without this property aquatic life would not be
able to survive in frozen lakes in the winter. We will return to other unique properties of water when we explore liquids in
more depth.
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Figure F11-5-3. The structure of ice (two perspectives). With its intricate network of hydrogen bonds (marked in yellow),
these ice structures maintain large voids within the solid which results in ice having lower density than liquid water. Click
on the image on the right to explore this 96-molecule model in 3D.
Because of their relative strength and directionality, hydrogen bonds are extremely important in biology and
biochemistry. Proteins, (the key structural elements of living organisms), enzymes (responsible for anabolism and
metabolism), receptors (crucial to cell signaling) and DNA (the repository of the genetic code of living organism) all
maintain their structures and function because of hydrogen bonding. Life as we know it depends upon hydrogen bonding.
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12 Gases
In this Lesson we explore the properties of gases. In the gas phase molecules move freely from collision to collision,
and the thermal motion (kinetic energy) overpowers the intermolecular interactions. The kinetic molecular theory model
(KMT) describes the movement of molecules, and provides the theoretical underpinning for our understanding of the
macroscopic properties of gas.
12-1 Kinetic molecular theory
The simple kinetic molecular theory model (KMT) assumes that perfectly elastic collisions occur between non-interacting
spherical atoms or molecules. It explains that pressure is the result of the force with which the moving particles collide with the
walls of the container. Temperature is a measure of the average kinetic energy of the particles. Particles with the average kinetic
energy move with a root-mean square speed (urms). The distribution of speeds of the moving particles is determined by the
temperature and mass of the particles. For particles of the same mass, molecules move faster and have a larger range of speeds
at higher temperatures. At lower temperatures the molecules move slower on average and have a narrower distribution of speeds.
At any given temperature, more massive particles move slower and have a narrower range of speeds while smaller particles move
faster and have a wider distribution of speeds.
12-2 Effusion and diffusion
Effusion (gas escaping through a small hole) and diffusion (the spreading out of gas through space) are measurable
properties that are explained well by kinetic molecular theory. The rates of both processes are proportional to the root-mean
square speed of the molecules involved, and therefore are inversely proportional to the square root of molecular mass. Effusion or
diffusion rates can be used to determine the molecular mass of a gas sample by experimental comparison with a gas of known
molecular mass.
12-3 PVT relationships
A sample of gas is described by four state variables, pressure (P), volume (V), temperature (T) and the amount of gas
particles, usually expressed in moles (n). These variables completely define the state of a gas and dictate its physical
experimentally measurable properties. We can explore the relationship between these variables by keeping two variables constant
while manipulating the others. Under constant pressure and temperature, the volume of gas is directly proportional to the number
of moles (Avogadro's law). At constant temperature and an unchanging number of moles of gas, the pressure is inversely
proportional to the volume (Boyle's law). With the pressure and number of moles kept constant, the volume of gas is directly
proportional to temperature (Charles's law). All of these PVT relationships can be derived directly from KMT.
12-4 The ideal gas law
When we combine the individual PVT relationships, the result is the ideal gas law: PV = nRT, where R represents the gas
constant. The ideal gas law can be used to calculate any of the variables of state, given the other values. In this way we can
calculate that the volume occupied by one mole of gas at standard temperature and pressure of 1 atm and 0 °C, (STP) is 22.41 L.
The ideal gas law may be rearranged to show the relationship between gas density and its molecular mass which can be used to
determine the molecular masses of gas molecules. Dalton's law of partial pressures, which states that the individual components
of a mixture of gases behave as if they were alone in the container, is another example of the ideal gas law in action.
12-5 Real gases
Real gases deviate in behavior from the ideal gas law. These deviations are due to intermolecular forces, which cause a
decrease the pressure of the gas, and the actual volume of molecules themselves, which results in a larger value for the total
volume of the gas compared to what the ideal gas law predicts. The van der Waals equation is an equation of state that corrects
for these deviations using experimentally determined molecular parameters a and b, which are measures of intermolecular
interactions and molecular volume, respectively. The smallest deviations from the ideal gas law are observed for small gas
molecules (with small molecular volume and weak intermolecular interactions) and at high temperatures (where the thermal
motions dominate the intermolecular interactions).
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12-6 Atmosphere
The Earth’s atmosphere is a mixture of gases, the composition of which is expressed in molar fractions or ppm (parts per
million). The density and pressure of the gases diminish exponentially with increasing altitude. The trends in temperature changes
also depend on the altitude, but are strongly affected by energy delivered by Sun, and the processes and reactions initiated by
arriving photons. The temperature decreases in the troposphere, increases in the stratosphere, decreases in the mesosphere, and
increases again in the thermosphere.
12-7 Atmospheric temperature
As Sun's energy travels through the atmosphere it gets partially absorbed by gas molecules. In the thermosphere the
highest-energy photons are absorbed, leading to formation of ions (photoionization) and high-energy excited states. Eventually,
the absorbed energy is converted to kinetic energy so that the temperature increases with increasing altitude in this layer. The
ozone cycle is responsible for the increase in temperature with increasing altitude in the stratosphere. In the troposphere, the
greenhouse effect traps part of the energy irradiated by Earth's surface, increasing Earth's equilibrium temperature. The
atmosphere acts as a filter, protecting life on Earth’s surface from harmful high-energy photons generated by the sun.
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12-1 Kinetic molecular theory
Gas particles move in straight lines with constant speed between collisions, during which they
redistribute and preserve their total kinetic energy
As we described in the Chapter 11, all gas particles (atoms or molecules) are in constant motion. They collide with each
other and with the walls that contain them; kinetic energy is transferred and redistributed through these collisions. These
motions constitute the thermal energy of the sample. Thermal motions play an important part in phase transitions, heat
transfer processes, and in the energetics of chemical reactions. To better understand these processes we need a
quantitative model that can illuminate how a system of particles behaves and how it responds to changes in temperature
and pressure. The kinetic molecular theory (or KMT) of gases provides us with such a model.
The assumptions of the kinetic theory are quite simple:
1. Gases consist of a large number of very small particles (molecules or atoms) in constant, random motion. The
molecules are separated by distances that are far greater than their sizes. In other words, the total volume of
all molecules is negligible compared to the volume of the gas sample.
2. The molecules exert no forces on each other (i.e. there are no intermolecular interactions), and between
collisions they move in straight lines with constant velocities.
3. All collisions with other molecules and with the walls of the container are elastic, i.e., no kinetic energy is lost
during the collisions.
Figure F12-1-1. The gas is a collection of a large number of particles
colliding with each other and with the walls. This 2-dimensional model
depicts the motion of a collection of helium atoms at 2000 atm and 25
°C. The speed of the particles shown here is 1012 times slower that
the actual speed. To lower the pressure to 1 atm, the volume of this
box would have to be increased by 2000.
We can develop these assumptions mathematically and show that there is a connection between the behavior of
gas particles and the experimentally measurable quantities: P, V and T. Based on these simple assumptions a profound
understanding of the behavior of all gases is obtained. In the picture that emerges, the gas is a collection of a large
number of randomly moving particles (atoms or molecules) constantly colliding with each other and with the walls of the
container in elastic collisions. The collisions between particles result in changes to the direction of their movement and
their speed, but the average kinetic energy is preserved.
Pressure results from collisions of gas particles with the walls of the container
In the collisions with the walls the particles rebound and transfer their momentum to the area of the wall (A) with which
they collided. The momentum transferred per unit time is equivalent to a force F acting at that area, and the force acting
on a given area is defined as pressure (P = F/A). That pressure is directly proportional to the number of particles (N) in
the container (the more particles, the more collisions with the walls), inversely proportional to the volume (V) of the
container (the smaller the container the more often the particles collide with the walls), directly proportional to the mass
(m) of the particles (more massive particles transfer more momentum upon collision), and directly proportional to the
square of their average speed (u) (the faster the particle moves, the greater the momentum it transfers to the walls). That
average speed is the mean-square average (average of squares) and it is the speed of a particle that has the average
kinetic energy. We can write this relationship as shown in equation E12-1-1, where the factor of 1/3 comes from the fact
that the speed components along the 3 axis directions (x, y, and z) are equivalent:
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1
PV =
3
2
N mur ms
E12-1-1
This important first results of KMT means that the pressure of the gas is the result of the collisions of its molecules
with the walls of the container. Indeed, there is a simple relationship between the macroscopic variables (P, V) and the
unique molecular parameters (the mass of the gas molecules and their mean-square speed).
Notice that both sides of the equation above have dimensions of energy. You may recall the PV work term from our
early considerations, and observe that the right side is (within a multiplication factor) related to the kinetic energy of the
particles (ε = ½mu2). We can combine these ideas.
Temperature is a measure of the average kinetic energy of particles
Although we have not yet introduced it formally, you may recall the ideal gas law from your previous chemistry studies:
PV = nRT. In the ideal gas law n is the number of moles of gas, R is the gas constant, and T is the absolute temperature.
We also take into account two other relationships. Avogadro's number NA can be expressed as n/N = 1/NA. The
Boltzmann constant kB, which is the per-molecule equivalent of R, can be expressed as kB = R/NA. Substituting these
expressions in E12-1-1, we obtain:
ϵav =
3 RT
=
2 NA
3
2
kB T
E12-1-2
This is another major result, as it tells us that temperature is a measure of the average kinetic energy of gas
particles. Thus, temperature, that nebulous yet commonly used term, now has a very concrete meaning related to the
thermal motions of molecules.
Finally, since we are chemists, and we think about masses of molecules in terms of their molecular mass, ℳ (in
kg/mol for SI unit consistency), we take into account that Nm/n = ℳ, and find that:
1
RT =
3
2
Mur ms
E12-1-3
or
−
−
−
−
−
3RT
ur ms = √
E12-1-4
M
Speeds of gas molecules are described by the Maxwell-Boltzmann distribution
with the most probable speed being lower than the root-mean-square speed
As described above, the root-square-mean speed, urms, is the speed of a particle with the average kinetic energy of the
entire sample. This result highlights another important relationship: all particles move faster at higher temperatures, and
lighter particles move faster than heavier particles at the same temperature.
Though we obtained expressions for the average kinetic energy and average speed (urms) of collections of gas
particles (molecules or atoms), individual particles move with different speeds. Individual particles can also change their
speeds during collisions, even though momentum is preserved. It is informative to examine the entire distribution of
speeds of a sample of N2 molecules at three different temperatures as shown in Figure F12-1-2. This function describing
allocations of molecular speeds is called a Maxwell-Boltzmann distribution. Note that the curve is not symmetrical, and
trails off at higher molecular speeds. The consequence of this is that the most probable speed ump (the peak maximum) is
not equal to the average speed uav (which is shifted to the right).
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Figure F12-1-2. The Maxwell-Boltzmann distribution of molecular speed for N2 molecules (ℳ = 0.028 kg/mol) at 100 K,
300 K and 1000 K. The most probable speed, ump, at each temperature corresponds to the peak values (green circles).
The average speed uav (light pink circles) is slightly offset to the right, and the root mean square speed urms (black circles)
is furthest to the right. Click on "Averages" button to display the average speeds. You may modify the temperature and
mass of the gas particles in any plot, or add more gases to the collection.
The distributions of speed are a bit like the exam-score distributions we obtain by plotting the number of students
receiving each possible exam score. As happens often with exam scores, the distribution of speeds is not symmetrical,
and the most probable speed (ump) does not equal the average speed (uav), or the root-mean square speed (urms), but the
differences between them are small. The number of particles that are moving at a given range of speeds is proportional to
the area under the distribution curve. Therefore, we can see that at a fixed temperature, a small number of particles move
very quickly, and a small number of particles move very slowly. Most move at the most probable speed (the peak ump).
The speed at the peak maximum increases with temperature. At any temperature the distribution is biased toward the
higher speeds (uav > ump), and it grows broader at higher temperatures; this indicates that the range of speed increases
with the temperature. Such distributions stress the random nature of thermal motions, which we must treat using statistical
methods, with different kinds of averages and means.
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12-2 Effusion and diffusion
Gas particles with lower masses move faster on average than particles with higher masses
Now that we have learned how the speed distribution changes for one gas at different temperatures, (F12-1-2), let’s
examine different gases at the same temperature (Figure F12-2-1). Since all gases must have the same average kinetic
energy at any given temperature, the bigger molecules (with larger molecular mass ℳ) must have a smaller urms, and vice
versa. This same logic applies to the whole speed distribution. The lightest molecules have the highest speeds and the
broadest distributions, while the heaviest molecules have the lowest speeds and the narrowest speed distributions.
Figure F12-2-1. The Maxwell-Boltzmann distribution of molecular speed for different gas particles (O2, H2O, He, H2) at the
same temperature of 300 K. The lightest molecules move the fastest and have the broadest distribution of speeds, while
the heaviest are the slowest and have a narrow speed distribution. The most probable speed, ump, for each gas
corresponds to the peak value (green circles). The average speed uav (light pink circles) is slightly offset to the right, and
the root mean square speed urms (black circles) is furthest to the right. Click on "Averages" button to display the average
speeds. You may adjust the molecular mass and temperate or add another gas to your collections of plots.
Lighter gas molecules effuse and diffuse faster than the heavier gas molecules
The difference in speed distributions for gases with different masses affects how gases behave in two common situations.
In effusion, gases escape through a tiny hole; in diffusion, they spread through space or through another substance (a
gas, for example). Since both processes depend on the speed of the gas particles, we can predict with confidence that at
a set temperature, lighter molecules will effuse or diffuse faster than heavier molecules.
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Figure F12-2-2. The effusion of a gas through a pinhole
redistributes the gas molecules evenly throughout the volume of
the containers (left). Time-lapse photographs of He- and Ar-filled
balloons over a 36 h period show how gas effuses out of the
balloons. Helium atoms escape the balloon about 3 times faster
than Ar atoms.
Thomas Graham studied rates of effusion even before the kinetic molecular theory was fully developed. He
observed that the ratio of the rates of effusion for two different gases (r1 and r2) is inversely proportional to the square
roots of the ratio of their molar masses (ℳ 1 and ℳ 2). Graham’s Law is given in E12-2-1:
r1
r2
−
−
−
−
M2
= √
M1
E12--2-1
In effusion, the hole is so small that only an individual particle can flow through it without collisions with other
molecules. The only way for a particle to get to the other side is to strike the hole exactly, and the probability of this event
increases with higher speed. Faster movement means more collisions per second with the wall, and therefore a better
chance to hit the target. The rate of effusion should then also be dependent on speed; it is proportional to the root-meansquare speeds, urms (E12-1-6). For two different gases we can write:
r1
r2
=
ur ms (1)
ur ms (2)
=
−
−
−
−
−
3RT
√
M1
−
−
−
−
−
3RT
√
M2
−
−
−
−
M2
= √
E12-2-2
M1
The kinetic molecular theory gives the empirical Graham's law a nice mechanistic underpinning; we can understand
the inverse relationship of the effusion rate ratios by thinking about how effusion is dependent on speed. Effusion
experiments can be used to determine the ratio of the molecular masses of two gases, and if ℳ for one of them is known,
the other can be easily calculated.
Figure F12-2-3. The diffusion of gases, shown as the mixing of two different gases. With time, the molecules of both gases
equally distribute between the two chambers after a hole is opened.
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Although Graham's law can also approximate the ratio of the diffusion rates of two gases under identical conditions,
diffusion is a bit more complicated than effusion. As gases mix and diffuse into one another there are frequent collisions
that affect the paths of traveling particles. A distinguishing feature of diffusion is that it results in mixing or mass transport,
without requiring bulk motion. Particles mix through their random thermal motion.
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12-3 PVT relationships
Gases exert pressure on any surface with which they are in contact
A sample of gas is described by four variables: pressure (P), volume (V), temperature (T) and the amount of gas particles,
usually expressed in moles (n). These variables completely define the state of a gas and its physical conditions. In the
previous section, we explored the kinetic molecular theory of gases. Now we will use this theory to explain the
macroscopic behaviors of gases, i.e., the functional dependencies among these four variables.
As we have already observed, pressure is defined as the force (F) acting on a given area (A):
F
P =
E12-3-1
A
Gases exert pressure on any surface with which they are in contact. That pressure is due to the constant barrage of
gas molecules colliding with the surface. Anything that affects the number of collisions per unit time (such as temperature,
volume, and number of particles) affects the pressure. The SI unit of pressure is the pascal (1 Pa = 1 N/m2 where N is
newtons and m is meters). Pressure can also be described using units of the bar (1 bar = 105 Pa).
On Earth’s surface, we are in a very unique position: we live surrounded by a mixture of gases (air) that we call the
atmosphere. The atmosphere protects us from damaging energy and high-energy matter particles arriving from our solar
neighborhood, is the medium for our weather, and provides the oxygen necessary for us to live. We will explore the
atmosphere in greater detail later, but for now we will concentrate on describing atmospheric pressure.
Let’s imagine a column of air with a 1 m × 1 m cross-section, stretching from Earth's surface to the outer reaches of
our atmosphere (Figure F12-3-1). This column of air has a mass of about 104 kg, and because of the gravitational pull of
Earth, it applies a force of F = 104 kg × 9.8 m/s2 to the 1 m2 area, translating into 1 × 105 Pa of pressure. This example
attaches a value to atmospheric pressure, but we need to remember that the weight of the air column actually makes the
air denser closer to the Earth’s surface, which in turn leads to more collisions per second, which are responsible for the
observed pressure. Atmospheric pressure depends on both altitude and the weather, as both of these factors affect air
density. Standard atmospheric pressure is taken as a typical pressure at sea level and is defined as 1 atmosphere (1 atm
= 1.01325 × 105 Pa = 1.01325 bar). Because the first pressure measurements were made using mercury, pressure is
often expressed in mmHg, also called torr (1 atm = 760 mmHg = 760 torr).
Figure F12-3-1. Atmospheric pressure. The gravitational pull on gas molecules causes the air density to increase at lower
altitude Higher gas density means more collisions per unit time per area and thus higher pressure.
Boyle’s law quantifies the relation between volume and pressure of a sample of gas
at constant temperature
Pressure within a confined container of gas is a bit easier to study than atmospheric pressure. It has long been known that
the pressure of a gas is related to its volume. Consequently, many early experiments on gases were directed at probing
how the volume of a gas depends on other variables, as volume is relatively easy to measure. Typically, two variables are
kept constant while the relationship between the other two is explored; this results in three gas laws: Boyle’s law,
Avogadro’s law, and Charles’s law.
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Boyle's law describes the relationship between the volume and pressure for a constant number of moles of gas at a
constant temperature. The experimental setup used to demonstrate this is shown below in Figure F12-3-2; the volume of
the gas is adjusted with a piston, and the pressure is read on a pressure gauge. The results show that there is an inverse
relationship between pressure and volume: as the volume of a fixed amount of gas at constant temperature decreases its
pressure increases.
Figure F12-3-2. Boyle's law experiment measuring the inverse relationship between the volume of a gas and its pressure.
The volume is adjusted by pushing the piston down. The temperature and the number of gas particles (n in units of moles)
remain constant.
The experimental results of Boyle’s law can be connected easily to the predicted behavior of gases using the kinetic
molecular theory. At constant temperature the root mean square speed (urms) does not change. Therefore, when the
volume is decreased the molecules travel a shorter distance between collisions, and there are more collisions per unit
time: the pressure increases with the number of collisions.
The data obtained from Boyle’s experiment are usually plotted in one of the two ways shown in Figure F12-3-3: with
V as a function of P, or with P as a function of 1/V. Because P and V are inversely proportional, when comparing the same
sample of gas at two sets of volume and pressure, or two gases at constant temperature and number of moles of
particles, the relationship in Boyle’s law can be written as follows:
P1 V1 = P2 V2
E12-3-2
Figure F12-3-3. Boyle's law plots at several temperature values (the temperature is constant for each separate
experiment). The volume of the gas is plotted as a function of pressure (left), and as a function of the reciprocal volume
(1/V) (right).
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Avogadro’s law relates the number of moles of gas to the volume it occupies
at constant pressure and temperature
Avogadro's law states that under constant temperature and pressure, the volume of a gas is directly proportional to the
number of moles of gas (Figure F12-3-4). This law explains how equal volumes of all gases contain the same number of
molecules when T and P are held constant, and why the ratios of volumes of reacting gases and their products can be
expressed with a simple whole number. It eventually led to the concept of the mole, and the number that we now know as
Avogadro's number (NA = 6.02214 × 1023).
This relationship between volume and moles can also easily be predicted using kinetic molecular theory (E12-1-1):
the mean-square speed is constant at constant T, and if P is also constant, V is directly proportional to N (N is the number
of particles, which is proportional to the number of moles, n). If the number of moles of particles in the container increases,
the volume must also increase in order to maintain a constant pressure.
Figure F12-3-4. Avogadro's law. The same volume of gas at constant temperature and pressure contains the same
number of molecules.
Since the volume and the number of moles are directly proportional, a plot of V versus n for a gas at constant T and
P will be linear with a positive slope, as shown in in Figure F12-3-5. And when comparing two gases 1 and 2 at constant
temperature and pressure, the relationship described by Avogadro’s law can be written as follows:
n1
V1
=
n2
V2
E12-3-3
Figure F12-3-5. Avogadro's law plotted for several values of P with T constant at 273.15 K (0 °C). The volume of gas is
directly proportional to the number of moles.
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Charles’s law prescribes that a volume of a sample of gas is proportional to
the absolute temperature if kept at constant pressure
Charles's law states that the volume of a fixed amount of gas, maintained at constant pressure, is directly proportional to
the absolute temperature. Kinetic molecular theory accounts for this observed relationship very well, as shown in Figure
F12-3-5. As the temperature drops, so does the kinetic energy of the particles (and their average speed). There are fewer
collisions per unit time and the collisions are less energetic. Since the number of moles of gas and the pressure must
remain constant, the only solution is to shrink the volume occupied by the gas.
Figure F12-3-6. Charles's law: the volume of a fixed amount of gas at constant pressure is directly proportional to
temperature.
Since the volume is directly proportional to temperature, a plot of V versus T (in °C) for a gas at constant P and n will
be linear with a positive slope, as shown in in Figure F12-3-7. And when comparing the same sample of gas at two
different sets of temperature and volume, or two gases at constant pressure and number of moles of particles, the
relationship described by Charles’s law can be written as follows:
V1
T1
=
V2
E12-3-4
T2
If we examine the Charles’s law plots for a few different constant pressures, we find an interesting phenomenon. As
the temperature is lowered, all the lines converge and can be extrapolated to a volume of zero. We cannot actually carry
the cooling that far since gases liquefy, but the result still has a profound significance; it means that there is a lowest
possible temperature of absolute zero (0 K or −273.15 °C).
Figure F12-3-7. Charles's law: the volume of a fixed amount of gas at constant pressure is proportional to temperature.
The extrapolated lines converge to zero volume at −273.15 °C, which is absolute zero (0 K).
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At absolute zero, the thermal energy of motion vanishes, in agreement with the definition of temperature derived
from kinetic molecular theory (E12-1-4). In a way, this gives us an anchor point for the temperature scale. Units of
measure, however, are another issue. After all, we do not talk about temperature in terms of average kinetic energy.
Instead, temperature is measured indirectly, by measuring a physical property that is linked to changes in the thermal
energy, such as liquid thermal expansion (conventional liquid-filled thermometers) or electric conductivity (thermocouple
thermometers). Historically, the temperature scales were arbitrarily set by selecting two well-defined thermal states. For
example, the Celsius scale is based on the freezing and boiling points of water at 1 atm; the difference in thermometer
readings was then divided into 100 degrees. The absolute scale in kelvin uses degrees of the same size as the Celsius
scale (T(K) = T(°C) + 273.15).
The true power of kinetic molecular theory is its ability to predict the macroscopic properties of gases and their
dependence on the variables of state, without even needing to test any specific examples. For instance, how does the
pressure of a gas relate to temperature when volume and the number of moles are held constant. An increase in
temperature means an increase in the average kinetic energy and urms. Since there is no change in volume, there are
more collisions with the walls per unit time, and the collisions are more forceful, increasing the gas pressure. This is one of
the reasons that closed gas containers should not be heated, as they may explode. The relationship between the gas
pressure and temperature is know as Gay-Lussac's law.
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12-4 The ideal gas law
The ideal gas law combines all the variables of state into one equation
The laws we just examined showed that volume is inversely proportional to pressure (when n and T are held constant),
proportional to temperature (when n and p are constant), and proportional to the number of moles (when P and T are
invariant). We can combine these relationships into a general law, called the ideal gas law, where R is the gas constant, or
proportionality constant.
E12-4-1
P V = nRT
An ideal gas is defined as any gas that obeys the kinetic molecular theory postulates. Specifically, the volume of
molecules must be significantly smaller than the volume occupied by the gas, and the molecules of gas cannot have any
intermolecular interactions. Many real gases satisfy these requirements closely enough that the observed parameters
differ only slightly (a few percent) from those calculated by the ideal gas law.
Table T12-4-1. Values of the gas
constant, R
The proportionality constant R in E12-4-1 is one of the fundamental physical
constants. The value of R depends on the units used for the variables of state,
and the most common values and units are collected in Table T12-4-1. The
equivalent of the gas constant "per-molecule" is the Boltzmann constant (kB
= R/NA).
Units
Value
L·atm/(mol·K)
J/(mol·K)
cal/(mol·K)
m 3·Pa/(mol·K)
L·torr/(mol·K)
0.08206
8.314
1.987
8.314
62.36
The molar volume of an ideal gas at standard temperature and pressure is 22.41 liters
The ideal gas law is useful in solving for any variable of state when the rest are known. For example, we can find the
volume of one mole of gas at 0 °C (273.15 K) under 1 atm of pressure:
nRT
V =
P
L ⋅ atm
= 1 mol × 0.08206
mol ⋅ K
273.15 K
×
1.0 atm
= 22.41 L
E12-4-2
The conditions of 1 atm and 0 °C are called the standard temperature and pressure (STP), and the volume occupied
by one mole of gas under STP conditions is called the molar volume of an ideal gas (VSTP). Many common gases have a
VSTP within less than 2% of the ideal gas value (F12-4-1).
Figure F12-4-1. STP volumes of some common gases. In many cases the STP molar volumes are very close the ideal
gas value, but some gases show substantial deviations (click on the image to see more examples).
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Measurements of gas density allow for determination of its molecular mass
The ideal gas law can also be related to the density of a gas (d = mass per unit volume) by dividing both sides of E12-4-1
by V and RT and multiplying by ℳ (the molar mass):
nM
d=
PM
=
V
E12-4-3
RT
The number of moles multiplied by the molar mass (in g/mol) gives the number of grams of gas, and when divided
by the volume of gas it results in the density of the gas. Therefore, the density of a gas is proportional to pressure and
molar mass, and inversely proportional to temperature. This relationship can be stated as:
M =
dRT
E12-4-4
P
Using this relationship, the molecular mass of an unknown gas can be determined by measuring its density at
constant temperature and pressure. To illustrate the method, let’s calculate the molecular mass of air. A flask of a known
volume is weighed evacuated (with a vacuum pump), and then refilled with air and weighed again. With the temperature
and pressure recorded, the density, and therefore, the molar mass can be calculated (Figure F12-4-2).
Figure F12-4-2. At 295 K, a 1 L flask evacuated with a
vacuum pump (left) is opened to the air (at 761 mmHg)
and weighed again (right). The air inside has a mass
of 1.18 g; thus, the gas density is 1.18 g/L. The molar
mass of air is then calculated (E12-4-4) to be 28.5 g/mol.
Dalton’s Law states that the pressure of a mixture of gases equals the sum of the partial
pressures
of the components
Gases mix without limit, forming homogeneous gas solutions. For ideal gases, there are no intermolecular interactions,
and the gases in the mixture should behave independently. We can define the partial pressure of each gas as the
pressure that it would exert if it were alone in the container. The partial pressure of each gas in the mixture is related to
the total pressure by Dalton's law of partial pressures; the total pressure Pt of a mixture of gases equals the sum of the
partial pressures of all component gases. Rearranging the ideal gas law provides the following relationship where nt is the
total number of moles of gas in the mixture:
RT
Pt = P1 + P2 + P3 + ⋯ = nt
E12-4-5
V
To calculate the total pressure using Dalton’s law, we can consider each gas in the mixture to be an ideal gas. If we
have a mixture of three gases we can then write the following:
RT
P1 = n1 (
RT
)
and
V
P2 = n2 (
RT
)
and
V
P3 = n3 (
)
V
In the mixture, all gases are at the same temperature and occupy the same common volume. The total pressure (Pt)
is therefore determined by the total number of moles (nt) of all gases present:
RT
Pt = P1 + P2 + P3 = nt
V
RT
= (n1 + n2 + n3 )
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Each gas's contribution to the total pressure depends on its mole fraction in the mixture. In general, mole fractions
(X) are defined as the number of moles of a selected component, divided by the total number of moles (X1 = n1/nt). For the
first gas we can write:
RT
P1
Pt
n1
V
=
RT
nt
=
n1
E12-4-7
nt
V
Rearranging this equation lets us solve for the pressure contributed by the first gas:
P1 = (
n1
nt
E12-4-8
) Pt = X1 Pt
For example, the mole fraction of N2 in air is XN2 = 0.78 and the mole fraction of O2 is XO2 = 0.21 (the remaining 1%
are other gases, which we will neglect). We can say that nitrogen contributes 78% of the atmospheric pressure in the air.
Thus. the estimated average molar mass of air (ℳ t) is calculated as follows, which is very close to our experimental
determination (F12-4-2):
g
Mt = X1 M1 + X2 M2 = (0.78 × 28
mol
g
) + (0.21 × 32
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12-5 Real gases
Plots of compressibility factors illustrate deviations from ideal gas behavior
at high pressures and low temperatures
The ideal gas law has its limitations. One way to test the limits of its applicability is to plot PV/RT (called compressibility
factor) for one mole of gas over a wide range of pressures. The ideal gas law predicts that PV/RT should equal 1.0 (for 1
mole of gas) for all values of pressure. Real gases deviate from this prediction as shown in Figure (F12-5-1).
Figure F12-5-1. Deviations from the ideal gas law for several gases at 300 K (left), and for nitrogen gas at different
temperatures (right).
For most gases the deviations are quite small at low pressures, but become quite apparent at higher pressures. The
magnitude of the deviations also depends on temperature, with smaller deviations observed at higher temperatures.
These deviations can be explained by two main factors: real gas molecules have finite sizes, and real gas molecules do
interact through intermolecular forces. The volume of the molecules may become a significant part of the total volume,
especially at high pressures where the amount of empty space diminishes as the gas particles are forced together. Under
high pressures, the value of PV/RT is greater than the expected value of 1 (for example, see H2 or N2 in F12-5-1). At
constant temperature and with an unchanging number of moles, this means that the apparent volume of a gas sample is
greater than would be predicted by the ideal gas law. The intermolecular attractions affect the pressure. If a molecule
about to strike a wall experiences an attraction from a nearby molecule, it will slow down a bit and hit the wall with a
smaller impact and less momentum. The intermolecular attractions make the apparent pressure smaller than that
predicted by the ideal gas law.
The van der Waals equation takes into account molecular volume and intermolecular forces
in order to quantify deviations from ideal gas behavior
The Dutch physicist Johannes Van der Waals recognized in the late 1800’s that the ideal gas law could be corrected to
account for molecular volume and intermolecular attractions. The resultant van der Waals equation is given below:
2
(P +
an
V
2
) × (V − nb) = nRT
E12-5-1
Van der Waals introduced two parameters specific to each gas: a and b. The pressure term P is adjusted upward by
adding a term (containing a) that takes into account the strength of the intermolecular attractions, and encompasses the
polarity and polarizability of gas molecules. The attractive IMFs increase proportionally to n2/V2 (the molar density
squared) as the probability of collisions between pairs of molecules increases; collisions increase as the molar density of
each of the colliding molecules increases. The volume is adjusted downward by subtracting the volume occupied by the
molecules, with b describing the molar volume (i.e. the volume occupied by the molecules themselves).
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Table T12-5-1. Van
der Waals constants
for some common gas
molecules
Gas
b
a (L2(L/mol) atm/mol2)
Ne
He
H2
H 2O
O2
Ar
N2
Kr
CO 2
CH 4
Xe
Cl 2
CCl 4
0.0171
0.0237
0.0266
0.0305
0.0318
0.0322
0.0391
0.0398
0.0427
0.0428
0.0510
0.0562
0.1383
0.211
0.0341
0.244
5.46
1.36
1.34
1.39
2.32
3.59
2.25
4.19
6.49
20.4
The van der Waals constants for various gases are collected in Table T12-5-1. We can
consider the values of a in terms of the molecular attractions that we explored previously.
London dispersion forces are the only intermolecular forces that are present for atomic noble
gases. These gases have small values of a, with He being the least polarizable and having the
weakest attractions. The interactions are stronger for polar molecules (H2O) and large
molecules (CCl4), which are highly polarizable due to the size of their electron clouds. In fact,
carbon tetrachloride is a liquid at STP (it has a boiling point of 77 °C) which is consistent with
its large value of a. You may recall that dipole-dipole and dispersion forces are collectively
referred to as van der Waals forces.
The volume correction (nb) depends not only on the size of the molecules, but also how close they can get together.
For atoms (or for molecules approximated as spheres), their van der Waals radii define the closeness of their approach;
this is the closest they can come to each other before the repulsive forces start increasing.
The discussion above explains how the van der Waals forces and radii made it into our presentation. The constants
(a and b) are obtained by measuring the macroscopic prosperities (P, V and T) of real (i.e. non-ideal) gases, yet they
provide the insight into the nanoscopic world—into the sizes and interactions of individual molecules. Inversely, kinetic
molecular theory allows us to understand the macroscopic trends. For example, we can now readily predict that the
smallest deviations from ideal behavior will be found for small molecules with weak intermolecular interactions (He, for
example), and the largest deviations will be found for large, polar, or polarizable molecules (like CCl4). The deviations will
be smaller at high temperatures (where the thermal motions dominate the attractions), and at low gas densities (where the
volume correction is small and there are fewer molecules per volume).
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12-6 Atmosphere
Earth's atmosphere is a mixture of many gases,
with nitrogen and oxygen being the main components
The atmosphere is a layer of gases that surrounds a body (a planet, for example) that has sufficient gravitational pull to
hold the mass of gases. Earth's atmosphere is a mixture of gases composed mainly of nitrogen (78%) and oxygen (21%)
with much smaller fractions of CO2 and noble gases, and traces of other components. The composition of air at sea level
is summarized in Table T12-6-1. Water vapor (humidity) is highly variable, depending on weather, and is not included in
the summary.
T12-6-1. Composition of dry air at sea level
Gas
Nitrogen (N 2)
Oxygen (O 2)
Argon (Ar)
Carbon dioxide (CO 2)
Neon (Ne)
Helium (He)
Methane (CH 4)
Krypton (Kr)
Hydrogen (H 2)
Nitrous Oxide (N 2O)
Xenon (Xe)
Content
Molar Mass
(mole fraction)
(g/mol)
0.78084
0.20946
0.00934
0.00039445
0.00001818
0.00000524
0.00000179
0.00000114
0.00000055
0.000000325
0.00000009
28.013
31.998
39.948
44.0099
20.183
4.003
16.043
83.80
2.0159
44.0128
131.3
As you can see in the table, mole fractions are used to describe the composition of air. The mole fraction is the
ratio of the moles of the component of interest to the total number of moles. Since most components are present at very
small mole fractions, parts per million (ppm) are a convenient way to list these values. If a number was listed as a percent,
this would describe how many parts per hundred; a percent is obtained by multiplying the fraction by 102. Similarly, ppm
values are obtained by multiplying the molar fraction by 106. For example, Ne is present at 18.18 ppm. This translates to
18.18 × 10−6 for the mole fraction. Indeed, the mole fractions are the measure of the concentration of gases in the
gaseous solution (a solution is a mixture that is homogenous on the atomic or molecular level). In gases, when fractions
are expressed in ppm, they refer to ratios of volume (for gases under constant T and P, the volume is proportional to the
number of moles). In liquid solutions, the units of ppm would refer to ratios of masses.
The atmospheric pressure and density decline exponentially with altitude
but the temperature decreases or increases in adjacent layers
The atmosphere changes with altitude. The density of the gases and the number of collisions per second drop
exponentially as altitude increases, and as a consequence, so does the pressure. The temperature also changes as a
result of various chemical and physical processes occurring at different altitudes. Reversals in the direction of temperature
change serve as the boundaries between different layers (Figure F12-6-1). The troposphere extends up to about 12 km
above the surface. In this layer of the atmosphere, temperature drops as altitude increases. Airplanes fly around 10 km
above the surface, which would be close to the boundary between the troposphere and the stratosphere. The tallest
mountain on Earth (which is Mount Everest at 8.85 km) almost reaches the maximum altitude at which humans can still
function (barely) without supplementary oxygen. Note that at the summit the pressure is about 30% of the standard
atmospheric pressure, as is the partial pressure of O2. The troposphere contains about 75% of the mass of our
atmosphere.
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Table T12-6-2. Changes in air properties with increasing altitude.
Altitude
Density
Pressure
Collision frequency
Temperature
(km)
0
5
10
50
100
200
(kg/m 3)
1.28
0.74
0.41
10 −3
10 −7
10 −10
(atm)
1
0.5
0.25
10 −3
10 −6
10 −10
(sec −1)
10 10
5 •10 9
2 •10 9
5 •10 6
10 3
1
(K)
298
256
224
270
200
850
Figure F12-6-1. Regions of the atmosphere showing pressure and density drop at higher altitudes. The inversions of
temperature trends creates natural boundaries for different regions(spheres).
Above the troposphere is the stratosphere, where the temperature increases with increasing altitude. Above the
stratosphere are the mesosphere (where temperature again decreases with increasing altitude) and the thermosphere
(where temperature again increases with increasing altitude). Due to density and temperature differences, the gases mix
slowly between layers.
The change in temperature patterns in the different layers is due to radiation energy and chemical reactions initiated
by radiation arriving from our Sun. These reactions are called photochemical reactions, as they are initiated by photons.
We will discuss these processes in more detail in the next Lesson.
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12-7 Atmospheric temperature
Some light energy is absorbed as it travels through the atmosphere towards Earth's surface
As we have just learned the temperature of the atmosphere changes with increasing altitude in a non-monotonous way; it
decreases in the troposphere, increases in the stratosphere, decreases in the mesosphere, and increases again in the
thermosphere (Figure F12-7-1). Indeed, the inversions of temperature trends serve to define the boundaries between the
atmospheric layers. This pattern of alternating temperature trends is due to various photochemical reactions and
processes initiated by photons arriving from our Sun.
Figure F12-7-1. The temperature profile of the atmosphere is given by the purple curve. The names of the regions of the
atmosphere are given on the right-hand side of the figure. The boundaries between regions are determined by reversals in
the direction of the temperature change with altitude. The black wavy lines indicate the depth of penetration of solar
radiation into the atmosphere before it is absorbed.
These Sun-energy driven processes are an integral part of Earth's energy balance that controls our climate. We
have introduced some of the pertinent basic ideas in the context of our study of electromagnetic radiation and
spectroscopy. Now that we have learned about the structure of gas molecules, the properties of gases, and Earth's
atmosphere, we can explore the energy conversion processes in more detail.
The wavelength distribution of Sun light as it enters the atmosphere is shown in yellow in Figure F12-7-2. About
10% of the energy is delivered in the form of ultraviolet (UV) radiation, 40% enters as visible light, and about 50% is
infrared (IR) radiation. Very high energy (X-ray or gamma) and very low energy (radio) photons are a very small fraction (<
1%) of the energy flux. The red curve shows the analogous spectral distribution at sea level. The differences between
these distributions is the energy absorbed by the atmosphere which amounts to 19% of the total energy arriving from our
Sun. At Earth's surface UV, visible, and IR constitute 3%, 44%, and 53% of the energy delivered, respectively. Thus, our
atmosphere acts as a filter, protecting life on Earth’s surface from harmful high-energy photons generated by the sun.
While the light travels towards Earth's surface, it interacts with molecules of atmospheric gases; it gets absorbed, and
converted to other forms of energy, ultimately ending up as thermal energy (kinetic energy of gas particles). The
atmospheric temperature changes depend on how far the energy delivered by Sun is able to penetrate the atmosphere
before it is absorbed and what are the available processes leading to the energy conversions.
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F12-7-2. Distribution of light wavelengths reaching Earth. This plot compares the distribution of solar radiation (i.e.
intensity and wavelength) outside the atmosphere (in yellow) to the distribution at sea level (in red). The dips in the red
curve marked in blue are the result of the absorption of infrared radiation by water in the atmosphere, the dip marked in
green is due to radiation absorbed by atmospheric CO2.
Ionization reactions in the thermosphere are the main pathway for converting high-energy
radiation
into kinetic energy
High energy wavelengths (photons with λ < 100 nm) are absorbed at high altitudes (above 80 km) in the thermosphere. At
these altitudes the gas density is extremely low and molecular collisions are relatively infrequent (Table T12-6-2). The
energetically-dominant processes are direct collisions between photons and gas molecules. If the colliding photons have
sufficient energy, photoionization may take place, with the photon ejecting an electron from the molecule to form an ion, as
shown in the example of N2 molecules:
N
2
+
+ hν ⟶ N
2
−
+ e
C12-7-1
The energy of the photon (calculated as E = hν = hc/λ) must be equal to or greater than the ionization energy in
order for this to happen. This process is analogous to the ionization of atoms. For example, the ionization energy of N2 is
1495 kJ/mol, and we can calculate the maximum wavelength required for this reaction as follows:
E =
λ=
NA hc
E12--7-1
λ
NA hc
= 80 nm
1495 kJ/mol
E12-7-2
This calculation indicates that the wavelength of the absorbed radiation has to be below 80 nm in order to initiate
photoionization in nitrogen gas. Recall that shorter wavelengths correspond to higher energy photons. Table T12-7-1
collects some photoionization processes that occur in the thermosphere, as well as the maximum wavelength needed for
each.
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Table T12-7-1. Ionization energies (IE) for processes occurring in the
thermosphere
Thermosphere process
N 2 + hν → N2+ + e−
O 2 + hν → O2+ + e−
O + hν →O+ + e−
NO + hν → NO + + e−
IE (kJ/mol)
λ max (nm)
1495
1205
1313
890
80.1
99.3
91.2
134.5
λmax is the longest wavelength of photons having sufficient energy to
ionize atoms or molecules
Molecules may also absorb energy in the same way atoms do: when a photon (with energy equal hν) is absorbed
by a molecule, an electron is promoted to a higher-energy molecular orbital, resulting in an excited state (marked with an
asterisk). For example, in the case of nitrogen gas in the thermosphere:
∗
N
2
+ hν ⟶ N2
C12-7-2
The ions formed in the photoionization processes and the excited state molecules have excess potential energy that
is eventually converted to kinetic energy (i.e. heat) upon collisions. Even if these unstable species have a significant
atmospheric lifetime because the concentration of gases at this altitude is low (and thus collision frequency is also low),
they ultimately collide with other particles; ions recombine with electrons and excited states relax to the ground state. The
energy is converted to kinetic energy of translation, vibrations, and rotations, or it is used in chemical reactions. For
example, if a nitrogen molecule in an excited state collides with an oxygen molecule in the upper atmosphere, the
following reaction can take place:
∗
N2 + O
2
⟶ N O + O
2
C12-7-3
All these process described above convert the energy of the absorbed photons into thermal energy, increasing the
temperature of the gases in the thermosphere. The effect diminishes with diminishing altitude as more and more highenergy photons are absorbed when light travels toward Earth and the density of the atmosphere increases providing more
opportunities for absorption. At about 80 km, essentially all high-energy photons (λ < 100 nm) are absorbed: the upper
atmosphere's mission to protect life on Earth from being bombarded by dangerous radiation is accomplished!
The region below the thermosphere is called the mesosphere. The temperature in the mesosphere decreases with
altitude since little or no absorption of solar radiation occurs in this region of the atmosphere. The concentration of gases
in the mesosphere is low and chemical reactions requiring collisions between two species are still relatively slow. In
addition, high-energy radiation has already been absorbed in the thermosphere.
Absorption of UV light is responsible for the warming of the stratosphere
The region below the mesosphere is called the stratosphere. The temperature in the stratosphere increases with
increasing altitude. This increase in temperature is attributed to the ozone cycle, a process by which ultraviolet (UV)
radiation is absorbed as ozone decomposes and reforms in the atmosphere.
The concentration of ozone in the atmosphere varies with altitude (like all atmospheric gases) but its concentration
peaks in the stratosphere at ~ 10 ppm. Ozone is formed in the stratosphere in a two-step process that begins with the
photo decomposition of O2 (see C12-7-4). Photodissociation is the process wherein photon energy is used to break
chemical bonds. A molecule of O2 can be broken apart into two oxygen atoms if the energy of the incoming photon is
greater than or equal to the bond dissociation energy (495 kJ/mol for O2), absorbing short wavelength UV radiation (λ ≤
242 nm).
O
2
+ hν ⟶ O + O
C12-7-4
In the second step, an oxygen atom combines with an oxygen molecule to form ozone, O3, which is an allotrope of
oxygen; a different physical form (molecular structure) in which an element can exist. Ozone is less stable than O2 by
about 142 kJ/mol, and can undergo photodissociation when it absorbs light with λ ≤ 320 nm.
O + O
2
⟶ O
3
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O
3
+ hν ⟶ O
2
+ O
C12-7-6
By combining the processes in this cycle, we see that two UV photons are absorbed, while all of O2 that
decomposes is reformed and all of O3 that forms decomposes. The result of this cycle is conversion of solar UV radiation
into heat without changing the concentrations of O2 or O3. This process is responsible for the warming of the stratosphere
while absorbing UV light (up to 320 nm). Such wavelengths cause sunburns and skin cancers. The ozone layer filters out
a significant amount of the harmful UV light arriving from outer space, diminishing its intensity at the Earth’s surface.
The temperature in the troposphere decreases linearly with increasing altitude
The troposphere is the region closest to Earth's surface. Most of the light from our Sun, mainly visible light but also some
UV light and a large portion of IR radiation passes through the atmosphere and is absorbed by the ground and oceans; it
adds up to 51% of the energy arriving from Sun to the edge of the atmosphere. The Earth's surface radiates this heat back
out in the form of IR radiation (Figure F12-7-3). However, significant part of this IR radiation is absorbed by so called
greenhouse gases that include H2O, CO2, CH4 and O3 in the lower atmosphere. These molecules absorb the radiation
energy and send it out in all direction, both downward toward Earth's surface and upward where more molecules absorb it,
and repeat the process. The net effect is the "trapping" of energy close to the surface, the so called greenhouse effect.
The temperature of the troposphere is greatest at sea level because the Earth's surface is the body emitting IR radiation,
and the concentration of the greenhouse gases which absorb it is greatest near sea level.
Figure F12-7-3. The distribution of radiation emitted from earth as observed from outside the atmosphere is shown in red.
The curve depicts the expected distribution of light that would be emitted by Earth' surface to maintain a stable planetary
temperature. The area in yellow corresponds to the light absorbed by atmospheric gases, mainly water, CO2 and
methane. The bands above the curve indicate the regions of the spectrum in which each of these molecules absorb
significant amounts of energy.
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13 Liquids
Liquids are an example of a condensed phase, wherein molecules are tightly packed and in constant contact with
each other. The strengths of intermolecular interactions between molecules in liquids are comparable to the energy of
their thermal motions. Heat exchange with the surroundings, leading to changes in the average kinetic energy, may lead to
phase changes. Removal of heat converts liquids into solids, while addition of heat frees the molecules from their mutual
interactions as they transition into the gas phase.
13-1 Phase changes
The transition from the solid to the liquid phase is called fusion (or melting). It is an endothermic process. The reverse
exothermic process is called freezing. The transition from liquid to gas phase is called vaporization; it is an endothermic process.
Condensation, the reverse of vaporization, is exothermic. Additionally, solids may directly transition into the gas phase in an
endothermic process called sublimation. The reverse exothermic process is called deposition. Each of the transitions is an
isothermal process. The heat exchanged with the surroundings during the process is called the enthalpy of the transition. The
enthalpy of sublimation (ΔHsub) is the sum of the enthalpies of fusion and vaporization (ΔHfus and ΔHvap). The magnitude of the
enthalpy of phase transitions is the same in both directions, but of opposite sign.
13-2 Heating curves
The addition of heat to a sample increases its temperature (and its average kinetic energy) until the temperature of a phase
transition is reached. The amount of heat required to increase the temperature of a substance by one degree (1 oC or 1 K) is
defined as the heat capacity, and can be expressed per mole (as molar heat capacity) or per gram (as specific heat). Each phase
of the substance has its own characteristic heat capacity. At the transition point all heat absorbed is used to disrupt intermolecular
forces, and the temperature stays constant until all molecules have transitioned to the new phase. Plotting the temperature as a
function of heat absorbed yields a heating curve, which illustrates the changes taking place as a sample transitions from phase to
phase. The transitions can be run in reverse by cooling the sample. Most often, the heat of vaporization (or conversely the heat
given off during condensation) represents the greatest heat exchange with the surroundings, because it is during this process that
the majority of intermolecular forces are completely disrupted (or reinstated, in the case of condensation).
13-3 Vapor pressure
Some molecules in the liquid phase have sufficient energy to escape into the gas phase even at temperatures below the
boiling point. In a closed container at constant temperature, a dynamic equilibrium is reached when the rate of escape into the gas
phase equals the rate of return to the liquid phase. At equilibrium, the gas phase exerts a pressure defined as the vapor pressure
at that temperature. The equilibrium vapor pressure depends on the strength of the intermolecular interactions present; liquids with
weak interactions between molecules have a high vapor pressure and are volatile (they evaporate easily). As temperature
increases, the vapor pressure increases. The temperature at which the vapor pressure equals the pressure outside the container
is called the boiling point.
13-4 Phase diagrams
A plot of the phases and their coexistence curves as a function of temperature and pressure constitutes a phase diagram.
The coexistence curves represent conditions at which two phases exist in equilibrium. The intersection point of the three
coexistence curves is called the triple point, where all three phases are in equilibrium simultaneously. The liquid-gas coexistence
curve ends abruptly at the critical point, above which the liquid and gas phases are indistinguishable and the substance exists as a
supercritical fluid. A gas cannot be liquefied at temperatures above its critical point temperature. In the area between the curves
only one stable phase may exist. The solid phase occupies the low temperature-high pressure region of the phase diagram, while
the gas phase occupies the high temperature-low pressure section; the liquid phase lies in between.
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13-5 Properties of liquids
The intermolecular interactions between molecules of a substance determine not only the phase at given temperatures and
pressures, but also other physical properties. The interactions of molecules within pure substances are called cohesive forces.
The interactions of molecules with surfaces with which they come into contact are called adhesive forces. One manifestation of
cohesive forces is viscosity, or a resistance to flow, which is related to the ease with which molecules can break some of their
intermolecular forces and move past each other. Another manifestation of cohesive forces is surface tension. Molecules on the
surface of the liquid have fewer attractions to other molecules and are therefore less stabilized. The liquid tries to minimize its
surface area pulling the surface molecules closer together. The balance between cohesive and adhesive forces is what causes
mercury to form a convex meniscus in glass containers (stronger cohesion), while water forms a concave meniscus (stronger
adhesion). Because of the strong glass-water attraction, molecules of water "climb" the walls of a test tube; the only force holding
the liquid in is the force of gravity. This strong adhesion is what causes capillary action in small diameter tubes.
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13-1 Phase changes
Fusion, vaporization, and sublimation are endothermic processes, whereas
freezing, condensation, and deposition are exothermic
We have just learned how substances transition from one phase of matter to another, if the average kinetic energy of their
molecules changes sufficiently. This change affects the balance between the energy of the intermolecular attractions and
the energy of the thermal motions. Thus, if a substance changes its phase, it has to exchange heat with the surroundings.
Under conditions of constant pressure, the energy change equals a change in enthalpy (ΔH).
The phase transition from solid to liquid, called fusion (or melting) is endothermic as shown in Figure F13-1-1. It
requires an increase in the thermal energy of the particles to disrupt the intermolecular forces between them. The amount
of energy required to melt a substance is called the enthalpy of fusion (ΔHfus), or sometimes the heat of fusion. We will
use the terms "enthalpy" and "heat" interchangeably, as we are going to deal with processes taking place under constant
pressure (usually an atmospheric pressure of 1 atm). The amount of heat exchanged in the phase transition is an
extensive property of the substance, and it is usually given on a per-mole basis.
The transition in the reverse direction, from liquid to solid, is called freezing. It is an exothermic process; the
intermolecular attractions gained by organizing the particles into the close-packed arrangement of the solid phase lower
the energy of the system. The enthalpy of the process is still called the heat of fusion, except it has a negative algebraic
sign in accordance with conventions for exothermic heat transfer.
Figure F13-1-1. Enthalpy changes during phase transitions. Melting,
vaporization, and sublimation are endothermic (orange). Energy in the
form of heat is needed to break the intermolecular attractions between
the particles of the system. Freezing, condensation, and deposition are
exothermic (green). The increased intermolecular attractions (which
are the result of bringing the particles back into increased contact in
the condensed phases) lower the energy of the system. The enthalpy
of the system increases as we move to the top of the chart, and
decreases moving to the bottom
The liquid to gas (vapor) transition is called vaporization. It is an endothermic process. The heat required, called the
enthalpy of vaporization (ΔHvap), is used to completely break all intermolecular attractions in the substance. The particles
are freed from each other, and begin to move as separate entities in the gas phase. The gas phase is the highest-energy
phase for any substance. The reverse process is called condensation; particles return to the condensed liquid phase in an
exothermic process and intermolecular attractions are restored, lowering the energy of the system.
The solid phase can change directly into the gas phase through a process called sublimation. Sublimation is
endothermic, as all intermolecular interactions between particles are broken in the transition. The heat exchanged in the
process is called the enthalpy of sublimation (ΔHsub). The reverse process is called deposition; since the restored
intermolecular interactions lower the energy of the substance, deposition is exothermic. As can be readily appreciated
from the graph in Figure F13-1-1, the enthalpy of sublimation is equal to the sum of the enthalpies of fusion and
vaporization (ΔHsub = ΔHfus + ΔHvap). As you may recall, enthalpy is a state function and is path-independent. It does not
matter whether the solid is converted directly into the gas, or it is first converted into the liquid phase then into the gas; the
overall change in the enthalpy of the system must be the same.
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The magnitude of the phase change enthalpy (ΔHsub, ΔHfus, or ΔHvap) is an excellent measure of the strength of the
intermolecular forces in a substance. Large values indicate strong intermolecular interactions, as shown in Figure F13-1-2.
For example, water has strong hydrogen bonds in addition to dipole-dipole and dispersion forces, and so has a larger
heat of vaporization than that of butane (with only dispersion forces). Mercury represents a case of metallic bonding, and
has a greater heat of fusion and vaporization than any of the non-metallic liquids shown. Examination of the chart also
shows that heats of vaporization are bigger than heats of fusion for individual substances. This general trend is due to the
fact that fusion is a transition between the solid and liquid condensed phases: the molecules remain in close contact, and
only some intermolecular interactions are broken. In the vaporization transition from liquid to gas (or sublimation directly
from solid to gas), the molecules have to move from a condensed phase to the gas phase, breaking all intermolecular
interactions in the process. These phase changes therefore require more energy to accomplish.
Figure F13-1-2. Phase-transition enthalpies for several substances. For each substance, the heat of sublimation is equal
to the sum of the heats of fusion and vaporization. Heats of vaporization are larger than heats of fusion. This is because
all intermolecular interactions in liquid are broken upon vaporization, but many of the interactions in solids remain after
melting since molecules are still in close contact.
Heating a sample leads to an increase in temperature until a transition temperature is reached,
at which point the heat is used to disrupt intermolecular interactions at constant temperature
Let us examine phase changes at the molecular level. When heat is added to a solid, the thermal motions of the particles
(atoms, ions, or molecules) increase. Since the particles are densely packed and cannot move far, they instead vibrate
faster and with larger amplitudes. This increased average kinetic energy is registered as an increase in the temperature of
the solid. At a certain temperature some of the particles have sufficient kinetic energy to break loose from these
interactions and start moving past each other. The solid starts to melt. Further addition of heat does not lead to an
increase in temperature; instead, the supplied energy is used to break intermolecular attractions throughout the solid. This
process of fusion is isothermal, meaning the temperature remains constant until the entire solid has melted; the
temperature at which this happens is called the melting point. The melting point is a characteristic property of the pure
substance, and it is a measure of the strength of the intermolecular interactions in the solid.
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If more heat is added after all of the solid has melted, the temperature of the resulting liquid starts increasing,
indicating the growing kinetic energy of the particles. At this point some of the particles close to the surface of the liquid
will have enough kinetic energy to escape into the vapor phase. At higher temperatures, even more particles escape and
the pressure of the gas above the liquid, called vapor pressure, goes up. In a closed container kept under constant
pressure (atmospheric pressure of 1 atm, for example), no particles can leave the system. Some of the particles of the
vapor may hit the surface of the liquid and reenter the liquid phase. When the rate of escape is equal to the rate of reentry
into the liquid phase, equilibrium is established between the liquid and vapor phases. With increasing temperature, more
and more particles move to the vapor phase, and when the vapor pressure in the container reaches the outside pressure,
the substance begins to boil. This temperature is called the boiling point, and its value is indicative of the strength of the
intermolecular interactions in the liquid. At the boiling point even the particles below the surface have sufficient energy to
break their intermolecular attractions and form the vapor. When a liquid changes into a gas there is an immense change in
volume (more than 1000-fold for water, for example); bubbles of vapor form throughout the whole liquid and rise to the
surface. The temperature of the boiling liquid remains constant until all liquid is transformed into vapor. The energy
provided during this phase transition is used to break all of the remaining intermolecular interactions. Finally, when all
particles are in the gas phase, providing more heat increases the kinetic energy of the particles, leading to an increase in
the temperature of the vapor.
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13-2 Heating curves
Heat capacity is the amount of heat needed to increase the temperature of a sample by one
kelvin
Let's explore the phase changes described in the previous section in a quantitative fashion, using water as an example.
We will start with 1 mol (18 g) of solid water (ice) at −50 °C, and we want to convert all ice to water vapor (steam) at 150
°C.
First, we have to provide enough energy to warm the solid to the temperature at which fusion takes place, which is 0
°C. The amount of heat required to raise the temperature of the ice by 1 K or 1 °C is defined as the heat capacity (given
the symbol C) of ice. In fact, for pure substances (like ice) the heat capacity is either called the molar heat capacity (Cm) if
given per mole, or the specific heat capacity (Cs), (sometimes just called specific heat) if given for 1 gram of the
substance. The amount of heat (q) needed to raise the temperature of n moles or m grams of a substance by a certain
number of degrees of temperature is calculated using Equation E13-2-1:
q = nCm ΔT = mCs ΔT
E13-2-1
In general, each phase of a substance has its own unique Cm, and Cs can be calculated by dividing Cm by the
molecular weight. For ice these values are Cm(ice) = 37.6 J/(mol·K) and Cs(ice) = 2.09 J/(g·K). Since we wish to bring our
ice sample from −50 °C to 0 °C, the change in temperature is ΔT = 50 °C = 50 K. Using E13-2-1 and the value of Cm(ice)
we can calculate that we need 1.88 kJ of heat for this change in temperature. We can observe the temperature change as
a function of the heat delivered to our sample in a plot called the heating curve (Figure F13-2-1a). The first line segment
(A-B in F13-2-1a) corresponds to heating the sample of ice from −50 °C to 0 °C. An alternative presentation, with
temperature as the x variable (Figure F13-2-1b), shows the increasing enthalpy of the sample. The slope of the same A-B
segment in F13-2-1b corresponds to Cm, since we are heating exactly one mole of ice in our example.
Figure F13-2-1. The heating curve for one mole of water being converted from −50 °C ice to 150 °C steam. Plot on the left
(a) presents the changes in the temperature of the sample as a function of added heat, while the plot on the right (b)
shows the increasing enthalpy of the sample as a function of the increasing temperature as heat is added. The heat
delivered from the surroundings is the heat gained by the sample.
The heat of transition is the amount of heat required to change the phase of the sample
at the temperature of transition
Having heated our sample of ice to 0 °C (point B), fusion begins and the temperature remains constant until the whole
sample has melted (B to C in Figure 13-2-1). The heat required to accomplish this process is the heat of fusion, which for
water is ΔHfus = 6.02 kJ/mol. In general, the heat q exchanged with the surroundings during a phase transition is
calculated using the enthalpy for that specific phase transition (ΔHtrans) in Equation E13-2-2:
q = nΔHf us
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Since we are melting one mole of ice we can use the heat of fusion for water to calculate that the total heat required from
B to C is 6.02 kJ.
After all of the solid ice melts, further heating of the liquid water is required to bring it to the boiling point at 100 °C, at
which point the next phase transition begins. The heat required to reach the boiling point is now expressed by the heat
capacity of liquid water, where Cm(water) = 75.3 J/(mol·K) or Cs(water) = 4.18 J/(g·K), and ΔT = 100 °C = 100 K. Using
Equation E13-2-1, the amount of heat required is 7.53 kJ. The heating of water corresponds to segment C-D, and the
slope of that segment in F13-2-1b corresponds to the molar heat capacity of water.
At 100 °C water boils and the phase transition to vapor takes place. The temperature does not change during the
phase transition. We must use the heat of transition, ΔHvap = 40.67 kJ/mol, to calculate the heat required (E13-2-2); 40.67
kJ of heat is needed to evaporate the whole 18 g sample. The vaporization corresponds to segment D-E in Figure F13-21.
Finally, the water vapor produced in the last step is heated up to 150 °C, which is our final temperature. The heat
capacity of steam is Cm(steam) = 33.1 J/(mol·K) or Cs(steam) = 1.84 J/(g·K), and ΔT = 50 °C = 50 K which allows us to
calculate that we need 1.66 kJ of heat to accomplish the final step (E13-2-1).
The heat of vaporization is generally the largest energy expenditure
when converting a sample in the solid state to one in the gas phase
To summarize, we have transferred 1.88 kJ of heat from the surroundings to our sample to warm the ice, 6.02 kJ to melt it,
7.53 kJ to heat the resulting water to the boiling point, 40.67 kJ to vaporize it, and 1.66 kJ to heat the resulting steam to
the final temperature. Our total heat expenditures were 57.76 kJ; a large fraction of this energy (70%) was spent on
converting the liquid into a gas (this energy corresponds to ΔHvap). The fact that the majority of our applied heat went
towards vaporization attests to the very strong intermolecular attractions between water molecules that have to be
broken before the molecules can enter the gas phase. Overall, the heat added from the surroundings has converted
H2O(s) into H2O(g), increasing the enthalpy of the sample (Figure F13-2-1b). The vapor has higher energy than ice.
The heating process can be reversed by cooling the sample. The path followed starts with steam at the right side of
Figure F13-2-1, and proceeds along the same path toward ice on the left side.
We encounter the heat transfer associated with phase transitions in everyday situations. We cool our drinks by
adding ice; the heat needed to melt the ice is taken from the drink, lowering its temperature. In hot weather we sweat, and
the evaporating water withdraws heat (needed for evaporation) from our skin, providing a mechanism to regulate our body
temperature. Refrigerators cool by withdrawing heat from the food compartment and using it to evaporate liquids with low
boiling points (usually under reduced pressure). The resultant gas is then compressed back into a liquid and recycled.
There is one final point to consider. If you look back at our phase change diagram (Figure F13-1-1) and compare it
to the heating curve (Figure F13-2-1), you may notice that they appear to be inconsistent. The phase change diagram in
F13-1-1 is schematic, showing all possible phase transitions, but neither the temperature nor pressure is specified. Thus,
the diagram does not take into account the heating or cooling processes required to reach the respective temperatures of
phase transitions. In contrast, the heating curve depicts experimental results at a fixed pressure (typically 1 atm). Once we
fix the pressure of an experiment, the number and type of possible phase transitions and temperatures at which they
occur are also fixed. For diagrams illustrating one substance such as in F13-1-1 or comparisons between substances
such as in F13-1-2, the heats of phase transitions (ΔHfus, ΔHvap, and ΔHsub) at standard temperature (298.15 K) are
used. These standard heats are obtained by extrapolations from the values measured at the actual temperatures of the
transitions.
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13-3 Vapor pressure
Vapor pressure represents a dynamic equilibrium between molecules in the liquid phase
and molecules in the gas phase of a substance
As we described the molecular view of phase transitions in the previous section, we noted that some molecules close to
the liquid’s surface had sufficient kinetic energy to escape into the vapor phase long before the liquid reached the boiling
point temperature. The pressure of the vapor phase above the liquid is called the vapor pressure. To describe vapor
pressure, let’s consider an example in which we place a sample of liquid in an evacuated empty flask that is held at a
constant temperature below the boiling point. At first the inter-phase movement of a small fraction of molecules is mainly
from the liquid to the vapor phase. With time, as the number of molecules in the gas phase increases, the return of
molecules back to the liquid becomes more probable. Since more energetic molecules have a greater chance to escape
into the gas phase, some self-sorting of molecules with various kinetic energies occurs. On average, the gas phase
molecules would be expected to have higher kinetic energy than the liquid-phase molecules. As a result of evaporation,
the liquid would normally cool down a bit. However, since our experiment is run at constant temperature by providing heat
from the surroundings to maintain the temperature, the average kinetic energies of both phases remain the same.
Eventually, a dynamic equilibrium is established. Equilibrium is defined as the point in a physical or a chemical process
where the forward rate is equal to the reverse rate. In our example, at equilibrium the rate of escape from the liquid phase
and the rate of return to the liquid phase are equal. Even if molecules constantly travel between the two phases, at
equilibrium the ratio of molecules in the two phases remains constant. Equilibrium vapor pressure has been established
within the flask as shown in Figure F13-3-1.
Figure F13-3-1. Liquid added to an evacuated flask reaches a dynamic equilibrium with its vapor. The pressure of the
vapor pushes up the column of mercury in the side arm allowing for vapor pressure measurement.
If the liquid is left in an open container, some of the molecules do not return back to the liquid phase; instead, they
diffuse away and equilibrium is never reached. In this case the liquid evaporates, and eventually all of the molecules
leave, resulting in an empty container. Liquids that evaporate readily under ambient conditions are called volatile. Volatile
substances have higher vapor pressures than non-volatile compounds. Vapor pressure (at a given temperature) and
volatility depend on the strength of the intermolecular interactions between liquid-phase molecules. The stronger the
interactions, the lower the vapor pressure, and the less volatile the substance.
Vapor pressure increases exponentially with temperature
Changes in temperature affect the vapor pressure; if we increase the temperature, a larger fraction of molecules gains
sufficient speed to escape the surface of the liquid. You should recall that a collection of gas molecules at a given
temperature has a characteristic speed distribution. Molecules in liquids have a very similar distribution of speed and
kinetic energy; at higher temperatures there are more molecules with a kinetic energy above the threshold value needed
to escape the liquid, as shown in Figure F13-3-2. Thus, we expect that the vapor pressure of a liquid will increase with
increasing temperature, which is consistent with kinetic molecular theory.
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Figure F13-3-2. Distribution of the kinetic energy of liquidphase molecules at two different temperatures. At the
higher temperature there are more molecules with
sufficient energy to escape the liquid phase.
Figure F13-3-3 illustrates several examples where the vapor pressure of each liquid increases with increasing
temperature, forming curves with distinct exponential shapes. When the vapor pressure reaches the external pressure the
liquid boils, and vapor bubbles form throughout the volume of the liquid. The temperature at which this happens when the
external pressure is 1 atm is called the normal boiling point, but a liquid can boil at different temperatures if the external
pressure is different from standard atmospheric pressure. For example, water boils at 70 oC at the high altitudes of Mount
Everest, or at 120 oC in the average pressure cooker. It takes significantly longer to cook food on Mount Everest since the
boiling water has less kinetic energy. On the other hand, the pressure cooker significantly shortens the cooking time.
The normal boiling point of each compound in Figure F13-3-3 occurs at the point where its vapor pressure curve
crosses the 1-atm line; they increase from left to right. As you may recall, boiling points increase with the strength of
intermolecular attractions in liquids. The compounds in F13-3-3 illustrate this relationship very well. The vapor pressure
increases with temperature, slowly at first at low temperatures, and then rapidly as the temperature approaches the boiling
point of a liquid. The exponential shape of the vapor-pressure curve has its basis in the molecular kinetic theory,
wherein the fraction of molecules that have sufficient speed to escape the liquid phase is an exponential function of the
temperature.
Figure F13-3-3. Vapor pressure as a function of temperature for several substances. Compounds with increasing boiling
points follow the trend of increasing intermolecular interactions among the molecules in the liquid. These
interactions increase with the sizes of the molecules (increasing the dispersion forces), with their polarity, and with their
ability to form hydrogen bonds.
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13-4 Phase diagrams
Phase diagrams represent pure phases and the boundary lines between them,
which meet at the triple point where all three phases coexist at equilibrium
We have just explored the equilibrium between the liquid and the gas phase. Similar dynamic equilibria exist between the
solid and the liquid phase, and between the solid and the gas phase under some range of temperature and pressure. As
we already mentioned, the phases and the equilibria between them are often presented in the form of phase diagrams. A
general phase diagram is shown in Figure F13-4-1, with the three major phases and the boundary curves between them
as functions of pressure and temperature. This graph is not really a standard 2D plot; it should not be interpreted that
pressure is simply a function of temperature, but rather we must imagine it as a "flattened" 3D plot, where the state of the
substance is described by the two variables T and P.
Figure F13-4-1. A general phase diagram. The colored lines are the coexistence curves. The orange curve between the
liquid and gas phase, starting from the triple point and ending at the critical point, represents the P-T relationship predicted
by the Clausius-Clapeyron equation. A gas cannot be liquefied at temperatures above its critical point since gas and liquid
are no longer distinguishable at these conditions and the supercritical fluid that is formed has properties of both phases.
The yellow solid-liquid coexistence curve is typically nearly vertical, and most commonly slants to the right, reflecting
higher stability of the solid phase under high pressure. The two coexistence curves cross the third one (the blue solid-gas
coexistence curve), at the triple point, where all three phases coexist in equilibrium. In the areas between the curves, only
one phase can exist. Crossing the coexistence curves corresponds to a phase transition. The line at 1 atm is drawn
because the normal melting and boiling point are located where it crosses the corresponding coexistence curves.
We can now recognize that one curve of the diagram (the orange liquid-gas coexistence curve) represents the
Clausius-Clapeyron function, and describes the liquid-vapor equilibrium. Crossing this curve corresponds to a phase
transition (vaporization or condensation). A similar boundary curve (blue) separates the solid and gas phases. Transitions
across this curve correspond to sublimation and deposition. The solid-liquid coexistence curve (yellow) is nearly vertical,
and in most cases leans slightly to the right. In general, the denser phase (usually the solid) is more stable at high
pressure. Passage across this curve corresponds to melting (fusion) or freezing. The three curves meet at the triple point,
where the three phases coexist in equilibrium. Only two phases are in equilibrium along the curves themselves, and only
one phase is present in the areas between the curves. The solid phase occupies the upper left corner of the plot
(corresponding to high pressure and low temperature). The gas phase occupies the lower right region of the diagram
(corresponding to low pressure and high temperature). The liquid phase occupies the center.
The crossing point of the 1-atm line with the solid-liquid boundary curve corresponds to the normal melting point.
When this curve (yellow line) is nearly vertical it means that melting points are not strongly influenced by pressure.
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The intersection of the 1-atm line and the liquid-gas boundary curve represents the normal boiling point. The heavy
slant of this boundary indicates that boiling points are strongly dependent on pressure. The liquid-vapor curve ends
abruptly at the critical point. At this point, and at pressures and temperatures above this point, the liquid and gas phases
are no longer distinguishable, and we are left with a supercritical fluid that has properties of both phases (fluid is a name
encompassing both gases and liquids). The highest temperature at which condensation can occur is called the critical
temperature. Above this temperature the gas cannot be liquefied, regardless of the pressure applied. The kinetic energy of
molecules above this temperature is always greater than any intermolecular attractions; the greater the attractions, the
higher the critical temperature. The pressure needed to liquefy a gas at the critical temperature is called the critical
pressure. These two values establish the placement of the critical point
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The phase diagram for water is atypical, because the solid-liquid coexistence line bends to the
left
since ice is less dense than water
The phase diagram of water differs significantly from most phase diagrams, and is shown in Figure F13-4-2. We use the
logarithmic scale for pressure and the linear scale for temperature to better illustrate the ranges of the phase changes.
The water phase diagram is anomalous due to its solid-liquid coexistence curve, which slants to the left. This slant
indicates that at higher pressures, liquid water is favored as the more stable phase. We know that the denser phase of a
substance is more stable under high pressure, and liquid water is indeed denser than ice. The extensive hydrogenbonding network creates voids within ice crystals, and is responsible for its lower density. There are very few substances
that show a similar trend in density; most solids are denser than the corresponding liquid.
Figure F13-4-2. Phase diagram for water. We use a
logarithmic scale for pressure in order to see all phases to
the full extent. The normal melting (mp) and boiling point
(bp) are found at the intersection of the 1-atm line and the
coexistence curves.
Water has a triple point at a temperature very close to 0 °C, but at a low pressure (0.006 atm). At pressures below
that of the triple point, there is an equilibrium established between the solid and gas phases. This property of water is
often used to freeze-dry various materials. The water is removed by sublimation at low pressure. The advantage of this
method is that the materials do not have to be heated to remove the water, thus protecting the integrity of any heatsensitive components (for example nutrients or medicines present in the substance). The 1-atm line crosses the solidliquid boundary at 0 °C and the liquid-gas coexistence curve at 100 °C; these are the familiar normal melting and boiling
points of water, respectively.
No liquid phase is observed in the phase diagram for carbon dioxide at ambient pressure,
because the triple point lies above atmospheric pressure
For most substances, including water, the 1-atm line lies between the pressures of the triple and the critical points. For
such substances, all three phases can be observed at normal pressure by changing the temperature. There are some
substances for which the triple point is above the 1-atm line, including carbon dioxide (CO2). The phase diagram of CO2 is
shown in Figure F13-4-3. In such substances, only the solid and the gas phase are observed under atmospheric pressure.
Solid CO2 is called dry ice, and sublimes at −78 °C at 1 atm; it transitions directly into the gas phase without an
intermediate liquid, thereby justifying its name.
Figure F13-4-3. Phase diagram for CO2. We use a
logarithmic scale for pressure in order to see all phases
to the full extent. The 1-atm line shows that CO2 can only
exist as a solid or a gas at this pressure, and the
sublimation point (sp) is −78 °C (195 K).
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Carbon dioxide has an additional interesting property; its critical point occurs at an accessible temperature and
pressure, under conditions that do not result in too much expense. Therefore supercritical CO2 can be produced at a
reasonable cost, and used as a convenient nontoxic solvent. One common use is to extract the caffeine from coffee
beans. After dissolving the caffeine in the supercritical liquid, the solvent is removed by simply lowering the pressure and
converting the CO2 back to the gas phase. The gas can then be compressed and reused, leaving the extracted caffeine
behind.
Many substances have multiple versions of the solid phase under different pressures. For example, there are at
least 9 different forms of ice. In addition, some liquid phases may exist in more than one form; liquid crystals exhibit a
different ordering of molecules in the liquid phase. Some of these interesting variations will be explored in the secondsemester of general chemistry.
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13-5 Properties of liquids
Viscosity is a reflection of the strength of the cohesive forces that provide resistance to flow
The intermolecular attraction forces that make molecules of one substance stick together are called cohesive forces. They
are an intrinsic property of a substance that find their source in the electronic structure and shapes of the molecules. We
have explored many liquid properties that depend on the strength of such cohesive forces, such as vapor pressure,
boiling points, or ΔHvap. Intermolecular forces in liquids are not only apparent in phase transitions; another commonly
encountered manifestation of cohesive forces is viscosity. Viscosity is a measure of resistance to flow, and is usually
measured in kg/m·s. Some liquids such as water or gasoline flow easily, while others, such as glycerin (F13-5-1) or
molasses, flow very slowly. Indeed, the viscosity of molasses is so high that it is often used to describe sluggishness (“as
slow as molasses”). Viscosity is related to the ease with which molecules can move past each other. Molecules with
elongated shapes and multiple contact areas for interactions become entangled and slip past each other slowly. In
general, viscosity decreases with increasing temperature for a given substance; the increased kinetic energy facilitates the
breaking of intermolecular interactions.
Figure F13-5-1. Relative viscosity of glycerin (left
cylinder) and water (right cylinder). The glycerin
molecules have strong intermolecular interactions
(multiple hydrogen bonds), making it difficult for one
molecule to slip past another. The falling marble is
slowed down by the slow motions of molecules getting
out of the way.
Surface tension results from minimization of the surface area of a liquid,
since the strongest cohesive forces exist in the bulk
Another manifestation of cohesive forces is surface tension which is a measure of the energy needed to create the
surface area of a liquid. Molecules on the surface of a liquid have fewer partners to interact with than molecules below the
surface. The surface molecules are exposed to air (or other gases), with which they have no interactions of any
significance. Compared to the molecules in the bulk sample, the surface molecules have higher energy since they are not
stabilized by as many intermolecular interactions and are attracted inwards the liquid. To minimize this excess energy, the
system will respond by minimizing the surface area, resulting in a lower number of molecules with less than optimal
interactions. The end result is the formation of a sort of "flexible skin" on the surface of the liquid. You may have seen a
water strider or another insect walking on water (F13-5-2) and utilizing surface tension. The same phenomenon is
responsible for the formation of nearly perfect spherical droplets by many liquids with strong intermolecular interactions;
this includes water (which has strong hydrogen bonding) and mercury (which has metallic bonding). Of all shapes, the
sphere has the smallest surface to volume ratio, so spherical droplet formation minimizes the number of molecules on the
surface.
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Figure F13-5-2. Surface tension. The molecules at the surface of the liquid are attracted inwards and experience fewer
intermolecular attractions when they are exposed to gas-phase molecules instead of liquid-phase molecules. These
surface molecules are less stable, so the system tries to minimize the number of such molecules by adopting the shape
that has the smallest surface area. This creates a flexible "skin" on which a water strider can move efficiently, without
getting wet.
The capillary effect and menisci depend on the balance of cohesive forces within the liquid and
adhesive forces between the liquid and the surrounding solid surfaces
Often, the properties of a substance depend on the balance between the cohesive forces and the adhesive forces.
Adhesive forces are interactions of molecules with surfaces with which they come into contact. Consider the menisci of
water and mercury in a glass test tube, as shown in Figure F13-5-3. Water forms a concave meniscus, and its surface
curves downward. The intermolecular interactions between water molecules are weaker than the interactions between
water molecules and the glass surface of the tube, so the adhesive forces are stronger than the cohesive forces. The
system tries to maximize the stronger interactions by increasing its contact surface. Molecules of water cling to the walls
of the test tube as far up as possible; the only force holding the liquid in is gravity. In contrast, mercury forms a convex
meniscus, with an upward-curved surface. The cohesive forces within mercury are metallic bonds, which are stronger than
the adhesive forces between mercury and glass. This system tries to minimize the contact surface, and the molecules
avoid interacting with the walls of the test tube as much as possible. Similarly, water spreads out on a piece of glass,
wetting the whole surface and maximizing the adhesive forces, while mercury on glass stays beaded in the form of
spherical droplets flattened by gravity.
Figure F13-5-3. Menisci of mercury and water (click on a picture for alternative view). In mercury the cohesive forces
(weak metallic bonds) are stronger than the adhesive forces to glass causing mercury to minimize contact with the glass
while maximizing intermolecular interactions within the liquid. As a result a convex meniscus forms. Water’s adhesive
forces with glass are stronger than its cohesive forces. Water maximizes its area of contact with glass, climbing the walls
and forming a concave meniscus. In narrow tubes, these adhesive forces give rise to capillary action, wherein a column of
water can climb the tube until the attraction is overpowered by gravitational forces.
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The balance between adhesive and cohesive forces determines the behavior of two substances when they come
into contact. The adhesive forces between water and glass are strong enough that in glass tubes of small diameter (called
capillary tubes) the water can climb to substantial heights, until gravity balances the increased attraction (F13-5-3). Such
capillary action is partially responsible for delivering water and nutrients to the upper parts of plants and trees. The narrow
tube walls in plants (xylem) have surfaces containing organic compounds with large numbers of −OH groups and oxygen
atoms that form hydrogen bonds with water molecules, increasing the adhesive forces.
The counterbalance of cohesive and adhesive forces determines surface interactions
between immiscible liquids or liquids and solids
Adhesive and cohesive forces also affect whether spherical droplets are formed, or a substance spreads out into a thin
layer when coming into contact with another substance. When water droplets are introduced onto the surface of oil, they
remain droplets (F13-5-4); the cohesive forces within water are stronger than the adhesive forces of water with the
hydrocarbon molecules of the oil. Another example of this balance of forces is the beading of rain droplets on a freshly
waxed car (wax often contains long hydrocarbon chains). In contrast, oil droplets placed on the surface of water spread
evenly over the whole surface. In this case the cohesive forces within oil are weaker than water-oil adhesive forces.The
relative strengths of adhesive and cohesive forces are of paramount importance when making glue (strong adhesion
required), or non-stick surfaces (weak adhesion needed).
Figure F13-5-4. From left to right: (a) Beading of water droplets in oil. The water's cohesive forces are stronger than any
water-oil adhesive forces. Water droplets minimize their contact area with oil by remaining spherical. Only when the
droplets touch the glass at the bottom of the dish do they start to spread out (water-glass adhesive forces are stronger
than water-water cohesive forces). (b) Spreading of oil on water. The oil-oil cohesive forces are weaker than the oil-water
adhesive forces. The spreading oil pushes away the small speck of camphor (or pepper powder ) floating on the water's
surface.
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14 Solutions
As we’ve just explored pure substances, let’s now turn our attention to mixtures. Homogenous mixtures composed of
only one phase are called solutions. The major component of the mixture is called the solvent, while the minor
components are called solutes. Solutions may be solids, liquids or gases. Liquid solutions, and especially aqueous
solutions, are most prevalent. Solutions have physical properties distinct from both the solvent and the solutes alone. The
properties of a solution are uniform throughout the sample, as the components are mixed at the molecular level.
14-1 Dissolution process
Gases mix completely and spontaneously, since mixtures are statistically favored over pure components. In general, the
statistical factors at play are related to the entropy, or randomness of the system. Spontaneous processes are favored by an
increase in entropy. The formation of a solution in condensed phases is also favored by increased entropy, but the solute-solute
and solvent-solvent attractions must be broken and replaced with solvent-solute interactions. If the enthalpy change associated
with these alterations in intermolecular attractions is favorable (corresponding to an exothermic process), the dissolution is
spontaneous. If the enthalpy change is weakly endothermic, the entropy-driven process is still spontaneous, but if enthalpy is
strongly endothermic, no solubility is observed.
14-2 Electrolytes
Electrolytes are substances that dissociate into ions when dissolved in water. Strong electrolytes (including ionic salts, strong
acids and strong bases) dissociate completely in water. Weak electrolytes (including weak acids and weak bases) dissociate only
partially in water (only a small fraction of the total sample of molecules dissociate). Non-electrolytes (including most covalent
compounds) do not dissociate into ions at all when dissolved in water, preserving their molecular integrity.
14-3 Concentration
The concentration of a solute is expressed as a ratio of the amount of solute compared to the amount of solvent and it may
be quantitatively expressed in various ways. The amount of solute can be given as either mass or number of moles, and the
amount of solvent can be given as the mass, number of moles, or volume. The most commonly used measures of concentration
are mass fraction, mass percent, molar fraction, molarity, and molality.
14-4 Solubility
Solubility is defined as the amount of solute that will dissolve in a solvent, and is determined by the nature of the
intermolecular interactions, temperature, and (for gases) pressure. Ionic and polar substances are more likely to be soluble in
polar solvents, while nonpolar solvents will most likely dissolve non-polar solutes. This generalization of solvent-solute interactions
can be summarized as "like dissolves like". Gas solubility is proportional to the partial pressure of the gas above the solution, and
for gases the solubility diminishes with increasing temperature. In contrast, the solubility of ionic or covalent substances usually
increases with increasing temperature, although exceptions are common. A solution in equilibrium with a solute in a different
phase is referred to as saturated. Solutions with solute concentrations approaching saturated are called concentrated, while those
on the low end of the concentration range are called dilute.
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14-1 Dissolution process
Dissolution depends on entropic effects, which are always favorable,
and enthalpic effects, which may be favorable or unfavorable
So far in our exploration of the physical properties of gases and liquids we have concentrated on pure substances.
However, the forms of matter around us are most often heterogeneous or homogenous mixtures. In heterogeneous
mixtures, the ingredients are not dispersed uniformly, and properties change from one region of the sample to another. In
homogenous mixtures the components are uniformly mixed at the atomic or molecular level, and we observe consistent
properties and composition throughout the whole sample. We call such mixtures solutions. The major component in a
solution is called the solvent and the minor components are called solutes. We usually think of a solution as a liquid that
contains a dissolved substance, which could be a solid, another liquid, or a gas. In fact, there are also solid solutions,
including brass (zinc in copper), steel (carbon in iron), or sterling silver (copper in silver), as well as gaseous solutions
(such as air). Indeed, we already encountered gaseous solutions when we discussed diffusion, partial pressures, and
the atmosphere. Liquid solutions are the most common, and we will focus on them in this Lesson. We will be particularly
concerned with aqueous solutions, in which water serves as the solvent.
Gaseous solutions are simple to make and are always homogeneous; gases mix spontaneously without any energy
supplied by the surroundings. Imagine that we have a gas in one container connected to another identical but empty
container. If we open a stopcock between the containers, after a while the gas molecules will be distributed equally
between the containers. This is the most statistically probable distribution. Any aggregation of an excess of molecules in
one of the containers is highly unlikely. Similarly, if the two containers are occupied by different gases, each container will
hold an identical mixture of gases after a period of time with the stopcock open. Both components will be distributed
equally between the containers.
This statistical tendency of gases to interdisperse is related to a thermodynamic quantity called entropy. Entropy can
be simplistically described as the amount of randomness in a system. Spontaneous processes, in general, are associated
with a balance between two different thermodynamic parameters; entropy and enthalpy of the system. Increasing entropy
favors spontaneity (increasing randomness), and decreasing enthalpy favors spontaneity (decreasing energy and
increasing stabilization). While entropy always increases upon mixing (the system becomes more dispersed), the enthalpy
change can either be favorable (exothermic) or unfavorable (endothermic).
Although the entropy increase upon mixing always favors the formation of a solution in the gas phase, the situation
is more complex in liquid solutions. The entropy of the system still increases upon mixing, but the enthalpy change is
influenced by the strong intermolecular attractions in condensed phases that may restrain molecules from mixing.
Solute-solute and solvent-solvent interactions are broken during dissolution,
while solute-solvent interactions are formed
Figure F14-1-1 shows that the energetic balance of breaking those forces and establishing new intermolecular solventsolute attractions constitutes the enthalpic component of solution formation:
Solute-solute interactions between the molecules of the solute must be broken so individual molecules can be
distributed among the solvent molecules (ΔH1). This process is always endothermic and requires energy.
Solvent-solvent interactions between molecules of the solvent must be overcome to make space for the
molecules of the solute (ΔH2). This process is always endothermic and requires energy.
Solute-solvent interactions form as the molecules mix, lowering the energy of the solution (ΔH3). This process
is always exothermic and releases energy.
The first two processes are endothermic since they break attractive interactions. The third is exothermic since new
attractions are formed. The extent to which a given substance is able to dissolve in another is determined by the relative
magnitudes of the enthalpies involved (F14-1-1).
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Figure F14-1-1. Enthalpies of solution formation with exothermic process on the left and endothermic process on the right.
ΔHsol = ΔH1 + ΔH2 + ΔH3. The process of separating the solvent and solute particles (ΔH1, ΔH2) is an endothermic
process and causes enthalpy of the system to increase. The mixing of the solvent and solute (ΔH3) is either endothermic
(because of repulsion between the solvent and solute) or exothermic (because of favorable interactions being formed).
If the newly created interactions between the solvent and the solute molecules dominate (ΔHsol = ΔH1 + ΔH2 + ΔH3
< 0), the overall enthalpy change is favorable. Since the entropy of mixing is always favorable, the dissolution will be
spontaneous (F14-1-1a).
If the newly established interactions are not able to compensate for the broken interactions between the solvent and
solute (ΔHsol = ΔH1 + ΔH2 + ΔH3 > 0, F14-1-1b), the dissolution may still occur if the increase in entropy is large enough
to make the process favorable. In such a case the dissolution process is endothermic, but still spontaneous; we call such
dissolutions entropy-driven.
When the latent interactions between the solute and solvent molecules are weaker than the existing intermolecular
interactions in the pure solvent and the pure solute that would need to be broken to produce the solution (ΔHsol = ΔH1 +
ΔH2 + ΔH3 >> 0), the dissolution will not take place. We then say that the substance is insoluble in the given solvent.
Solubility can be understood by considering the specific intermolecular interactions
made and broken during the dissolution
Consider a solution of cyclohexane (C6H12) in heptane (C7H16). Both hydrocarbons are liquids at 25 °C and are
completely miscible (they have no limit of solubility). London dispersion forces are the dominant intermolecular forces in
both liquids. The forces are stronger in heptane, as shown by its higher boiling point (98 °C for heptane vs. 80 °C for
cyclohexane). One would expect the addition of cyclohexane to heptane to be slightly endothermic, as the rod-like
heptane molecules have more contact areas that facilitate polarization, while cyclohexane molecules are more spherical
in shape, with limited contact area (Figure F14-1-2). The net effect is the lowering of the number of attractive interactions if
cyclohexane is added to heptane. Indeed, a 20% molar solution of cyclohexane has ΔHsol = +0.6 kJ/mol, and the
endothermic dissolution is entropy-driven.
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heptane
ethanol
cyclohexane
water
Figure F14-1-2. Structures of heptane, cyclohexane. ethanol and water (MEPs are on ±200 kJ scale). The immiscible
layers of water (bottom) and cyclohexane (top) in a test tube. The menisci of both liquids (pointed out by arrows) are
clearly visible.
The dissolution of cyclohexane in ethanol (CH3CH2OH) is more endothermic (ΔHsol = +2.8 kJ/mol for the 20%
solution), as the cost of breaking the hydrogen bonds in ethanol to accommodate molecules of cyclohexane goes up. This
enthalpy cost is still easily balanced by the entropy of mixing; cyclohexane and ethanol are miscible.
In contrast, cyclohexane is essentially insoluble in water (F14-1-2). The intermolecular interactions in water are
dominated by dipole-dipole interactions, specifically by hydrogen bonding, while dispersion forces are essentially the
only attractions between the molecules in cyclohexane. The two intermolecular forces are quite dissimilar in their nature;
the costs of breaking the interactions within the solvent (water) and the solute (cyclohexane) cannot be compensated by
any strong solvent-solute interactions. As a result, the two liquids are immiscible, and no solubility is observed.
Water is not inhospitable to all solutes. Solutes that have similar intermolecular forces to water are quite soluble.
Ethanol is completely soluble in water, as the dominant dipole-dipole and hydrogen bonding intermolecular interactions in
both are the same. Indeed, the mixing of ethanol and water is favorable and exothermic (ΔHsol = −10.0 kJ/mol).
Water has a special ability to dissolve ionic substances because it is polar. You should recall that ions are attracted
to each other by the strong electrostatic forces in ionic solids. When the substance is dissolved, water must be able to
provide sufficient ion-dipole attractive interactions to the dissolved solute to both compensate for the energy needed to
separate the ions, and also to break apart the hydrogen-bonded structure of water (at least locally, in the immediate
vicinity of the ions). We previously analyzed these ion-dipole attractions for Na+ and Cl−; the ions in solution are
surrounded by a solvation sphere of water molecules (F14-1-3). We often refer to the stabilizing interactions between the
solute(s) and the solvent as solvation, and solvation in water is called hydration, especially if the formation of structured
assemblies is involved.
Figure F14-1-3. Ion-dipole interactions for Na+(aq) and Cl−(aq) ions in water. The cation attracts the negative end of the
water dipole (oxygen), while the anion is hydrogen-bonded to H–OH groups.
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In the case of sodium chloride, dissolution in water is weakly endothermic (ΔHsol = +3.9 kJ/mol). Other ionic solids
may dissolve in water through even more endothermic entropy-driven processes, such as the dissolution of ammonium
nitrate (ΔHsol(NH4NO3) = +26.4 kJ/mol). Some ionic solids dissolve in exothermic processes, such as magnesium sulfate
(MgSO4) (ΔHsol(CuSO4) = −66.5 kJ/mol). Not all ionic solids are soluble in water; in some cases the ion-ion attractions are
so strong that the molecules of water cannot dislodge and solvate individual ions when they are part of the solid lattice.
Figure F14-1-4. Exothermic dissolution of CuSO4 (left) and endothermic dissolution of NH4NO3 (right). In the exothermic
process the heat released by the system (by the dissolving copper sulfate) is transferred to the water, increasing the
temperature of the solution. In the endothermic process, the heat needed to dissolve the ammonium nitrate (which is
entropically driven) is withdrawn from the water, lowering the temperature of the solution.
It is also not surprising that ionic solids are much less soluble in solvents that have significant organic residues (such
as hydrocarbon chains) that interact mainly through dispersion forces. For this reason, many ionic solids are not very
soluble in ethanol (CH3CH2OH), even though it is polar and has the ability to hydrogen bond. For example, 36 g of NaCl
will dissolve in 100 mL of water at 25 oC, but only 1.4 g will dissolve in 100 mL of methanol, and only 0.06 g will dissolve in
100 mL of ethanol at the same temperature. Ionic solids are totally insoluble in pure hydrocarbon solvents, such as
heptane or cyclohexane; the intermolecular forces are too dissimilar in the solute and the solvent.
Dissolution can be reversed through solvent removal without changes in solute identity
A final word must be said about the difference between dissolution and chemical reactions. In the dissolution process the
original solute can be recovered, chemically unchanged, after the solvent is removed. Even though the dissolution of ionic
solids leads to solvated individual ions (such as the solvated sodium and chloride ions formed in the dissolution of NaCl),
the original ionic solid is reformed after the solvent is evaporated. The same is true of liquid solutions; a liquid solvent can
be separated from a liquid solute by fractional distillation, where the mixture is held at the boiling point of one of the
components so that it is selectively evaporated leaving the other pure component behind.
In contrast, when iron metal is dissolved in hydrochloric acid, a chemical reaction takes place, giving us a solution
that contains Fe3+ and Cl− ions and producing hydrogen gas (C14-1-1). This is not a simple dissolution, it is a chemical
reaction:
2 Fe(s) + 6 HCl(aq) ⟶ 2 FeCl (aq) + 3 H (g)
3
2
E14-1-1
If we evaporate the water from the resultant solution, we obtain the ionic solid iron(III) chloride hexahydrate, FeCl3 •
6H2O, which may be further dehydrated into anhydrous FeCl3. These solids are chemically different from the original iron.
The chemical reaction that took place changed the chemical identity of the species involved.
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14-2 Electrolytes
Electrolytes are substances that dissociate into ions in solution,
which enables the solution to conduct electricity
When sodium chloride dissolves in water, we get a solution containing solvated Na+(aq) and Cl−(aq) ions, which move
freely within the liquid. Since flow of electricity simply requires the movement of charged particles, such a solution can
conduct electricity if two metal electrodes are immersed in it and a potential is applied. Ions carry the charge between the
electrodes and allow current to flow; an indicator light will light up if included in the circuit.
deionized water
tap water
acetic acid (aq)
ammonia (aq)
HCl (aq)
methanol (aq)
Figure F14-2-1. Electrical circuit set up to test conductivity of aqueous solutions. The relative magnitude of the flowing
electric current is shown by the position of yellow light on the 1-10 scale at the bottom of the digital measuring device.
Pure water ("deionized") does not have a significant number of ions present and does not conduct electricity. A solution
with a relatively small number of ions allows a limited current to flow. A solution with a large number of ions is fully
conductive.
In general, substances that dissolve in water and dissociate into ions are called electrolytes. A strong electrolyte is
one that dissociates completely in water to form ions. Substances that dissociate partially in water are called weak
electrolytes. In these substances, only a small fraction of the molecules dissociate into ions when dissolved in water; most
of the molecules of a weak electrolyte retain their molecular integrity. Substances that do not dissociate into ions at all are
called nonelectrolytes. Conductivity tests like the one above (F14-2-1) can determine to which class a substance belongs.
Ionic solids, strong acids, and strong bases are all strong electrolytes
that dissociate completely in solution
All ionic solids that are soluble in water are strong electrolytes. Upon dissolution, ionic solids dissociate completely into
ions, producing solutions with high conductivity. Ionic solids often consist of metal/non-metal combinations of elements
and are called salts. Sodium chloride (table salt) is one example of a soluble ionic solid that dissociates according to the
following equation, where the presence of water is implied by the aqueous phase designation (aq) in the products:
+
NaCl(s) ⟶ Na
(aq) + Cl
−
(aq)
C14-2-1
In ammonium salts, a protonated amine plays the role of the cation, combining with an appropriate anion. Such salts
are formed if the lone pair in ammonia or an organic amine bonds a proton. Ammonium salts dissociate into ions in
aqueous solution and are strong electrolytes. The simplest example of such a salt is ammonium chloride, NH4Cl.
Another class of ionic solids are the strong bases; these are defined as the hydroxides of the groups 1 (1A) and 2
(2A) elements, with the exception of Be(OH)2 and Mg(OH)2 (which are not strong bases or strong electrolytes). The strong
bases are all strong electrolytes, and dissociate completely in aqueous solution to produce hydroxide ions and the
appropriate metal cation. The group 2 hydroxides illustrate that a substance does not have to be very soluble to be a
strong electrolyte. For example, the solubility of calcium hydroxide, Ca(OH)2, is only about 0.17 g per 100 mL of water, but
the amount that is dissolved completely dissociates into ions.
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Some covalent compounds also dissociate in water. Acids, which are a class of compounds defined as proton
donors, dissociate in water, producing solvated protons in the form of H3O+(aq), called hydronium ions. The hydronium ion
is frequently represented just as H+(aq). Hydrogen chloride is a covalent substance, but when dissolved in water it
becomes hydrochloric acid that dissociates completely according to the following equation:
+
HCl(aq) + H O(l) ⟶ H O
2
3
(aq) + Cl
−
C14-2-2
(aq)
Acids that dissociate completely are called strong acids, and they are strong electrolytes. There are only seven
strong acids: hydrochloric acid (HCl), hydrobromic acid (HBr), hydroiodic acid (HI), perchloric acid (HClO4), chloric acid
(HClO3), nitric acid (HNO3) and sulfuric acid (H2SO4).
Weak electrolytes are molecules that dissociate partially when they dissolve
Any other substances that dissociate producing protons can do it only partially, i.e. only a small fraction of the molecules
are dissociated in solution. Such substances are called weak acids. Since the number of ions formed constitutes only a
small fraction of the dissolved substance, weak acids are also weak electrolytes. Examples of weak acids include
hydrofluoric acid (HF), phosphoric acid (H3PO4), hydrocyanic acid (HCN), formic acid (HCOOH), acetic acid (CH3COOH)
and other organic carboxylic acids. The weak acid acetic acid is commonly known as vinegar; it is only about 1%
dissociated, as shown in the following equation:
−
+
⇀
CH COOH(aq) + H O(l) −
↽
− CH COO (aq) + H O (aq)
3
2
3
3
C14-2-3
Notice that we use a different arrow notation for the dissociation of HCl (a strong acid) and the dissociation of acetic
acid (a weak acid). In the case of HCl the arrow points to the right, indicating that, for all practical purposes, all molecules
of HCl have dissociated into ions. There are no undissociated HCl molecules floating around—the process is complete. In
the case of acetic acid, we have two arrows, one pointing to the right as before, and another one pointing to the left. This
double-arrow symbol designates an equilibrium, where both the forward and the reverse process are taking place at the
same time. In such solutions there are both individual ions and undissociated molecules present, and they readily
interconvert. Some molecules of acid dissociate, and some ions of opposite charge come together to reform the neutral
molecule. This is an example of dynamic equilibrium, a phenomenon we have already encountered in our discussion of
vapor pressure, and which we will explore in more detail in Chapter 18.
Amines are another class of compounds that produces ions in equilibrium with undissociated molecules. Amines
are considered weak bases and weak electrolytes. The lone pairs of amine nitrogens are able to accept protons from
water in aqueous solution, forming the protonated amine (i.e. the ammonium cation) and a solvated hydroxide ion.
Protonated amines are weak acids. Ammonia offers the simplest example of such reaction:
+
−
⇀
NH (aq) + H O(l) −
↽
− NH (aq) + HO (aq)
3
2
C14-2-4
4
This reaction is written as an equilibrium, similarly to the deprotonation reaction of weak acids; both the forward and
reverse processes are taking place simultaneously. Ammonia is only partially protonated in water (about 1%); it is a weak
electrolyte. Other weak bases include organic amines such as methylamine (CH3NH2), triethylamine ((CH3CH2)3N),
aniline (C6H5NH2), and pyridine (C5H5N); all are weak electrolytes.
Nonelectrolytes retain their molecular structure in solution
Substances that maintain their molecular integrity when they dissolve and do not dissociate into ions are called
nonelectrolytes. Most covalent compounds, with the exception of acids and weak bases, do not dissociate into ions when
dissolved in water. Such covalent nonelectrolytes include many groups of organic molecules, such as alcohols (for
example methanol and ethanol), simple ketones (acetone, for instance), aldehydes (for example formaldehyde), and
carbohydrates such as table sugar or glucose (C6H12O6).
C H
6
12
⇀
O (s) + H O(l) −
↽
− C H
6
2
6
12
O (aq) + H O(l)
6
2
C14-2-5
What happens when a substance dissolves in a solvent affects the properties of the resulting solution. Some of
these properties (for example conductivity) depend on the identity and the number of species in the solution, but in many
cases just the number of solute particles (molecules or ions) that exists in the solution determines the solution's
characteristics. In either case, the ratio of solute particles to solvent molecules is expressed by various measures of
concentration, some of which we will explore in the next Lesson.
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14-3 Concentration
Mass fractions, mole fractions, and molarity are the most common ways to express
concentration
of solutes in solvents
The concentration of a solution describes the amount of solute in relation to the amount of solvent or solution. It may be
quantitatively expressed in various ways.
Mass fraction is the ratio of the mass of solute per total mass of the solution (Equation E14-3-1). Usually, the mass
fraction is multiplied by 100 to give mass percent or weight percent of the solute. If the fraction is very small, it is more
convenient to use parts per million (ppm) or parts per billion (ppb) instead of percent (parts per hundred). You may recall
that we introduced these fractions when we talked about mixtures of atmospheric gases. In the case of gas solutions,
they refer to ratios of volumes (or moles) rather than to ratios of masses (which are used for liquid and solid solutions).
mass f raction =
mass of the component
× 100 = mass% = weight%
E14-3-1a
total mass
mass f raction =
mass of the component
× 10
6
= ppm
E14-3-1b
= ppb
E14-3-1c
total mass
mass f raction =
mass of the component
× 10
9
total mass
Mole fraction was also previously introduced, as a measure of the concentration of gases. Molar fractions are
usually used for gas solutions, but also have limited applications in condensed phase solutions. "Xi" is the symbol used for
the mole fraction of the component of interest as compared to the total moles in the solution (E14-3-2):
Xi = molef raction =
moles of the component
E14-3-2
total moles
Molarity is the most commonly used concentration measure for liquid solutions; molarity is, defined as the number of
moles of solute per 1 L of solution and is given units of upper case M, which represents mol/L (E14-3-3). A 1 M solution is
called a “1 molar solution.”
M = molarity =
moles of solute
E14-3-3
volume (L) of solution
Depending on convenience or tradition, chemists may routinely use many different expressions of concentration. For
example, in the previous section we specified the solubility of some substances in g per 100 mL of solvent—yet another
way to convey concentration. The ability to convert between various concentration measures is an important skill,
requiring some practice. The scheme in Figure F14-3-1 outlines the connections between the quantities involved.
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Concentration conversions and dilution calculations are routinely based on molar quantities
Figure F14-3-1. Scheme illustrating connections between the various quantities involved in calculating concentrations.
Let's illustrate various ways to convert concentrations. We will use a concentrated solution of hydrochloric acid as
our example, which is 36.0% HCl by weight and has a density of 1.18 g/mL. The figure of 36.0% corresponds to the mass
percent measure of concentration (E14-3-1). A 100-g sample of this solution contains 36.0 g of HCl and 64.0 g of water,
which corresponds to 0.987 moles of HCl (36.0 g divided by 36.46 g/mol) and 3.55 moles of H2O (64.0 g divided by 18.01
g/mol). Thus, the molar fraction (E14-3-2) of HCl in the solution is XHCl = (0.987/(0.987+3.55)) = 0.218. A 1 L (1000 mL)
sample of this solution weighs 1.18 kg. Since 36.0% of that mass is HCl, the 1 L volume contains 425 g or 11.6 moles of
HCl. Thus, the molarity of the solution is 11.6 mol/L (E14-3-3).
Often, chemists need to prepare dilute solutions from a more concentrated solution (often called the "stock"
solution). Let's say we want to prepare 1.00 L of 1.00 molar (1 M) HCl solution using the HCl solution from our example
above. Calculations of such dilutions are based on the idea that the number of moles of the solute must remain constant
when the sample is diluted with solvent (water). The number of moles of solute is calculated by multiplying the molarity of
the solution by the volume (n = MV). In dilutions, the number of moles must be the same before and after dilution (E14-34).
E14-3-4
n = Minit Vinit = Mf inal Vf inal
In our hydrochloric acid example, Minit = 11.6 mol/L, Mfinal = 1.00 mol/L, Vfinal = 1.00 L, and simple arithmetic gives
us Vinit = 0.0862 L. We need to dilute 86.2 mL of our stock solution with water to reach a total volume of 1.00 L in order to
obtain a 1.00 M solution of HCl.
Note that in the manipulation of chemical solutions two different processes can take place: transfer, or dilution. The
transfer of a solution involves moving a given volume of solution from container number 1 to container number 2. The
volume and the number of moles of solute may change, but the concentration does not change (V1 ≠ V 2, n1 ≠ n2, M1 =
M2). The unchanging value of molarity allows the calculation of either the volume or number of moles in the second
container. In contrast, the dilution of a solution involves the addition of water (or solvent) to one container of the solution,
which does not affect the number of moles of solute present. The volume and the concentration of the solution changes,
but the number of moles of solute does not change before and after the dilution (V1 ≠ V2, M1 ≠ M2, n1 = n2). The
unchanging number of moles of solute is the source of equation E14-3-4 above.
In the case of strong electrolytes, we know that these compounds dissociate completely into ions in solution, and the
concentration of each ion in the final solution can be calculated by using the stoichiometry of the dissociation process. In
order to calculate the concentration of chloride ions in 1 L of 2.0 M solution of calcium chloride, we must first write the
dissolution process:
CaCl (s) ⟶ Ca
2
2 +
(aq) + 2 Cl
−
(aq)
C14-3-1
From the balanced reaction we can see that there are two mole of chloride ions formed for every one mole of
calcium chloride dissolved. In 1 L of a 2.0 M solution of CaCl2 there are 2 moles of CaCl2. Therefore, there are four moles
of dissociated chloride ions in the 1 L volume, and the concentration is 4 mol/1 L = 4.0 M chloride ions.
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14-4 Solubility
Saturated solutions contain the maximum amount of solute that will dissolve,
dilute solutions contain less, and supersaturated solutions contain more than this amount
All gases and some liquids are completely miscible (soluble in each other without limits). The atmosphere supplies an
illustration of the first case, and we presented a couple of examples of miscible liquids in the previous section. What
these cases have in common is that they do not involve dissimilar intermolecular interactions; the IMFs are practically
non-existent in gases, and they are very similar in both the solvent and the solute for miscible liquids. However, in many
cases, especially for solid-state solutes, the strength and type of the intermolecular interactions impose limits on solubility.
Returning to our classical example of an ionic solid dissolving in water, we notice that at room temperature (25 °C)
only 36 g of NaCl dissolves in 100 mL of water. When we add solid NaCl to water (F14-4-1), initially the water molecules
extract individual ions from the crystalline solid, solvating each one in turn as previously described. As the number of
solvated ions increases, so does the probability of some of them colliding with the surface of undissolved crystals and
being recaptured.
Figure F14-4-1. Dissolution of NaCl in water. The individual ions are carried off the surface of the crystal and solvated,
leading to the dissolution of the solid. A dynamic equilibrium is reached when enough solid is added so that the dissolution
and crystallization rates are the same. At this point the solution is saturated and adding more solid will cause it to
precipitate.
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The rebuilding of the crystal is called crystallization. At a certain point a dynamic equilibrium is achieved, with the
rates of dissolution and crystallization being equal (C14-4-1). At this point we have a saturated solution; no more solute
will dissolve in the available amount of solvent at that temperature. That maximum amount of solute that will dissolve in a
given solvent and temperature is called its solubility, i.e. solubility values are equilibrium values for saturated solutions in
contact with some undissolved solid.
Under certain conditions it is possible to obtain a supersaturated solution, which contains more of the solute than its
solubility limit at the given temperature. Since crystallization requires precise orientation of molecules in a 3D
arrangement, the formation of the first crystals (sometimes called seed crystals) may be slow. Such solutions are
unstable, and any perturbation (such as the addition of an external seed, a speck of dust or another solid impurity) may
spark rapid crystallization of the excess dissolved solute (F14-4-2).
Figure F14-4-2. Rapid crystallization (real speed) of a
supersaturated solution of sodium acetate. The saturated
solution is made at 100 °C. It contains 170 g of solute per 100
mL of water. It is slowly cooled to 25 °C where the solubility is
only 46 g per 100 mL. Thus, there is 124 g (per 100 mL) of
excess sodium acetate above its solubility limit at 25 °C. The
sodium acetate rapidly crystallizes when a seed crystal is added.
We can now reexamine some of the descriptive adjectives we used to qualitatively describe concentrations of
solutions. The solubility of a substance is equivalent to the concentration of the saturated solution at the given conditions
of temperature and pressure. A concentrated solution is close in concentration to the saturated solution, while a dilute
solution is on the other end of the concentration range. Both of those terms are strictly relative, and can be meaningfully
applied to a specific substance, or used to compare substances of similar solubility. For example, at 25 °C, a saturated
solution of NaCl has a solubility of 5.4 mol/L, while a saturated solution of cesium sulfate has a solubility of only 0.06
mol/L, a concentration that would be considered "dilute" for sodium chloride.
“Like dissolves like” is the empirical pattern that polar solutes typically dissolve
in polar solvents and nonpolar solutes typically dissolve in nonpolar solvents
Solubility is strongly influenced by the type and strength of the intermolecular interactions, especially between the solute
and solvent molecules: the stronger the interactions, the greater the solubility. We previously noted these trends when we
discussed the enthalpy of the dissolution process. We noted, for example, that simple alcohols like ethanol are
miscible with water. A closer examination shows that the solubility of an alcohol drops off as the hydrocarbon chain
grows longer (T14-4-1). The London dispersion interactions become more and more dominant over the hydrogen-bonding
forces with increasing chain length, and the alcohol becomes less similar to water and less soluble.
Table T14-4-1. Solubility of alcohols in water and cyclohexane.
Alcohol
CH 3OH
CH 3CH2OH
CH 3CH2CH2OH
CH 3CH2CH2CH2OH
CH 3CH2CH2CH2CH2OH
CH 3CH2CH2CH2CH2CH2OH
CH 3CH2CH2CH2CH2CH2CH2OH
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Solubility in
water
Solubility in
cyclohexane
in g per 100 mL
in g per 100 mL
∞
∞
∞
8.06
2.82
0.62
0.17
3.84
∞
∞
∞
∞
∞
∞
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Dissolution of alcohols in hydrocarbon solvents follows the opposite trend (Table 14-4-1). Long chain alcohols are
miscible with cyclohexane, as dispersion forces are dominant in both the solvent and the solute. However, the solubility
diminishes for methanol with only one CH3 group attached to the OH group. You may recall that cyclohexane is insoluble
in water, and that methanol is the alcohol most similar to water. One can imagine what kind of structural changes we
would need to make to cyclohexane to make the modified molecule soluble in water. Indeed, such modified molecules
already exist; glucose is essentially a hydrocarbon that has -OH groups covering the whole molecular surface (F14-4-3).
The intermolecular forces within glucose and water are thus very similar. It is not surprising that glucose is very soluble in
water (91 g per 100 mL at 25 oC).
water
glucose
acetone
acetic acid
DMSO
Figure F14-4-3. Structures of water, glucose, acetone, acetic acid, and dimethylsulfoxide, showing the similarities in the
polarities of their functional groups. Glucose is very soluble in water, and acetone, acetic acid, and DMSO
(dimethylsulfoxide) are miscible with water. All MEPs are at ±200 kJ scale.
In general, molecules that have similar polarity to water are expected to be soluble in water. For example, acetone,
acetic acid, and dimethylsulfoxide (all liquids at 25 °C) are completely miscible with water (F14-4-3). Their ability to form
hydrogen bonds also heavily influences solubility in water.
The examination of multiple examples of solvent-solute interactions gives us the helpful rule, "like dissolves like".
Thus, ionic and polar substances are more likely to be soluble in polar solvents, while nonpolar solvents will most likely
dissolve nonpolar solutes. If the intermolecular interactions broken and formed upon mixing are approximately the same,
dissolution is probable.
Figure F14-4-4. Illustration of "like dissolves like." Iodine,
I2, a nonpolar solid, dissolves in (d) a nonpolar solvent
(heptane), but is insoluble in (b) a polar solvent (water).
Copper sulfate pentahydrate (CuSO4 · 5H2O), an ionic
solid, dissolves in (a) water (a polar solvent), but is
insoluble in (c) heptane (a nonpolar solvent)
The solubility of a gas diminishes with rising temperature
and is proportional to the gas’s partial pressure above the solution
The solubility of a substance is different under different conditions. The solubility of gases is affected by pressure, and the
solubility of all substances depends on temperature. In general, the solubility of a gas in a solvent depends on the partial
pressure (Pg) of that gas above the solution; increases in Pg lead to increases in solubility. At equilibrium, at a given
pressure, the rates of gas molecules entering and leaving the solution are the same. Increasing the partial pressure of the
gas above the solution increases the rate at which gas molecules strike the liquid surface. On average more molecules
enter the solution than escape it, and so a new equilibrium is established with a higher concentration of gas molecules
dissolved in the solution. This relationship is described by Henry's law:
Cg = kPg
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Figure F14-4-5. Henry's law for the solubility of several gases in water at 25 °C. The slopes of the lines correspond to the
values of the Henry's law constants (k). k increases with increasing intermolecular forces, which, for molecules shown
here, are limited to dispersion forces. An exception is CO, which has a small dipole moment and exhibits somewhat
stronger interactions than N2. The increasing solubility of gases under higher pressure is used to prepare carbonated
beverages. High-pressure CO2 is used to fill the bottles with extra gas, which rapidly escapes when the bottle is opened.
Henry’s law states that the concentration of a gas in solution, Cg (usually expressed as molarity), is directly
proportional to the partial pressure of the gas above the solution (Pg). The value of the proportionality constant k, known
as the Henry's law constant, depends on the solute, solvent and temperature (see below). The Henry's law plots for
several gases are shown in Figure F14-4-5.
As a general trend, the solubility of gases decreases with increasing temperature (F14-4-6). This phenomenon is
commonly observed when carbonated beverages go flat when left open at room temperature. The solubility of the carbon
dioxide gas is lower at room temperature, so the CO2 escapes the liquid. Similarly, the solubility of O2 in water diminishes
with temperature, making the oxygenation levels needed for living organisms difficult to maintain in the warm waters of
lakes or streams.
Figure F14-4-6. Solubility of gases in water as a function
of temperature. For all gases, the solubility diminishes
with increasing temperature. Solubility correlates with
molecular sizes (dispersion forces) and is enhanced
somewhat for weakly polar molecules (CO, NO).
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The solubility of ionic compounds frequently increases with temperature
but some counterexamples exist
Figure F14-4-7. Solubility of ionic compounds as a
function of temperature. For most substances the
solubility increases with the temperature, but there are
multiple exceptions to the general trend.
The solubility of liquid and solid solutes generally increases with temperature (F14-4-7), in some cases quite
dramatically (see for example KNO3). However, there are many exceptions to that trend (see, for example, CdSeO4.
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15 Chemical Reactions
As we have learned over the last few lessons, intermolecular interactions affect phase transitions and other
properties of condensed phases, but do not change the participating molecules. Now we are ready to explore processes
in which molecules change their identities. It turns out that even stable molecules may engage in even stronger types of
interactions in which electrons are transferred, bonds are broken and remade, and energy is exchanged with the
surroundings. In such processes, called chemical reactions, reactants (the initial participants) are transformed into
products (new chemical species). The only constant in such reactions are atoms, whose number and type remain
unchanged even as their bonding arrangements are reconfigured.
15-1 Reaction types
Reactions are divided into categories depending on their outcome. The categories we introduce include combination,
decomposition, combustion, single displacement and double displacement. Chemical reactions are described by chemical
equations, listing reactants on the left and products to the right of the reaction arrow. The stoichiometric coefficients are the
numbers in front of reactants and products that express the molar ratios in which they participate in the reaction. In order to obey
the law of conservation of mass, the stoichiometric coefficients must be such that the number of each type of atom is the same on
both sides of the equation. Equations with coefficients that satisfy this condition are called balanced equations, and they constitute
the basis of all stoichiometric calculations.
15-2 Driving force
For a reaction to proceed as indicated by its equation, there must be some driving force. This driving force might be a
lowering of the system’s enthalpy (this happens when products are more stable than reactants), or an increase of entropy (when
products are statistically more likely than reactants). The driving force of double exchange reactions is most commonly manifested
in the formation of a precipitate, the formation of a weak electrolyte or nonelectrolyte, or the formation of a gas.
15-3 Redox reactions
Redox reactions are chemical processes in which electrons are transferred between reactants, resulting in reduction of one
and oxidation of the other. The reactant that loses electrons is oxidized, and the reactant that gains electrons is reduced. The
transferred electrons are accounted for by assigning oxidation states (or oxidation numbers) to each element in reactants and
products. In single exchange redox reactions the driving force is the transfer of electrons from metals of higher activity to metals of
lower activity, as ordered in the metal activity series.
15-4 Stoichiometry
Stoichiometry deals with relative quantities of reactants and products in chemical reactions. It is based on the stoichiometric
coefficients of balanced chemical equations and the proportionality between molecules and moles. It is used to determine the
amounts of reactants and products of a given reaction in mass, moles, or volumes. The stoichiometric calculations are used to
identify the limiting reactant, which is the reactant that determines the maximum theoretical yield of products. The amount of
product actually formed is commonly expressed as percent yield, which is calculated by dividing the actual yield by the theoretical
yield.
15-5 Example calculations
Stoichiometric calculations are applicable to all types of reactions, including combustion, elemental analysis, redox reactions,
solution reactions, titrations, and reactions of gases utilizing the ideal gas law.
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15-1 Reaction types
Combination, decomposition, combustion, and single and double displacement
are the common types of chemical reactions
A chemical reaction is a process that transforms one set of molecules, called reactants, into a different set of molecules,
called products. Chemical reactions encompass changes in the apportionment of bonding and lone-pair electrons
between atoms, and the resulting energy changes. The nuclei remain unchanged, preserving the atoms' identities.
Elements can only undergo transmutation into a new element in nuclear reactions, which will not be explored until the
second semester of general chemistry.
There are a wide variety of chemical reactions, from the relatively simple, such as the production of carbon dioxide
bubbles when an Alka-Seltzer tablet dissolves in water , to the complex reactions that take place in our eyes and brain
as we read this sentence. Some go almost unnoticed, like the constant oxidation reactions in our bodies, providing energy
that allows us to function. Others may be spectacular, like the 4th of July fireworks, which release energy in the form of
light, heat, and acoustic waves for our entertainment. All reactions can be represented by chemical equations, which are a
form of symbolic notation that captures the essence of the changes taking place. We will only explore a very small subset
of reactions to illustrate some basic concepts of chemical reactivity. Even with a limited scope, it helps to categorize
reactions into several groups that describe the net change. These categories are listed below with illustrative examples.
Combination: elements react to form compounds, or small compounds combine to make larger ones
2 Mg(s) + O (g) ⟶ 2 MgO(s)
C15-1-1
CaO(aq) + CO (g) ⟶ CaCO (s)
C15-1-2
2
2
3
Decomposition: a compound breaks apart into two or more smaller compounds
Δ
C15-1-3
PbCO (s) −
→ PbO(s) + CO (g)
3
2
Combustion: a compound (usually organic) reacts (usually violently) with oxygen, O2(g); complete combustion
means that the highest-oxidation state oxides are produced (CO2 for carbon, H2O for hydrogen).
C15-1-4
2 CH OH(g) + 3 O (g) ⟶ 2 CO (g) + 4 H O(g)
3
2
2
2
Single displacement: one element replaces another in a compound
C15-1-5
CuSO (aq) + Zn(s) ⟶ ZnSO (aq) + Cu(s)
4
4
Double displacement (metathesis): exchange of ions or atoms between two compounds
CaCl (aq) + 2 AgNO (aq) ⟶ 2 AgCl(s)↓ + Ca(NO ) (aq)
C15-1-5
HNO (aq) + KOH(aq) ⟶ KNO (aq) + H O(l)
C15-1-7
Na S(aq) + 2 HCl(aq) ⟶ H S(g)↑ + 2 NaCl(aq)
C15-1-8
2
3
3
3
2
3
2
2
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Chemical equations follow certain conventions to standardize their meaning. The reactants are always listed on the
left. The arrow translates to "react" and points to the products. You may recall from our previous encounter with such
symbols, that an arrow pointing to the right signifies a reaction that essentially goes in one direction; products are formed
from the reactants, and there is no significant reaction in the opposite direction. The chemical formulas of the reactants
and products are often supplemented by designation of their physical state in parentheses (g meaning gas, ℓ meaning
liquid, s meaning solid, and aq indicating an aqueous solution). For ionic, metallic, or extended solids the elemental
symbols or empirical formulas are used, sometimes specifying the allotropic form in parentheses (for example, K, NaCl,
C(graphite), etc). Occasionally, some additional symbols may be included such as "Δ" above the reaction arrow to indicate
heat has been added, an upward pointing arrow after the formula (↑) to indicate gas formation, or a downward pointing
arrow (↓) after the formula to indicate precipitation of a solid.
Balanced chemical equations document that no atoms are created or destroyed in chemical
reactions
Although listing the reactants and products on opposite sides of the arrow identifies the reaction unambiguously, the
complete equation must still be balanced. A balanced equation allows us to carry out the stoichiometric calculations
that describe the relative amounts of reactants and products, and to determine the percent yield of the products. A
balanced equation is a reflection of the law of conservation of mass; the number of atoms of each element must be equal
on both sides of the equation—no atoms can be created or destroyed in regular chemical reactions (nuclear reactions
notwithstanding). Balancing an equation can be achieved only by changing the stoichiometric coefficients (the numbers in
front of each chemical formula). Altering subscripted numbers within a formula changes the identity of the molecule, and
therefore changes the entire reaction rather than balancing it.
To balance a chemical equation we follow a systematic approach, balancing one element at a time and adjusting as
necessary. For example, the combustion of ethane (C15-1-9) produces carbon dioxide and water. It is unbalanced when
written with coefficients of 1 in front of each formula. For instance, there are 2 carbon atoms in the reactants but only 1
carbon atom in the products.
C15-1-9
CH CH (g) + O (g) ⟶ CO (g) + H O(g)
3
3
2
2
2
Indeed, we can start the balancing with carbon: we have 2 carbon atoms among the reactants on the left, so we
must also have 2 carbon atoms in the products. We need to place a "2" in front of CO2, the only carbon-containing
product. We can then move on to the hydrogen atoms; we have 6 of them on the left, and all must be included in water,
which is the only hydrogen-containing product. We need to put a “3” in front of H2O on the right to account for those 6
hydrogen atoms. The last element to balance is oxygen; we have total of 7 oxygen atoms on right side of the equation (2 x
2 in carbon dioxide and 3 in water). We need the same number on the left, but the oxygen atoms come in sets of two
since dioxygen is the reactant; we must have a coefficient of 7/2 in front of O2 (C15-1-10).
CH CH (g) +
3
3
7
2
O (g) ⟶ 2 CO (g) + 3 H O(g)
2
2
2
C15-1-10
Although C15-1-10 is balanced, chemists often prefer to use the smallest possible whole number coefficients instead
of fractions. Whole number coefficients can be obtained by multiplying all coefficients on both sides of the equation by two
(C15-1-11). The ability to multiply a balanced equation in this way illustrates the meaning of stoichiometry: it gives the
relative ratio of atoms or molecules involved in a reaction, with a focus on proportions and not on specific absolute
amounts. We will return to this concept later in the lesson.
2 CH CH (g) + 7 O (g) ⟶ 4 CO (g) + 6 H O(g)
3
3
2
2
2
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Figure F15-1-1. Stoichiometry of the combustion reaction of ethane. The balanced equation uses the smallest whole
number stoichiometric coefficients to reflect the whole numbers of molecules participating in the reaction.
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15-2 Driving force
Reactions of electrolytes are presented as net ionic equations with spectator ions omitted
Consider a double-exchange reaction between lead nitrate and potassium iodide carried out in water. The two compounds
are about to exchange their cations, creating lead iodide and potassium nitrate. The molecular equation, a balanced
equation showing the complete chemical formulas without indicating the ionic nature of the reactants and products, is
shown below.
Pb(NO )
3
2
+ 2 KI ⟶ PbI
2
C15-2-1
+ 2 KNO
3
However, both reactants are ionic solids and strong electrolytes and are completely dissociated into hydrated ions,
as should be indicated thusly (C15-2-2):
2 +
Pb
−
(aq) + 2 NO
3
+
(aq) + 2 K
−
(aq) + 2 I
C15-2-2
(aq) ⟶
At the starting point, we formally have four different kinds of ions floating in solution. Dissolving NaCl in water yields
two kinds of ions (Na+ and Cl−), which will reconstitute as NaCl(s) if the water is evaporated. Following this logic, if there
is no reaction, after the evaporation of water, we would get a mixture of all four ionic solids: two reactants, and two
products; that would be quite a messy exchange. However, when we actually pour a solution of potassium iodide into the
solution of lead nitrate, a yellow precipitate of lead iodide, PbI2, forms (F15-2-1).
Figure F15-2-1. Precipitation of lead iodide (PbI2) that is
created in a double exchange reaction upon the addition of
drops of potassium iodide to a solution of lead nitrate
It turns out that lead iodide is essentially insoluble in water. Taking the precipitation of one of the products into
account, we can now complete equation C15-2-2 as follows:
2 +
Pb
−
(aq) + 2 NO
3
+
(aq) + 2 K
−
(aq) + 2 I
(aq) ⟶
+
PbI (s)↓ + 2 K
2
−
(aq) + 2 NO
3
C15-2-3
(aq)
The equation above is a complete ionic equation showing all reactant and product ions and the insoluble solid of
PbI2 that separated from the solution. A closer examination of the equation indicates that we have some of the same ions
on both sides of the equation (K+ and NO3−). These ions actually do not participate directly in the reaction; they are
spectator ions. We can remove them from both sides (like in an algebraic equation), giving us the net ionic equation:
2 +
Pb
−
(aq) + 2 I
(aq) ⟶ PbI (s)↓
2
C15-2-4
Notice that the net ionic equation is still balanced. Charges on ions are balanced as well; the sum of the ionic
charges must be the same on both sides of the equations.
The net ionic equation is especially important: it shows only those species that actually take part in the reaction and
the chemical change that takes place. For example, from the net ionic equation above, we can tell that to make solid PbI2,
we need a source of Pb2+(aq) and I−(aq) ions. We will obtain the same result by using HI(aq) and lead acetate, for
example.
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The driving forces for exchange reactions include formation of a precipitate, gas,
weak or nonelectrolyte, and redox activity
This example demonstrates that for double exchange to work, it needs some kind of driving force that will assure that
products (as written in the chemical equation) actually form. Normally, the most obvious driving force is an exothermicity
of the process (ΔH < 0), i.e., the product stability exceeding that of the reactants. For example, the products of combustion
reactions of hydrocarbons, carbon dioxide and water, are much more stable than the reactants, hydrocarbons and oxygen.
Another driving force in reactions can be a favorable entropy change, analogous to the dissolution process. We will
explore these thermodynamic aspects of reactions in the next Chapter and in the second semester of general chemistry.
The driving forces at work in exchange reactions that we consider here are:
formation of a precipitate
formation of a gas
formation of a weak electrolyte or nonelectrolyte
redox (activity series)
You may observe that in the first three categories a product is removed from the reaction phase. Insoluble solids
separate from the aqueous phase, gases escape the solution, and the formation of nonelectrolytes (often water) involves
converting ionic species into covalent compounds. In all of these cases at least one of the products is removed from the
reaction mixture, driving the reaction to completion. In the reduction-oxidation (redox) processes, electrons are transferred
from a higher energy state in a more active metal to a lower energy state in less active metals.
To be able to predict the outcome of a double exchange reaction, we need to recognize which type of the driving
force is present. For example, for precipitation reactions, we need to know whether any of the ions involved in the reaction
form an insoluble salt. It turns out that the solubility of many salts follows similar patterns, which may be generalized into a
set of widely applicable guidelines (Table T15-2-1).
Table T15-2-1 Solubility guidelines for ionic solids in water
soluble
Guidelines
Exceptions
ammonium (NH 4+)
alkali metals (Li +, Na+, K+, Rb+, Cs+)
nitrates (NO 3−)
acetates (CH 3COO−)
perchlorates (ClO 4−)
halides (Cl −, Br−, I−)
sulfates (SO 42−)
Ag +, Hg22+. Pb2+
Ca 2+, Sr2+, Ba2+, Ag+, Hg22+, Pb2+
sulfides (S 2−) and hydroxides (HO−)
insoluble
carbonates (CO 32−) and phosphates (PO43−)
NH 4+, Li+, Na+, K+, Rb+, Cs+
Ca 2+, Sr2+, Ba2+
NH 4+, Li+, Na+, K+, Rb+, Cs+
A second driving force for exchange reactions is the formation of a weak electrolyte or a nonelectrolyte. This driving
force is most often encountered in acid-base neutralization reactions, wherein an acid reacts with a base to produce a salt
and water. For example, the molecular equation for a reaction between hydrochloric acid and sodium hydroxide is shown
in C15-2-5:
C15-2-5
HCl(aq) + NaOH(aq) ⟶ H O(ℓ) + NaCl(aq)
2
When the molecular equation is rewritten as a complete ionic equation, we can see that chloride ion and sodium ion
are spectator ions:
+
H
(aq) + Cl
−
+
(aq) + Na
−
(aq) + OH
+
(aq) ⟶ H O(ℓ) + Na
2
(aq) + Cl
−
(aq)
C15-2-6
The net ionic equation makes the key transformation more obvious; the net change is the formation of water by a
reaction between H+ and HO−:
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+
H
−
(aq) + OH
C15-2-7
(aq) ⟶ H O(ℓ)
2
Equation C15-2-7 represents the main feature of a neutralization reaction between any strong acid and any strong
base. The formation of a covalent nonelectrolyte molecule (water) from ions (H+ and HO−) provides the driving force for
this type of reaction by removing a pair of ions from the solution.
Formation of a gas is another driving force. In these types of reactions a gas is produced that escapes the solution
due to its limited solubility in water. For example, hydrogen sulfide gas forms when hydrochloric acid reacts with sodium
sulfide, as shown in the molecular equation C15-2-8.
2 HCl(aq) + Na S(aq) ⟶ H S(g)↑ + 2 NaCl(aq)
2
2
C15-2-8
The net ionic reaction (C15-2-9) resembles the neutralization reaction of equation C15-2-7, although sulfide ion is a
weak base. The product is a weak electrolyte, like water. It also happens to be a gas; it escapes the solution, driving the
reaction to completion.
+
2H
2 −
(aq) + S
C15-2-9
(aq) ⟶ H S(g)↑
2
Other gases often produced in exchange reactions include CO2, NO2, or SO2. Sometimes a weak acid is produced
first, with decomposition into gas in a subsequent step. For example, the reaction of hydrochloric acid with sodium
hydrogen carbonate (baking soda) produces carbonic acid (C15-2-10). This weak acid is a weak electrolyte, which
decomposes to produce carbon dioxide gas and water (C15-2-11).
HCl(aq) + NaHCO (aq) ⟶ H CO (aq) + NaCl(aq)
C15-2-10
H CO (aq) ⟶ CO (g) + H O(ℓ)
C15-2-11
3
2
3
2
2
3
2
The net ionic equation is obtained by combining these two steps, and removing the spectator ions and carbonic acid
(which is only an unstable intermediate):
+
H
−
(aq) + HCO
3
(aq) ⟶ CO (g)↑ + H O(ℓ)
2
2
C15-2-12
The exchange reactions presented above are all very similar; they can be thought of as acid-base neutralization
reactions. Their products are all weak electrolytes or nonelectrolytes, and are all covalent molecules that exist as gases or
liquids.
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15-3 Redox reactions
In a redox reaction one reactant gets oxidized, losing electrons,
while another reactant gets reduced, gaining electrons
A redox reaction is a type of reaction where oxidation and reduction of the reactants take place. When an atom, an ion, or
a molecule has lost one or more electrons, we say it has been oxidized.
Figure F15-3-1. Transfer of electrons in a redox reaction
The name oxidation is used because the first reactions of this
type studied involved oxygen, yielding oxygen-containing products.
Oxygen is an excellent oxidizing agent, as we can see by the
formation of rust on iron bridges parts, or the formation of green
copper oxide on copper roof tiles after long exposure to oxygen in the
air. However, the oxidation is only part of the process; when one
chemical entity loses electrons, another must gain electrons. Such
electron transfer from one species to another occurs in every redox
reaction. The species that gains electrons is said to be reduced.
Since reduction cannot occur without oxidation and vice versa, we can say that each species is acting as an agent of
the change that the other species undergoes. An oxidizing agent (or oxidant) oxidizes something else, but in the process
is itself reduced (it gains electrons from the other reactant). A reducing agent reduces something else, but in the process
is itself oxidized (it loses electrons to the other reactant).
The oxidation of magnesium is a fantastic example of a redox process. Shiny, silver-colored magnesium metal can
be ignited in air, creating a bright flame, as it is converted into a dull, gray magnesium oxide powder (C15-3-1).
2 Mg(s) + O (g) ⟶ 2 MgO(s)
2
C15-3-1
In this reaction each magnesium atom loses two electrons, and each oxygen atom gains two electrons. Thus,
oxygen is reduced and magnesium is oxidized; oxygen is the oxidizing agent, and magnesium is the reducing agent.
Oxidation numbers are used to account for electrons gained and lost
by atoms participating in redox reactions
To keep track of the gained and lost electrons we assign oxidation numbers (or oxidation states) to each atom in the
reaction. Oxidation numbers are assigned based on the difference between the number of valence electrons the neutral
atom would have, and the number of electrons assigned to it in a Lewis structure. The number of electrons assigned to
the atom includes all lone-pair electrons, plus half of the electrons shared with atoms of equal electronegativity, plus all
electrons shared with atoms of lesser electronegativity. In this model, for the purpose of determining the oxidation
numbers, the bonds between different atoms are treated as ionic; the atom with greater electronegativity is awarded all
electrons in a bond.
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Based on this rule, species in elemental form always have oxidation numbers equal to zero. Any monatomic ion has
an oxidation number equal to its charge. In compounds, nonmetals (as being more electronegative) usually have negative
oxidation numbers, but exceptions are known. Oxygen is usually −2 (with the exception of peroxides, where it is −1).
Hydrogen is usually +1 when bonded to nonmetals, and −1 when bonded to metals (when it is called a hydride). The
oxidation number of fluorine (the most electronegative element) is always −1 unless it is in elemental form. Other halogens
are commonly −1 as well, but in compounds with more electronegative elements they have positive oxidation numbers.
The sum of the oxidation numbers of all atoms must always add up to the overall charge of the species.
Using the guidelines above, we can rewrite our redox reaction from equation C15-3-1 while explicitly showing
oxidation numbers:
C15-3-2
The metallic magnesium and molecular oxygen both have an oxidation number of 0. In the MgO product the
magnesium ion (Mg2+) has a +2 oxidation number, while the oxide ion (O2−) has a −2 oxidation number. The overall
reaction can be divided into two half-reactions that separately describe the oxidation and reduction processes:
2 +
Mg(s) ⟶ Mg
−
O (g) + 4 e
2
C15-3-3
−
+ 2e
C15-3-4
2 −
⟶ 2O
The overall redox reaction (C15-3-2) is the sum of the two half reactions. However, in order for the overall redox
reaction to be balanced, the overall number of electrons lost and gained in the half-reactions must be the same. We
cannot lose only 2 electrons from magnesium and gain 4 electrons on oxygen. We must multiply all coefficients in the
oxidation half-reaction (C15-3-3) by 2 before we add the half reactions to get the overall redox reaction.
The activity series organizes metals in order of their reactivity with water
or their ability to lose electrons
In redox processes, the driving force for the reaction is based on the products being of lower energy than the reactants. In
single exchange reactions, this means that electrons are transferred from a higher energy state in a more active metal to a
lower energy state in less active metals. For example, in the reaction between zinc metal and copper sulfate, zinc ions
replace copper ions in solution:
Zn(s) + CuSO (aq) ⟶ ZnSO (aq) + Cu(s)
4
4
C15-3-5
In this reaction the zinc atom transfers two electrons to the copper ion, and zinc is oxidized. Its oxidation number
changes from 0 to +2, while copper accepts two electrons and is reduced; its oxidation number changes from +2 to 0. The
redox nature of this reaction is clearly apparent from the net ionic reaction:
Zn(s) + Cu
2 +
2 +
(aq) ⟶ Zn
(aq) + Cu(s)
C15-3-6
Zinc is a more active metal than copper; the valence electrons have higher energy in zinc metal than in copper
metal. This difference in stability provides the driving force for this reaction. Similarly, the difference in the activity of
different metals provides the driving force for other reactions of metals. Indeed, metals can be arranged according to their
decreasing activity, forming the activity series. The most active metals lose electrons most easily, and are listed at the top.
The inert metals that give up their electrons with great difficulty are listed at the bottom. The activity series is presented in
Table T15-3-1.
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Table T15-3-1. Activity series for metals
Element
Lithium
Potassium
Barium
Active Calcium
Sodium
Magnesium
Aluminum
Manganese
Zinc
Chromium
Iron
Cadmium
Cobalt
Nickel
Tin
Lead
Hydrogen
Copper
Silver
Inert Mercury
Platinum
Gold
Oxidation half reactions
Li → Li + + e−
K → K + + e−
Ba → Ba 2+ + 2e−
Ca → Ca 2+ + 2e−
Na → Na + + e−
Mg → Mg 2+ + 2e−
Al → Al 3+ + 3e−
Mn → Mn 2+ + 2e−
Zn → Zn 2+ + 2e−
Cr → Cr 3+ + 3e−
Fe → Fe 2+ + 2e−
Cd → Cd 2+ + 2e−
Co → Co 2+ + 2e−
Ni → Ni 2+ + 2e−
Sn → Sn 2+ + 2e−
Pb → Pb 2+ + 2e−
H 2 → 2 H+ + 2e−
Cu → Cu 2+ + 2e−
Ag → Ag + + e−
Hg → Hg 2+ + 2e−
Pt → Pt 2+ + 2e−
Au → Au 3+ + 3e−
You may notice that the alkali and alkaline earth metals (group 1 and 2) appear at the top of the table (they are
easily oxidized), which is consistent with their low ionization energy. Yet the activity series matches ionization energy
trends only very crudely. For example, lithium is more active than potassium, even though it has a higher first ionization
energy. Ionization energies measure the ease of removal of an electron from an isolated atom in the gas phase. However,
the activity series represents the ease with which an electron may be removed from a solid-state metal and transferred to
an ion in aqueous solution.
From the point of view of the driving force of a single-exchange redox reaction, we can conclude that a metal higher
in the activity series will reduce any metallic element below it and displace it from its compounds. Zn(s) will reduce Cu2+
(C15-3-5), and Cu(s) will reduce Ag+ , but no reaction will take place between Zn2+ and Cu(s), or between Cu2+ and
Ag(s).
The activity series includes hydrogen. All metals above hydrogen in the activity series react with water or aqueous
acids to reduce H+(aq) to H2(g). The most active metals (K, Na, and Li, for example) react violently with water, reducing H+
within the water molecules. Less active metals like magnesium and zinc react sluggishly with water; Zn, for example,
reacts imperceptibly slowly at room temperature. Increasing the temperature or adding acid can accelerate these
reactions. In general, higher temperatures accelerate chemical reactions by increasing the energy of the reactants. The
addition of an acid increases the concentration of H+ (which is actually in the form of H3O+) and facilitates the reaction.
Examples of these reactions are provided below:
0
+1
2 K(s) + 2 H
0
+1
2
0
+1
Mg(s) + 2 H
0
+1
+1
2
0
O(ℓ) ⟶ 2 NaOH(aq) + H (g)
C15-3-8
2
+2
2
C15-3-7
2
+1
2 Na(s) + 2 H
0
O(ℓ) ⟶ 2 K OH(aq) + H (g)
0
O(ℓ) ⟶ Mg(OH) (aq) + H (g)
+2
0
Zn(s) + 2 H Br(aq) ⟶ ZnBr (aq) + H (g)
2
E15-3-10
2
2
C15-3-10
2
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However, not all metals react with water. Metals that are below hydrogen in the activity series do not produce H2 in
water , even at higher temperature or in the presence of acids. Such metals are commonly called noble metals because of
their resistance to oxidation; they are often used in the production of jewelry because they do not tarnish in air. It is
possible to oxidize noble metals if the acid used is a strong oxidant. For example, the noble metal copper reacts with nitric
acid: copper is oxidized and the nitrogen of the nitrate ion is reduced, forming nitrogen dioxide. The oxidation states of Cu
and N are included below the reaction in C15-3-11.
0
+2
+5
+4
C15-3-11
Cu(s) + 4 H N O (aq) ⟶ Cu(NO ) (aq) + 2 H O(ℓ) + 2 N O (g)
3
3
2
2
2
Gold is also a noble metal, and the least active of all metals. However, it can be oxidized by a 1:3 mixture of
concentrated nitric and hydrochloric acid called aqua regia as shown in C15-3-12. The oxidation states of Au, N and H are
included below the reaction. Notice that even though H+ is a reactant, its oxidation number does not change (it is not part
of the redox process). In this case the nitrogen of the nitrate gets reduced, yielding nitrogen oxide.
0
+5
+1
−
Au(s) + N O
3
+
(aq) + 4 H
+3
+ 4 Cl
−
⟶ AuCl
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4
+1
+2
(aq) + 2 H 2 O(ℓ) + N O(g)
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15-4 Stoichiometry
Molar ratios of reactants and products expressed in balanced equations
are the basis of all stoichiometric calculations
Stoichiometry is the application of proportions to chemical reactions. It allows us to calculate and measure out the
amounts of reactants needed to carry out a reaction, and to predict the amounts of products formed. It all starts with a
balanced equation that describes the transformation taking place. For example, the coefficients in the equation for the
combustion of hydrogen to yield water tell us that 2 molecules of hydrogen combine with one molecule of oxygen to form
two molecules of water (C15-4-1).
C15-4-1
2 H (g) + O (g) ⟶ 2 H O(l)
2
2
2
Table T15-4-1 Stoichiometry of combustion of H2
Equation
2 H 2(g)
structures
molecules
mass (amu)
amount (mol)
mass (g)
+
O 2(g)
+
2 molecules of H 2
4.0 amu of H 2
2 mol of H 2
4.0 g of H 2
+
+
+
+
→
2 H 2O(ℓ)
→
1 molecule of O 2
32.0 amu of O 2
1 mol of O 2
32.0 g of O 2
→
→
→
→
2 molecules of H 2O
36.0 amu of H 2O
2 mol of H 2O
36.0 g of H 2O
We can express this information as the masses of elements participating in the reaction. Using atomic masses we
calculate that two molecules of hydrogen have a mass of 4.0 amu, while a molecule of oxygen has a mass of 32.0 amu.
When the reaction is over, we should have 36.0 amu of water.
Of course, it is impractical to measure or manipulate such small quantities. Even if we wanted to carry out the
reaction on a molecular scale, dealing with just a couple of molecules is extremely hard, if not impossible. We need to
convert the proportions expressed in the chemical equation to practicable scale. The mole is the perfect "conversion" tool.
On a macroscopic scale, the proportions are the same as on the microscopic scale; we have two moles of hydrogen (H2)
reacting with one mole of oxygen (O2) to produce 2 moles of water (H2O). Instead of dealing with individual molecules, we
deal with Avogadro's numbers of them.
We can measure out the 4.0 g of H2 and 32.0 g of O2 needed to produce 36.0 g of H2O. We are also not limited to
whole number molar quantities; we can double or quarter the scale of our reaction, or we can multiply the amounts by any
whole or fractional multiplier. The mass ratio of the two reactants will always remain 1:8. The power of proportion will
always deliver just the right relative amounts of reactants, and allows us to predict the amount of products that will be
formed from any given amount of reactants.
Basic stoichiometric calculations always rely on the proportion of moles of reactants and products, as derived from
the stoichiometric coefficients of a balanced chemical equation (F15-4-1). The number of moles is commonly obtained
from direct measurements of the masses of the substances involved. For gases, the number of moles can be obtained
from the ideal gas law, given the pressure, volume and the temperature. For a given volume of solution, the number of
moles may be obtained directly, if the molar concentration is known, or indirectly if the weight percent and mass of the
solution or its density are known. Whatever information is available, finding the solution is a matter of converting the given
values into moles using unit conversions and stoichiometry.
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Figure F15-4-1. Basic stoichiometric calculations always rely on the proportions of reactants and products. These are
derived from the stoichiometric coefficients of the balanced reaction equation. We convert experimentally measured
quantities (mass, concentration, P,V,T of a gas) to moles using appropriate connections.
A well-designed problem-solving strategy utilizing ICF tables facilitates all
stoichiometric calculations
Next we need to know how to use the information provided to solve stoichiometry problems. The real challenge in doing
this is mapping out a problem-solving strategy without being distracted by side-tasks. The "moles of reactants" to "moles
of products" ratio will always be at the center of our strategy (F15-4-1), while other calculations and conversions will play a
"supporting" role. To be successful we must organize the given information and find connections between this information
and the quantities we need to determine. The balanced chemical reaction, which provides the stoichiometric coefficients,
is the most important part of this strategy. Starting with the balanced reaction, we can develop a basic template that will
allow us to organize the given information and chart the steps we must execute to reach the solution.
Basic Template:
1. Write the balanced chemical equation for the reaction.
2. Make an ICF table (Initial, Change, Final) below the equation and enter the given information. Make note of what you want to determine as
well.
Start by including a row to record experimental information given or to be determined and a row for additional information needed to
make the connections to numbers of moles.
In this second row, note connections between measured quantities and moles.
I. When given mass, use formula weight (FW) to get numbers of moles.
II. For solutions, use molarity (M) and volume (V) to get numbers of moles.
III. For gases, use P, V, and T within the ideal gas equation to get numbers of moles.
Below this initial information, add more rows for the initial number of moles, (I), calculated from the experimental quantities, the change
in number of moles (C) as the reaction proceeds to completion, the change in experimental quantity, and the final number of moles (F) after
the reaction is done. You may supplement the table with additional rows converting the number of moles into the masses of reactants and
products. We will call this table the ICF table to indicate that it tracks the number of moles from their initial quantities to the final values.
3. Solve the problem. Fill in the table until you are able to solve the problem.
I. Use the connections to convert given measured quantities into moles, and the final number of moles back to experimentally
determinable quantities.
II. Use the stoichiometry of the balanced reaction (the mole ratios) to relate the number of moles of reactants to the number of moles of
products.
III. Use conservation of mass (total mass of reactants = total mass of products) when it is appropriate.
4. Make sure your answer is reasonable.
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As you peruse the examples provided to illustrate this problem-solving strategy, note that not all steps or
connections are used for every problem, just as in some problems not all of the information provided will be needed to
answer the questions asked. The ability to determine which information is relevant to the posed question is an important
skill that is best developed through abundant practice.
Consider this example of a combustion reaction:
Example 1. If a clean-burning engine burns exactly 1 gallon of gasoline, how many grams of CO2 will it
produce? Assume that the gasoline consists of only octane and that its density is d = 0.6929 g/mL.
The essence of the reaction that the example addresses is the conversion of octane (and oxygen) to carbon dioxide
(and water) as expressed in the balanced equation (C15-4-2):
2C H
8
18
(l) + 25 O (g) ⟶ 16 CO (g) + 18 H O(g)
2
2
2
C15-4-2
Our strategy for solving the problem will center around finding out how many moles of octane (C8H18) will burn, as it
—being the sole carbon-containing reactant—will determine the number of moles of CO2 produced, taking into account
the 2:16 molar ratio of the reactant (octane) to the product (CO2). The supporting calculations will include calculating the
number of moles of octane available, and the conversion of moles of CO2 produced into grams of CO2. The
complete calculations using our strategy outlined above are presented on a separate page, but the example brings into
focus two additional concepts that we need to consider. In our example, we assumed that we had an excess of oxygen,
and that the entire amount of octane was converted completely into the product. This may not always be the case.
The theoretical and actual yields of products are determined by
the amount of the limiting reactant
The reactant that is used up first in a reaction is called the limiting reactant (or limiting reagent); its amount limits the
amount of the product that can form in the reaction. Excess reagents are those present in quantities greater than
necessary to completely react with the limiting reagent. In our example, octane was the limiting reactant, and oxygen was
the excess reactant.
The limiting reactant sets the upper limit on the amount of product that the reaction can produce. This amount is
called the theoretical yield. The theoretical yield is the amount that will be produced if the reaction works perfectly, as
predicted by the reaction’s stoichiometry. A calculation based on the balanced equation and the amount of limiting
reactant available predicts a theoretical yield of products. In our example, we assumed that the octane was completely
converted to CO2 as the only carbon-containing product; we calculated the theoretical yield of carbon dioxide. In practice,
the actual yield, the amount actually obtained from the reaction, is less than the theoretical yield. This outcome may be
due to an incomplete reaction, side reactions (formation of other products), or mechanical losses during the separation of
the product from the reaction mixture. During hydrocarbon combustion, other carbon containing products may form, such
as carbon monoxide or soot (carbon particles); these are formed in side reactions that decrease the actual yield of the
expected product. We neglected such complications in our example by assuming there would be clean, complete
combustion.
In general, chemists calculate percent yield using equation E15-4-1. The percent yield is usually less than 100%,
and cannot ever be more than 100%. Efficient reactions have high yields (>90%).
actual yield
% yield =
× 100%
E15-4-1
theoretical yield
Many commercially important products, such as pharmaceuticals, are produced in multistep reactions. If the
synthesis (preparation from simpler components) involves just 7 steps and each has 90% yield, the desired product is
obtained in only 48% yield (0.97), meaning that over 50% of the materials are lost in the entire process. This simple
calculation illustrates the importance of finding reaction conditions that maximize the percent yield. For example, we might
change the temperature or add a catalyst.
To understand how to use the limiting reagent, let's consider another example:
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Example 2. Lithium oxide is used on the space station to remove water from the air via the following
reaction. If we need to remove 100.0 kg of water and 82.00 kg of Li2O is available, which is the limiting
reactant? How much product is formed?
Li O(s) + H O(g) ⟶ 2 LiOH(s)
2
2
C15-4-3
In this example, the mole ratio of the two reactants is 1:1. In cases like this we can determine the limiting reagent by
comparing the number of moles of each reactant. The limiting reagent will be the one with the smaller number of moles.
The complete calculations for this example are presented on a separate page. When the mole ratio of reactants is not
1:1, or if more than two reactants are involved, it is a bit harder to determine which is the limiting reagent.
There are two approaches to determining the limiting reactant. The first approach is to choose one of the reactants
(usually the first one in the balanced reaction) and determine the number of moles of the second reactant needed to react
completely with the first by using the stoichiometric mole ratio. If the number of moles of the second reactant is equal to or
greater than the number of moles needed to react completely, then the first reactant is the limiting reagent (it will be used
up completely). However, if there is not enough of the second reactant to react completely, then the second reactant is the
limiting reagent. The second approach is to assume in turn that each reactant is the limiting reagent and calculate the
number of moles (or mass) of one of the products. The limiting reagent is the one that produces the smaller amount of
product.
We will illustrate both of these methods in Example 3:
Example 3. Silicon carbide (SiC) is widely used in the production of car brakes (because of its
abrasiveness and high thermal endurance) and in the production of high-temperature semiconductor
electronics. It is made from sand (SiO2) and carbon at high temperatures. CO is also formed as shown in
C15-4-4. If 100.0 kg of sand reacts with 100.0 kg of C, and 55.0 kg of SiC are formed, what is the
percent yield?
SiO (s) + 3 C(s) ⟶ SiC(s) + 2 CO(g)
2
C15-4-4
Since we base the percent yield calculation on the amount of the limiting reagent available, we need to start by
determining which reagent is present in a limiting amount. Reaction stoichometry indicates that in this case, we need three
times as many moles of carbon as SiO2; our calculations (see this separate page), therefore, must account for this ratio.
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15-5 Example calculations
Stoichiometric calculations are central to quantitative analysis of chemical reactions
Stoichiometry calculations form the basis of quantitative examination of chemical reactions with regard to mass balance,
but their use is not limited to calculating the masses of products and their yields. We can apply these concepts to any
chemical reaction, including combustion, elemental analysis, redox reactions, solution reactions, acid-base reactions,
titrations, or reactions of gases (utilizing the ideal gas law). To be able to apply our problem-solving template to such
situations we have to understand the nature of the experiment so we can extract and interpret the numerical data
correctly.
Elemental analysis applies stoichiometry to determine compositions of compounds
For example, you may recall that when we discussed molecular composition we used elemental analysis based on the
combustion of a known amount of the sample. Now that we understand reactions better, let’s consider another example of
calculations directed toward finding the empirical formula from the experimental data.
Example 4. An alcohol is composed only of C, H, and O. A 4.00 g sample of the alcohol is completely
combusted with an excess of oxygen, producing 7.65 g of CO2 and 4.70 g of H2O. What is the empirical
formula of the alcohol?
To get the empirical formula we need to determine the molar ratio of the elements present, x:y:z in CxHyOz. In this
case we cannot start with the balanced equation for combustion as we do not know the formula of the alcohol, which is
precisely what we need to find out. We know that the complete combustion will convert all carbon atoms present in the
alcohol into carbon dioxide and all hydrogen atoms into water. We will not be able to directly learn about the oxygen
content because oxygen atoms are present in both reactants (the alcohol and O2) and both products (CO2 and H2O).
However, we have information on the identity (CO2 and H2O) and the mass of the products formed in the combustion.
Thus, we can calculate both the molar ratio of carbon to hydrogen and the mass of carbon and hydrogen in the products.
Applying the mass conservation law, we can then calculate the mass of oxygen in the sample as we know the mass of the
sample that was analyzed. The calculations in this case proceed in the "reverse" order; we start with masses of products
to learn about the composition of the reactants, but the use of stoichiometric proportions (F15-4-1) and the overall scheme
of "connecting" masses with moles remain the same.
Titrations illustrate concentration-based stoichiometry calculations for reactions in solution
The example above illustrates how molar ratios are the true underpinning of stoichiometric calculations. If molar
information can be obtained by other means, such calculations do not need to involve masses of reactants or products.
For example, consider a titration experiment where a known volume of solution with an unknown concentration is allowed
to react with a solution of known concentration called the titrant. The volume of the titrant needed to react completely with
the unknown can be measured, and the concentration of the unknown solution can then be determined using the
stoichiometry of the balanced reaction.
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Figure F15-5-1. Titration of an acid with a base. A sample of H2SO4 (aq) of
unknown concentration (the analyte) is analyzed. A base solution of known
concentration (the titrant) is added dropwise from a burette (a long, calibrated
tube with a stopcock at the bottom end). Phenolphthalein is added to the
H2SO4 solution and acts as an indicator by turning the solution pink when a
small excess of base is added beyond the equivalence point. In the example
shown, the volume of titrant used is 12.3 mL.
In an acid-base neutralization reaction experiment (F15-5-1), a strong base of known concentration is added dropwise to a solution of a strong acid of unknown concentration (or vice versa) and the volume needed to reach the
equivalence point (or endpoint) is measured. The titration equivalence point is the point when the amount of H+ and
HO− ions in the solution is equal, usually indicated by a change in the color of an indicator added to the solution.
Example 5. If 12.3 mL of 0.200 M NaOH solution is needed to neutralize 10.0 mL of H2SO4 solution,
what is the concentration of H2SO4?
Since we want to find the concentration of sulfuric acid, we want to use the balanced molecular equation rather than
the net ionic equation. Taking into account that sulfuric acid is a diprotic acid (it can give up two H+ ions), we need two
moles of NaOH for each mole of the acid to neutralize it completely. The numbers of moles present in solution are easily
obtained if the molar concentrations are known (n = V × M). In this case, the stoichiometric calculations follow the basic
template. The full solution is presented on a separate page.
Stoichiometric calculations involving gases utilize the ideal gas law
to quantify moles of gaseous reactants and products
Gas reactions are also amenable to stoichiometric calculations, and may involve volume or pressure rather than mass
quantities. In general, the ideal gas law provides sufficient accuracy for reactions near room temperature and under
relatively low pressures. In many cases, even further simplifications are possible. To illustrate, let’s consider another
example:
Example 6. A reaction between ozone (O3) and water was carried out at 40 °C and 780 mmHg as shown
in C15-5-1 below. How many liters of ozone are needed to produce 3 L of O2?
O (g) + H O(l) ⟶ H (g) + 2 O (g)
3
2
2
2
C15-5-1
Given the balanced equation and a specified temperature and pressure, one might be tempted to use the ideal gas
law to calculate the number of moles. However, that is unnecessary, as all measurements are of gases at constant P and
T. Under these conditions, the number of moles is directly proportional to the gas volume as stated in Avogadro's law (n =
V(P/RT)const). From the reaction stoichiometry we can set up a simple proportion: one mole of ozone produces 2 moles of
oxygen, and thus 1 L of ozone produces 2 L of O2. Therefore, we need 1.5 L (1 × 3/2 L) of ozone to produce 3 L of O2.
In other reactions involving gases the use of the ideal gas law may be needed. Such is the case in heterogeneous
reactions combining masses of solid species or concentrations of solutions with gas calculations. We always convert the
given information to moles, and use the stoichiometry of a balanced reaction. Let’s consider the following example:
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Example 7. Sodium azide, NaN3, is used in automobile airbags to produce a large volume of N2 gas
within milliseconds of a collision as shown in C15-5-2 below. How many liters of N2 gas at 735 mmHg
and 26 °C are produced from 126 g of NaN3?
2 NaN (s) ⟶ 2 Na(s) + 3 N (g)
3
2
C15-5-2
This calculation follows the established pattern. Since we are already given the balanced reaction, we follow our
standard template and first calculate the number of moles of sodium azide available and then, using reaction
stoichiometry, calculate the number of moles of N2 produced. Only in the final stages of our calculations we use the ideal
gas law to calculate the volume of gas formed under the specified conditions. As the result of our calculations illustrate,
a relatively small sample of solid NaN3 can very rapidly provide a large quantity of gas that fills the airbag, cushioning the
car occupants and minimizing the effects of a collision.
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16 Thermochemistry
Many of the reactions and physical processes that we have studied over the course of several lessons are
associated with the flow of heat between a system and its surroundings. Heat flow can be measured using calorimetry,
and the enthalpies of known reactions can be used to calculate or estimate the enthalpies of other processes with the help
of Hess's law.
16-1 Reaction enthalpies
If the products are more stable than the reactants, heat is released to the surroundings in an exothermic reaction.
Conversely, when a reaction is endothermic, heat flows from the surroundings into the system. Numerical information about the
heat exchanged with the surroundings may be included in balanced chemical equations describing the process, called
thermochemical equations. The enthalpy of a reaction is an extensive property, and depends on the state of reactants and
products. For the reverse reaction, the enthalpy has the same magnitude but opposite sign.
16-2 Calorimetry
The heat exchanged by reactions with the surroundings can be measured using calorimetry. If the heat capacity of the
immediate surroundings is known, the amount of heat transferred can be calculated from the change in the temperature of the
surroundings. Such measurements can be carried out under constant pressure in an insulated calorimeter open to the
atmosphere, or at constant volume in a bomb calorimeter.
16-3 Hess's law
Hess's law states that the sum of the enthalpies for a series of steps is the same as ΔH for the overall process. This
conclusion is based on enthalpy being a function of state. The change in enthalpy does not depend on the specific steps or the
number of steps, only on the initial and final state. Based on Hess's law, thermodynamic cycles can be constructed that allow us to
determine enthalpies for reactions that cannot be measured directly. These reactions can be incorporated into thermodynamic
cycles in which the enthalpies of other steps are known. Hess's law gives us the conceptual underpinning for using enthalpies of
formation (ΔH°f) and bond dissociation energies (BDEs) to evaluate the enthalpy for a large number of reactions based on a
limited amount of data.
16-4 Enthalpies of formation
The enthalpy of formation (ΔH°f) represents the amount of heat released or absorbed when 1 mole of a compound is made
from pure elements in their most stable form under the standard conditions of 298 K and 1 atm. Elements in their most stable form
at standard conditions are assigned enthalpies of formation equal to zero, and serve as a reference point on the energy scale. A
large number of standard enthalpies have been measured for inorganic and organic compounds. With the help of Hess's law,
these values can be used to calculate and predict the enthalpy of reaction without the necessity of carrying out the actual
measurements. In general, compounds or systems of the same composition that have a lower (more negative) enthalpy of
formation are more stable than those with a higher enthalpy of formation.
16-5 Bond energies
The amount of energy required to break a bond in such a way that the bonding electrons are equally divided between the
resulting fragments is called the bond dissociation energy (BDE). BDEs also represent values at standard conditions; in most
cases the average values for a given bond type are used. Reaction enthalpies may be estimated by comparing BDEs of bonds
broken and bonds formed. If the sum of the BDEs of bonds formed is larger than that for bonds broken, the reaction is exothermic.
The BDEs provide a link between the structural aspects responsible for bond strengths and reaction thermochemistry, which often
dictates reactivity.
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16-1 Reaction enthalpies
Spontaneous reactions typically have favorable enthalpy or entropy
In Chapter 15 we have learned about many physical processes and chemical reactions associated with energy changes.
In the vast majority of these processes, the energy transfer between the system of interest and the surroundings takes
place under constant pressure. Under such conditions, enthalpy is the preferred way to account for the heat flow and PVwork (if any). In many reactions volume changes are very small, and the change in enthalpy corresponds to the change in
the internal energy of the system. We cannot directly measure the total enthalpy, but we are only interested in changes of
enthalpy. Specifically, for a chemical reaction, its enthalpy is the difference in enthalpy between the product and the
reactants, ΔHrxn = H(products) − H(reactants). If the products are more stable (have lower enthalpy) then ΔHrxn < 0, the
energy is transferred to the surroundings, and the reaction is called exothermic. If the reverse is true and the reactants
are more stable (have lower enthalpy) than the products, then ΔHrxn > 0, the energy is provided by the surroundings, and
the reaction is deemed endothermic.
For example, a reaction between hydrogen and oxygen resulting in the formation of water is exothermic (F16-1-1).
Water is more stable (i.e. has lower enthalpy) than H2 and O2. The reaction is accompanied by a release of energy to the
surroundings.
Figure F16-1-1. Energy diagram of the exothermic reaction between H2 and O2. The enthalpy of the product (H2O) is
lower than the combined enthalpy of the reactants. In this reaction, the energy is released in the form of heat, light (as a
flame), and acoustic waves (result of high kinetic energy and velocity of gas molecules). The air temperature around the
exploding balloon is measured by a thermocouple (a wire thermometer) and the acoustic wave is registered by a smartphone app for illustrative purposes. The actual measurement of enthalpy change for this reaction can be done in a bomb
calorimeter.
Other reactions may be endothermic, giving products that are less stable than the reactants. Such reactions are
much less likely to take place unless they have some other driving force, such as an increase in the entropy of the
system. For example, the spontaneous reaction between barium hydroxide octahydrate and ammonium thiocyanate
results in the formation of barium thiocyanate, ammonia and water. The large number of small molecules produced
increases the disorder of the system, making the entropy change highly favorable. (F16-1-2). This increase in entropy is
similar to the dissolution process, where the transformation from an ordered crystalline solid to a large number of hydrated
ions in solution resulted in a favorable entropy change.
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Figure F16-1-2. Energy diagram of the endothermic reaction between barium hydroxide octahydrate and ammonium
thiocyanate that produces barium thiocyanate, ammonia, and water. The reaction is entropy-driven and the heat needed
for the reaction is provided by the surroundings. The temperature of the air around the solution drops due to the heat
being absorbed from the surroundings, which is sufficient to freeze a small amount of water under the beaker, fusing it to
the wooden base.
Thermochemical equations include enthalpies of reactions
We can now supplement our balanced chemical equation with the value of enthalpy associated with the specific reaction
defined by the equation. Since enthalpy is an extensive property (it depends on the amount of matter in the system), in
such thermochemical equations the value of enthalpy given corresponds to the numbers of moles specified by the
stoichiometric coefficients. For example, in the complete combustion of one mole of methane 802.3 kJ of energy is
released to the surroundings (C16-1-1).
CH (g) + 2 O (g) ⟶ CO (g) + 2 H O(g)
4
2
2
2
Δ Hr xn = −802.3 kJ
C16-1-1
Notice the negative sign of the reaction enthalpy; this is as expected for an exothermic process. In contrast, the
enthalpy of the reverse reaction has the same magnitude, but opposite sign (C16-1-2). Such a strongly endothermic
process is highly unlikely to be spontaneous. There is no entropic driving force as both sides have the same number of
molecules.
CO (g) + 2 H O(g) ⟶ CH (g) + 2 O (g)
2
2
4
2
Δ Hr xn = 802.3 kJ
C16-1-2
The enthalpy of the reaction depends on the states of the reactant and products. You may recall that phase
transitions change the energy of the system, and therefore the phase transition enthalpies must be taken into
consideration. For example, if liquid water is the product of methane combustion, the thermochemical equation is:
CH (g) + 2 O (g) ⟶ CO (g) + 2 H O(l)
4
2
2
2
Δ Hr xn = −890.3 kJ
C16-1-3
The additional 81.3 kJ (as compared to C16-1-1) of energy is released from the condensation of two moles of water.
If we start with a different amount of reactants, for example two moles of methane instead of one, we double the amount
of heat released. For the combustion of two moles of methane producing gaseous water we can write:
2 CH (g) + 4 O (g) ⟶ 2 CO (g) + 4 H O(g)
4
2
2
2
Δ Hr xn = −1604.6 kJ
C16-1-4
The relationships described by the thermochemical equations are often shown graphically in relative energy
diagrams. The diagrams relate information about the changes between different energy states of reactants and products,
but do not address the energy contents of any species listed. We will later learn how to establish a reference point for our
energy diagrams.
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Figure F16-1-3. Graphical representation of
energy changes for the combustion reaction of
one mole of methane. Exothermic processes are
shown with an arrow pointing downwards, from a
higher energy state to a lower energy state.
Before we continue, let's summarize what we have learned about enthalpy of reactions:
1. Enthalpy is an extensive property.
2. When we reverse a reaction (or process) the magnitude of the enthalpy is the same but the sign is reversed.
3. The enthalpy of a reaction (or process) depends on the state of the reactants and products.
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16-2 Calorimetry
The amount of heat exchanged between the reaction participants and the surroundings
is measured using a calorimeter
The measurement of the energy released or absorbed by a system during a reaction or any physical process is called
calorimetry (i.e., the measuring of calories). A calorie (cal) is an energy unit based on the heat capacity of water. By
definition, 1 cal will increase the temperature of 1.000 g of water by 1 °C (from 14.5 °C to 15.5 °C). One calorie is equal to
4.184 J. We did, in fact, already encounter heat capacities when we discussed heating curves in the context of phase
transitions. The idea behind calorimetry is that with known heat capacities we can measure heat flow by measuring
temperature changes. The basic equation that we use to calculate the heat transferred when heating a single phase is as
follows:
E16-2-1
q = m × C × ΔT
The heat released by a system must equal the heat absorbed by the surroundings, and conversely, the heat
absorbed by a system must equal the heat lost by the surroundings. In most cases, we define the system as the
molecules of reactants and products. Everything else in the direct vicinity of the system is defined as the surroundings,
including the solvent (for example, water) in which the reaction takes place, the thermometer, and the container.
Measurement of the temperature increase in a coffee-cup calorimeter is needed
to determine the heat of a neutralization reaction
Although an actual calorimeter that is used to obtain precise measurements is much
more sophisticated, we can illustrate its operating principles through the example of a
simple coffee cup calorimeter (Figure 16-2-1). We usually use nested coffee cups for
better thermal insulation. We want to make sure that all heat produced (or absorbed)
by our system of interest stays trapped in our measuring device, rather than being lost
to the surrounding air. The calorimeter is not sealed, but is open to the atmosphere so
that the reaction is run under constant pressure. Let's say we want to measure the
heat of the neutralization reaction between aqueous solutions of HCl and NaOH. We
have 50 mL of a 1.0 M solution of each reagent at 19 °C. First we pour the NaOH
solution into our cup calorimeter, which is equipped with a stirring rod, a thermometer
and a stopper. The HCl solution is added, and then we close the stopper, stir the
solution, and observe that the temperature rises to 26 °C.
Figure F16-2-1. Coffee cup calorimeter.
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The calculation of the heat transferred takes into account that we have a total of about 100 mL of an aqueous
solution with a density close to 1 g/mL. The specific heat of water (Cm) is 4.18 J/g-K. The temperature change was ΔT =
Tfinal − Tinitial = 26 °C − 19 °C = 7 °C = 7 K. Thus, the heat absorbed by our calorimeter (which is part of the surroundings)
is calculated using E16-2-1:
J
q = m × C × ΔT = 100 g × 4.18
g ⋅ K
× 7 K = 2.9 kJ
E16-2-2
The amount of heat released by our neutralization reaction (which is the system) is the same amount as that
absorbed by the calorimeter (the surroundings). Since our reaction mixture contained 0.05 mole of H+ and 0.05 mole of
HO− (n = 50 mL × 1.0 M), we must divide our total heat transferred by the number of moles to find the amount of heat
transferred per mole. Therefore, the heat of the neutralization reaction is ΔHrxn = −2.9 kJ/0.05 mole = −58 kJ/mol (C16-21).
+
H
−
(aq) + HO
(aq) ⟶ H O(l)
2
Δ Hr xn = −58 kJ/mol
C16-2-1
Of course, we can expect that our value for the enthalpy of the neutralization is approximate. We neglected any
temperature changes to the cup calorimeter itself, and we made a number of simplifying assumptions about the density,
the mass of the solution, and the specific heat of the solution (we assumed it to be pure water). Surprisingly, our value is
very close to the values obtained with much more sophisticated calorimeters. Indeed, depending on circumstance, the
heat of neutralization of many acid-base reactions is around −57 to −59 kJ/mol.
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16-3 Hess's law
Hess’s law is used to determine enthalpies of reactions that cannot be measured directly
Enthalpy is a state function. It does not depend on the path or the number of steps needed to convert the reactants into
products. The energetic outcome is determined simply by the enthalpy difference between the final and the initial states.
This conclusion is known as Hess's law: the sum of the enthalpies for a series of steps is the same as ΔH for the overall
process (F16-3-1).
A → B ΔH1
B → C ΔH2
-----------------------A + B → B + C ΔH1 + ΔH2 = ΔHrxn
A → C
ΔH1 + ΔH2 = ΔHrxn
Figure F16-3-1. Hess's Law energy diagram. The enthalpy change between A and C is the same regardless of whether
the transformation happens in one step from A → C, or through a series of steps as in A → B → C. Thermochemical
equations can be treated as algebraic equations; they can be added to or subtracted from each other and equivalent
terms present on both sides of the equations can be canceled.
Graphical representations of Hess's law (F16-3-1) are called thermodynamic cycles, and as the name illustrates, the
initial and final states are connected by two different paths forming a loop along which reactants can travel to get to the
product. Hess's law allows us to calculate the enthalpy of processes that cannot be measured directly. You may recall its
application to calculate lattice energies. In general, this means that a relatively small number of experimentally measured
enthalpies can be used to calculate ΔH for a vast number of reactions by using different combinations of enthalpy values.
The enthalpy of a reaction that cannot be measured directly is determined
algebraically or graphically using an appropriate thermodynamic cycle
Let's return to our example of the combustion of methane (C16-3-1). We know that the combustion of one mole of
methane releases 890.3 kJ of heat. In an independent experiment we can measure the combustion of one mole of carbon
monoxide to carbon dioxide: we find that it produces 283.0 kJ of heat (C16-3-2). How much heat is produced when one
mole of methane is combusted to form carbon monoxide (C16-3-3)? This measurement is experimentally impossible to
carry out, since controlling the degree of oxidation of carbon to form only CO (instead of CO2 or a mixture) in a
combustion reaction cannot be done.
CH (g) + 2 O (g) ⟶ CO (g) + 2 H O(l)
4
CO(g) +
2
1
2
2
O (g) ⟶ CO (g)
CH (g) + 1
4
2
2
1
2
2
O (g) ⟶ CO(g) + 2 H O(l)
2
2
Δ H1 = −890.3 kJ
C16-3-1
Δ H2 = −283.0 kJ
C16-3-2
Δ H3 = ?
C16-3-3
From an "algebraic" point of view we observe that in C16-3-3 we have one molecule (or mole) of methane on the left
side of the equation and one molecule (or mole) of carbon monoxide on the right side of the equation. Since we need to
have CO as a product, we need to subtract equation C16-3-2 from equation C16-3-1.
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CH (g) + 2 O (g)−CO(g)−
4
2
1
2
O (g) ⟶ CO(g) + 2 H O(l)−CO (g)
2
2
2
C16-3-4
In C16-3-4 the CO2 molecules on the product side cancel out, and subtracting half a mole of dioxygen from the 2
moles on the reactant side gives us 3/2 of a mole of dioxygen (2 O2 − ½ O2 = 1½ O2). Since CO is subtracted from the
reactant side, we can move the CO molecules to the product side (with a change of sign to the stoichiometric coefficient).
We then obtain the exact equation of interest laid out in C16-3-3. Since we subtracted the chemical equations to get the
overall reaction, we must also subtract the enthalpies to get the overall enthalpy. Thus, ΔH3 = ΔH1 − ΔH2 = −890.3 kJ −
(−283.0 kJ) = −607.3 kJ.
The same information can be presented graphically (F16-3-2) in the form of a thermodynamic cycle. All reactions
involved are exothermic, with arrows pointing downward toward systems with lower energy.
Figure F16-3-2. Hess's law applied to the combustion of methane. The enthalpy of combustion of methane to carbon
dioxide (ΔH1) and the enthalpy of combustion of carbon monoxide to CO2 (ΔH2) can be measured experimentally. The
enthalpy of combustion of methane to CO cannot be measured, but it can be calculated from the thermodynamic cycle
presented here. Note that the composition is the same on each energy level of the diagram (1 C + 4 H + 4 O). Energy and
stability comparisons can only be made for systems of exactly the same composition.
This example illustrates that enthalpies of reactions establish a relative stability scale for systems of the same
composition. It is directly apparent from the plot that carbon dioxide and water are more stable (have lower energy) than
methane and oxygen, with partial oxidation products (carbon monoxide) occupying the middle ground. If we could anchor
our relative scale to a preset reference point for energy, we would have an absolute energy scale. In the next section, we
will describe how this is done.
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16-4 Enthalpies of formation
The standard enthalpies of formation of compounds use pure elements
as a zero-enthalpy reference point
To compare the energy (and enthalpy) changes on a common scale, it is a chemical convention to use elements in their
most stable form as a reference. Since enthalpy depends on the conditions (P and T) and the state (gas, liquid, or solid) of
a system, a set of standard conditions has been designated to be 298 K (25 °C) and 1 atm of pressure. Notice that these
standard thermodynamic conditions are different than the STP for gases (T = 273.16 K or 0 °C, and 1 atm).
Under these standard conditions (designated by a superscript "°"), the standard enthalpy of formation for a
compound (ΔH°f) may be defined as the amount of heat absorbed or released when one mole of the compound is formed
from elements in their most stable form. Under standard conditions, the enthalpy of formation of a pure element in its most
stable form is assigned a value of zero.
The standard enthalpy of the formation of liquid water ΔH°f (H2O) is obtained by measuring the heat of combustion
of hydrogen:
H (g) +
2
1
2
∘
O (g) ⟶ H O(ℓ)
2
2
Δ Hr xn = −285.8 kJ/mol = Δ H
f
(H O)
2
C16-4-1
This process can be visualized with our standard energy graph (F16-4-1). The reaction is exothermic (with a
downward arrow), but our starting point is now defined as the zero-enthalpy level (the top level, where the elements are in
their most stable form). Thus the standard enthalpy of water is also defined. Similarly, the enthalpy of combustion of 1
mole of carbon (graphite) under standard conditions provides the standard enthalpy of formation of carbon dioxide (C16-42).
∘
C(s) + O (g) ⟶ CO (g)
2
Δ Hr xn = −393.5 kJ/mol = Δ H
2
f
(CO )
2
C16-4-2
Figure F16-4-1. Standard heats of formation of water and carbon dioxide. The standard heats of formation of elements or
elemental compounds are zero (by definition). The enthalpies of formation correspond to the heat of the combustion
reactions shown.
The method is not limited to reactions involving elements as reactants. Here we can again use our example of the
combustion of methane under standard conditions:
∘
CH (g) + 2 O (g) ⟶ CO (g) + 2 H O(l)
4
2
2
2
Δ Hr xn = −890.3 kJ
C16-4-3
Let's start by drawing an energy diagram for this reaction (F16-4-2). Methane is composed of one carbon atom and
four hydrogen atoms. We can analyze their combustions individually. We first combust one mole of graphite to get one
mole of carbon dioxide, which produces 393.5 kJ/mol of heat (see above). This is followed by the combustion of 2 moles
of hydrogen to produce water and 571.5 kJ of heat (2 moles × −285.8 kJ/mol), for a total heat transfer of –965.1 kJ. In
total, we have formed 1 mole of CO2 and 2 moles of H2O, which is exactly the same amount of the same products as
formed by the combustion of 1 mole of methane. Since the combustion of methane only gives −890.3 kJ/mol, we can
place its energy 890.3 kJ/mol above the energy of products of its combustion (CO2 and 2 H2O), or 74.8 kJ/mol below the
reference point (which is the pure elements). Using Hess's law, we have thus established the enthalpy of formation of
methane: ΔH°f (CH4) = −74.8 kJ/mol.
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Figure F16-4-2. Hess's thermodynamic cycle for the combustion of methane under standard conditions. The enthalpy of
combustion of methane is compared with the sum of the enthalpies of combustion of one mole of C(graphite) and two
moles of H2. The comparison shows that formation of methane from elements is exothermic by −74.8 kJ/mol. Therefore,
ΔH°f (CH4) = −74.8 kJ/mol. Click on the picture for a stepwise analysis of the thermodynamic cycle.
The standard enthalpies of formation are available for a large number of compounds
in thermodynamic tables
Table T16-4-1. Selected standard enthalpies of formation, ΔH°f (298 K, 1 atm)
Substance
Formula
acetic acid
acetylene
ammonia
aluminum oxide
benzene
bromine
calcium carbonate
calcium oxide
carbon dioxide
carbon monoxide
carbon tetrachloride
diamond
ethane
ethanol
ethanol
ethyl bromide
ethylene
formaldehyde
glucose
C 2H4O2(ℓ)
C 2H2(g)
NH 3(g)
Al 2O3(s)
C 6H6(ℓ)
Br 2(g)
CaCO 3(s)
CaO(s)
CO 2(g)
CO(g)
CCl 4(ℓ)
C(s)
C 2H6(g)
C 2H5OH(g)
C 2H5OH(ℓ)
C 2H3Br(g)
C 2H4(g)
CH 2O(g)
C 6H12O6(s)
ΔH° f
(kJ/mol)
−487.0
226.8
−46.2
−1675.7
49.0
30.907
−1207.1
−635.5
−393.5
−110.5
−193.3
1.9
−84.7
−235.1
−277.7
−97.6
52.3
−108.6
−1273.0
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Substance
Formula
hydrogen bromide
hydrogen chloride
hydrogen fluoride
hydrogen iodide
hydrogen peroxide
iron(III) oxide
methane
methanol
ozone
propane
silver chloride
sodium bicarbonate
sodium carbonate
sodium chloride
sodium hydroxide
sucrose
sulfur trioxide
water
water vapor
HBr(g)
HCl(g)
HF(g)
HI(g)
H 2O2(ℓ)
Fe 2O3(s)
CH 4(g)
CH 3OH(ℓ)
O 3(g)
C 3H8(g)
AgCl(s)
NaHCO 3(s)
Na 2CO3(s)
NaCl(s)
NaOH(s)
C 12H22O11(s)
SO 3(g)
H 2O(ℓ)
H 2O(g)
ΔH° f
(kJ/mol)
−36.2
−92.3
−268.6
25.9
−187.8
−824.2
−74.8
−238.6
142.3
−103.8
−127.0
−947.7
−1130.9
−410.9
−425.6
−2221.0
−395.2
−285.8
−241.8
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Analogous procedures have been carried out for a variety of compounds, and the standard enthalpies of formation
have been determined. Some values are listed in Table T16-4-1. A much more extensive list is available in our data tables
(the button on the top of the web page).
There are some important general observations that can be made by analyzing even this small subset of the data.
Although most enthalpies of formation are negative (indicating that the compounds are more stable than the elements
from which they are made) this is not universally true. There are multiple examples of stable compounds (for example
acetylene or benzene) that have positive enthalpies of formation. Such compounds do not decompose spontaneously into
their elements because there is no low-energy mechanistic path available, even though the elemental products are more
stable; the path to get to these products often requires a high-energy intermediate step.
You might also notice that less-stable forms of elements have positive enthalpies of formation (for example, diamond
is a less stable form of carbon than graphite). Different states of the same substance also differ in their heats of formation.
For example, liquid water has a lower enthalpy of formation than water vapor, and the difference is equal to the amount of
heat released upon the condensation of one mole of water vapor.
Enthalpies of reactions are calculated
by subtracting the sum of the enthalpies of formation of the reactants from those of the products
With the help of Hess's law, the heats of formation become a powerful tool for evaluating the thermochemistry of a
reaction. Let's once more return to the combustion reaction of methane, this time listing the heats of formation under each
species:
∘
CH (g) + 2 O (g) ⟶ CO (g) + 2 H O(ℓ)
4
2
2
Δ Hr xn
2
C16-4-4
∘
ΔH
− 74.8
f
0
− 393.5
2(−285.8)
− 890.3
(kJ/mol)
We have, in fact, already analyzed the thermochemistry of this reaction in our energy chart (F16-4-2). We were able
to find the enthalpy of formation of methane based on its heat of combustion and the heats of formation of carbon dioxide
and water. Using the same cycle we can rewrite our calculation in general terms as:
∘
ΔH
f
∘
(CH ) + 2 × Δ H
4
f
∘
∘
(O ) + Δ Hr xn = Δ H
2
f
∘
(CO ) + 2 × Δ H
2
f
E16-4-1
(H O)
2
We are essentially restating that the two paths must have the same enthalpy according to Hess's law. The first path
goes from elements to carbon dioxide and water through the combustion of methane, and the second path starts with
direct combustion of elements. We can rearrange our equation to express the standard enthalpy of combustion as a
function of enthalpies of formation of reactants and products.
∘
∘
Δ Hr xn = [Δ H
f
∘
(CO ) + 2 × Δ H
2
f
∘
(H O)] − [Δ H
2
f
∘
(CH ) + 2 × Δ H
4
f
(O )]
2
E16-4-2
The result indicates that the enthalpy of the reaction is equal to the difference between the enthalpies of formation of
products and enthalpies of formation of reactants multiplied by the appropriate stoichiometric coefficients (2 for water and
2 for oxygen). This result is, of course, general, as expressed in E16-4-3 where n and m stand for the appropriate
stoichiometric coefficients of all products and reactants, respectively:
∘
∘
Δ Hr xn = ∑ nΔ H
f
∘
(products) − ∑ mΔ H
f
E16-4-3
(reactants)
We can use this relationship to evaluate the enthalpy of one spectacularly exothermic reaction, the so-called
thermite reaction between aluminum powder and iron oxide (C16-4-5).
2 Al(s) + Fe O (s) ⟶ Al O (s) + 2 Fe(s)
2
3
2
3
C16-4-5
Δ Hr xn = ?
You may recall that aluminum is higher in the metal activity series than iron and it oxidizes at the expense of iron
oxide (which is reduced). Thus, our reaction is expected to be exothermic.
∘
Δ Hr xn = [Δ H
∘
f
∘
(Al O ) + 2 × Δ H
2
3
f
∘
(Fe)] − [2 × Δ H
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f
∘
(Al) + Δ H
f
(Fe O )]
2
3
E16-4-4
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The standard heat of formation of aluminum oxide is highly exothermic, ΔHof(Al2O3) = −1675.7 kJ/mol, but the heat
of formation for iron oxide is less so ΔH°f(Fe2O3) = −824.2 kJ/mol. The standard heats of formation of elemental metals
are zero by definition. Thus, the enthalpy of the reaction is ΔH°rxn = −1675.7 kJ/mol − (−824.2 kJ/mol) = −851.5 kJ/mol.
The heat produced during the reaction (F16-4-2) is sufficient to melt the iron formed in the reaction (mp = 1,535 °C).
Figure F16-4-3. The thermite reaction between
aluminum and iron oxide is highly exothermic. Once
initiated by a flame, it produces sufficient heat to melt
the iron formed in the reaction, which glows yelloworange as it drips into the sand trap below the
reaction vessel.
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16-5 Bond energies
Bond dissociation energy is the enthalpy required to break a bond in a homolytic fashion,
producing two radicals
In an earlier lesson on bonding we introduced the concept of bond dissociation energy (BDE), which is the energy
needed to break a given bond under standard conditions. BDE values correspond to a homolytic bond cleavage process
wherein the electrons in the broken bond are split equally between the fragments. Let's consider a couple of examples:
∙
Cl (g) ⟶ 2 Cl (g)
BDE = 242 kJ/mol
2
∙
H (g) ⟶ 2 H (g)
BDE = 436 kJ/mol
2
∙
∙
HCl(g) ⟶ H (g) + Cl (g)
BDE = 431 kJ/mol
C16-5-1
C16-5-2
C16-5-3
Enthalpies of reactions are calculated by subtracting BDEs of the bonds broken
from those of the bonds formed
The BDE values quoted above are the precise values for the specific bonds broken in the gas phase. With the help of
Hess’s law, we can use them to calculate the standard enthalpy of reaction between hydrogen and chlorine to form
hydrogen chloride.
H (g) + Cl (g) ⟶ 2 HCl(g)
2
2
C16-5-4
We must first provide enough energy to break H–H and Cl–Cl bonds (242 kJ/mol + 436 kJ/mol = 678 kJ/mol),
producing pairs of H• and Cl• radicals, which then combine to make the H–Cl bonds in two HCl molecules. Since we know
the H–Cl bond dissociation energy (C16-5-3), making two H–Cl bonds (the opposite process) releases 862 kJ/mol of
energy. We have successfully made stronger bonds on average (2 × 431 kJ/mol), at the expense of weaker bonds (242
kJ/mol + 436 kJ/mol). Since we released more energy in making bonds than we absorbed in breaking bonds, the overall
process is exothermic. The enthalpy of the reaction is obtained by subtracting the energy of the bonds formed from the
energy of bonds broken (242 kJ + 436 kJ − 2 x 431 kJ = −184 kJ). Notice that all BDEs are positive by definition, and we
need to place a negative sign in front of the BDEs of the bonds formed to properly account for the fact that bond making is
always exothermic. We can generalize this analysis:
Δ Hr xn = ∑ BDE(bonds broken) − ∑ BDE(bonds formed)
E16-5-1
We have calculated that reaction C16-5-4 is exothermic by −184 kJ. Since our reactants are elements in their most
stable form at standard state and we have produced two moles of HCl(g), the standard enthalpy ΔH°f(HCl) = −92 kJ/mol,
which is in perfect agreement with the value in Table T16-3-1.
Unfortunately, the exact values of BDEs are known only for a limited number of bonds. Instead, chemists use
average BDE values. Although not 100% accurate, these average values provide a convenient way to rapidly estimate the
enthalpies of reactions. Equation E16-5-1 has to be modified accordingly to take into account the approximations involved:
Δ Hr xn ≈ ∑ BD Eav (bonds broken) − ∑ BD Eav (bonds formed)
E16-5-2
Exact and average bond dissociation energies are available in thermodynamic tables
Even if the average BDEs are only approximate, the conclusions reached from their use may be strengthened by chemical
intuition. It is simpler to discuss the driving force in a reaction in terms of stronger bonds being formed at the expense of
weaker bonds, than in terms of the heats of formation of reactants and products. Bond breaking and formation have a
much more intimate connection to the structures of the molecules involved. Bond orders can be read from Lewis
structures and bond lengths can be approximated from atomic radii. In general, a bond’s strength (and its BDE)
correlates with its bond order and its length, providing us with a direct (if approximate) connection between molecular
structure and thermochemistry, which, in turn, often controls reactivity.
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Here we present again a collection of BDEs for single and multiple bonds (T16-5-1). We have marked the precise
BDEs for diatomic molecules in green, but most of the other BDEs are average values.
Table T16-5-1. BDEs, average single bond dissociation energies (in kJ/mol)
I
Br
Cl
S
P
Si
F
O
N
C
H
H
299
366
431
368
322
323
568
463
391
413
436
C
220
276
328
259
264
301
453
358
276
348
N
159
243
200
200
335
272
176
193
O
234
234
203
364
340
368
190
146
F
277
237
193
327
490
582
157
Si
234
310
464
226
P
184
264
319
218
253
242
S
Cl
208
218
Br
175
193
I
151
226
209
266
Multiple Bonds (BDE in kJ/mol)
Bond
BDE
Bond
BDE
N=N
418
C=C
620
N≡N
941
C≡C
815
C=N
615
C=O
745
C≡N
891
C≡O
1072
N=O
607
O=O
499
Enthalpies of reactions estimated from BDEs help in selecting more feasible reactions
and to better understand reactions on a molecular level
We will now demonstrate the use of average BDEs to estimate the enthalpy of two possible reactions between ethane and
chlorine, one forming ethyl chloride and HCl (C16-5-5) and the other producing methyl chloride (C16-5-6).
∘
C H (g) + Cl (g) ⟶ C H Cl(g) + HCl(g)
2
6
2
2
5
Δ Hr xn (A) = ?
C16-5-5
We need to identify the bonds broken and the bonds made in both reactions. The best way to do so is to draw the
Lewis structures of all species involved in the reaction. In the first reaction (C16-5-5), we see that the C–C bond remains
intact, but a C–H bond is broken in our organic substrate. The average BDE(C–H) is 413 kJ/mol (T16-5-1). The BDE of
the second bond broken, Cl–Cl, is 242 kJ/mol. The two formed bonds are C–Cl and H–Cl. The average BDE(C–Cl) is 328
kJ/mol, and BDE(H–Cl) = 431 kJ/mol. This estimate gives us ΔHorxn (A) = 413 kJ + 242 kJ − (328 kJ + 431 kJ) = −104 kJ.
∘
C H (g) + Cl (g) ⟶ 2 CH Cl(g)
2
6
2
3
Δ Hr xn (B) = ?
C16-5-6
By contrast, even if the condensed formula of ethane is given in the second reaction (C16-5-6), we recognize that
the C–C bond must have been broken in this case, since we have two single-carbon molecules as products. The average
BDE(C–C) is 348 kJ/mol (T16-5-1). The BDE of the Cl–Cl bond is 242 kJ/mol. The average BDE(C–Cl) is 328 kJ/mol, and
two such bonds are formed in the products. This estimate gives us ΔHorxn (B) = 348 kJ + 242 kJ − 2 x 328 kJ = −66 kJ.
The thermodynamic cycles corresponding to these calculations are shown graphically in Figure F16-5-1. Note that
we first formally break the bonds in endothermic processes, then make new bonds in exothermic processes (indicated by
downward arrows), which must be given negative signs, as is required for any exothermic process. The results indicate
that the first reaction (C16-5-5) is favored thermodynamically (more exothermic). Indeed, ethyl chloride and HCl are the
main products observed when ethane reacts with chlorine. This example shows how Hess's law can be used to predict the
thermodynamically favored reaction, which often (but not always) will be the one observed experimentally.
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Figure F16-5-1. The thermodynamic cycles for the reaction between ethane and chlorine. The cycles are based on BDEs.
ΔH1 = BDE(C−H) + BDE(Cl−Cl), ΔH2 = −(BDE(C−Cl) + BDE(H−Cl)), ΔH3 = BDE(C−C) + BDE(Cl−Cl), ΔH4 =
−2×BDE(C−Cl).
Consider another visually appealing reaction (F16-5-2): we can use "magic beans" and a few drops of water to start
a fire on a piece of cotton. The magic beans are pellets of potassium peroxide, K2O2(s), which decomposes in the
presence of water:
∘
K O (s) + 2 H O(l) ⟶ 2 KOH(aq) + H O (aq)
2
2
2
2
C16-5-7
Δ Hr xn = ?
2
Subsequently, the hydrogen peroxide formed in the first step decomposes to form water and oxygen.
∘
2 H O (aq) ⟶ 2 H O(l) + O (g)
2
2
2
2
C16-5-8
Δ Hr xn = ?
Figure F16-5-2. The reaction of potassium peroxide with water
on cotton. The reaction is initiated by the addition of a few
drops of water. The processes involved are exothermic and
produce oxygen; this is a sure way to set fire to any
combustible material (such as cotton) in the vicinity.
We can calculate the enthalpy of the first reaction (C16-5-7) using enthalpies of formation. Subtracting the enthalpies
of formation of the reactants from the enthalpies of formation of the products shows that the first step is mildly exothermic.
The necessary heats of formation can be found in our thermodynamic data table.
∘
K O (s) + 2 H O(l) ⟶ 2 KOH(aq) + H O (aq)
2
2
2
2
2
Δ Hr xn
C16-5-9
∘
ΔH
f
− 495.8
2(−285.8)
2(−482.4)
− 187.8
− 85.2
(kJ/mol)
The enthalpy of the second reaction (C16-5-7) may be calculated using BDEs or heats of formation. We will carry
out both calculations to see how they compare. Based on the enthalpies of formation, the second reaction is much more
exothermic than the first, and produces the O2 that supports the combustion of cotton.
∘
2 H O (aq) ⟶ 2 H O(l) + O (g)
2
2
2
2
Δ Hr xn
C16-5-10
∘
ΔH
f
2(−187.8)
2(−285.8)
0
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When using the BDEs to analyze the thermodynamics of the second step (C16-5-10), it helps to draw the Lewis
structure of hydrogen peroxide, H–O–O–H. This will help you realize that the simplest way to convert two molecules of
hydrogen peroxide into the products is to break all of the bonds. We must break four sets of O–H bonds, and two sets of
O–O bonds. Then we can reform four sets of O–H bonds (to make two molecules of water) and make one O=O double
bond to form the O2 molecule. Since we are using average bond energies, the O–H bonds in hydrogen peroxide and in
water are given the same value and cancel out. Thus, we are left with two broken O–O bonds (BDE = 146 kJ/mol) and
one newly made O=O bond (BDE = 499 kJ/mol) for a net difference of −207 kJ. The difference between the more accurate
calculation (the −196 kJ value based on enthalpies of formation) and the estimate based on BDEs (−207 kJ) is quite small
(less than 6%). This confirms that the use of average BDE values is an acceptable approximation when calculating the
heat of reaction.
We have learned a lot of chemistry, but there is so much more to explore! We mentioned entropy, but never
analyzed it. We talked about the rates of reactions or physical processes, but we never even mentioned the word kinetics.
We introduced acids and bases, but we have not defined pH. We mentioned that mass can be converted into energy, but
we never discussed nuclear reactions. We learned very little about metals, and almost nothing about solid-state structures
(except for some ionic solids). We did not have time to talk about the amazing new materials chemists cook up in their
labs. But we hope we got you interested in chemistry and instilled in you some basic concepts on which you can build. We
hope you stay with us on our journey through the chemical world in the second semester of general chemistry!
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