Chapter 2
Some definitions
Atoms-Atoms are the smallest particles that interact chemically. They are the smallest
particles that show the properties of an element. For instance gold atoms make up a bar
of the element gold. All the atoms which are elements are shown in the periodic table.
While oxygen molecules make up gaseous oxygen it is two oxygen atoms that make up
this molecule and the chemical properties of oxygen are in the oxygen atom.
Elements-Are substances made up of only one type of atom. Here “one type” means
chemically identical atoms. Elements are made up of several atoms that are called
isotopes.
amu or atomic mass unit-This is just a label or name for a number it is not a real unit. It
is in fact a ration of masses. It is the mass of anything to that of the mass of Carbon-12
isotope.
Nucleus-Is located at the center of an atom and generally contains protons, and neutrons.
Protons-These are positively charged particles. These are found in the nucleus. They
have a mass of nearly 1.0073amu.
Neutron-These are neutral or without charge. These are found in the nucleus. They have
a mass of nearly 1.0086amu.
Electron-Negatively charged. Found outside the nucleus. They have a mass of
0.000545amu.
Isotopes-These are atoms that have the chemical properties of a particular element but
different from each other by the number of neutrons in the nucleus of the atom. Isotopic
atoms also have the same number of protons and electrons. Some are extremely unstable
and radioactive. Examples of isotopes are carbon-12, carbon-13, and carbon-14.
Another example are the hydrogen isotopes: Hydrogen-1, Hydrogen-2 and Hydrogen-3.
All elements have isotopic atoms. Because of this the masses listed on the periodic table
are weighted averages of the different isotopes of an element. Please make note that the
mass given on the periodic table for an element is a weighted average of all stable
isotopes of that element.
Masses of Single Atoms and Particles Found in the Nucleus and Outside the Nucleus
The mass of all atoms and the particles that make up the atom are determined
relative to the Carbon-12 isotope using a mass spectrometer. Carbon-12 is given the
mass of exactly 12Amu or atomic mass units. With this in mind it has been
determined that the mass of the elementary particles that make up the atom are as
follows:
p+=1amu
no=1amu
e-=1/1850 amu
The Structure of the Nucleus of an Atom
The atom has a small dense or massive core called the nucleus with electrons somewhere
outside the nucleus. For hydrogen with 3 isotopes they have the following structures and
isotope symbols:
1
1
H has 1proton only in the nucleus.
2
1
H has 1 proton plus 1 neutron in the nucleus.
3
1
H has 1 proton plus 2 neutrons in the nucleus.
All have just a single electron. Almost all of the mass is in the nucleus of an atom.
The Mole and Avogadro’s Constant
Most chemists are not interested in the mass of a single atom or molecule. Atoms and
molecules are too small for practical lab work. That is why chemists use the mole. It
allows them to use their balance to do laboratory work. Chemical reactions depend on
the relative amounts (moles) of molecules or atoms so it makes sense to use relative
masses in the laboratory. Take as an example the following chemical reaction
2H2+O2=2H2O
This is the balanced equation for the formation of water from oxygen molecules and
hydrogen molecules. Now the equation also says that 2 molecules of hydrogen react with
1 molecule of oxygen to form 2 molecules of water. For most laboratory chemists this
equation is read 2 moles of hydrogen molecules react with 1 mole of oxygen molecules to
form 2 moles of water molecules. Let us imagine that we can use a balance to find the
mass of 1 mole of oxygen molecules and 2 moles of hydrogen molecules. What mass in
grams contains 2 moles of hydrogen molecules and what mass in grams contains 1 mole
of oxygen molecules?
When discussing the mole and Avogadro’s constant we are discussing the number of
particles in a given sample. A mole of anything whether it is jelly beans or atoms has an
Avogadro’s number of particles. Avogadro’s number is 6.02x1023 particles. This
number is critically dependent on the standard kilogram sample which is stored in France.
If the kilogram was originally defined to be a different value than it is now, Avogadro’s
number would have a different value. To see how this number, 6.02x1023 was derived
consider the following. A mole is the amount of substance that contains the same
number of atoms or molecules (chemical species) as there are atoms in 12grams of
the pure isotope Carbon-12. A mole of this isotope of carbon is defined to have a
mass of 12 grams exactly. Carbon found naturally is actually a mixture of several
isotopes so its relative mass is 12.01 on the periodic table. The value 12.01 is a weighted
average. The mass of all the stable isotopes of all the elements in the periodic table are
measured relative to Carbon-12. Note that the masses on the periodic table have no units.
It is only when we measure an element in grams with the same numerical value as in the
periodic table that we get moles with Avogadro’s number of atoms. The values in the
periodic table have no units because they are derived from ratios of mass. These
numbers are named atomic mass units or abbreviated u or amu and are not really units.
All of the elements listed on the periodic table come in different types called
isotopes. Different isotopes of a single element differ by the number of neutrons they
have. For instance what we call hydrogen comes as three isotopes: proteum, deuterium
and tritium. That means that hydrogen gas is a mixture of three isotopes. In fact
hydrogen gas is really a mixture of 2 isotopes because tritium is so unstable. Hydrogen
gas consists of a little bit of deuterium and mostly proteum. The chemical properties of
the 3 isotopes are identical. Each reacts to form water with oxygen. Deuterium has about
2x the mass of proteum and tritium has about 3x the mass of proteum. Remember that
there is very little deuterium in hydrogen gas so the average mass of a hydrogen atom
increases to a relative value of 1.01. This is also a weighted average. A mole of
hydrogen atoms is 1.01grams. A list of all the isotopes in existence is given at this web
site. http://physics.nist.gov/cgi-bin/Compositions/stand_alone.pl
Here is a way of looking at the average atomic mass; the masses listed in the periodic
table. We will use copper (Cu) as an example. Copper has 2 stable isotopes- Copper-63
and Copper -65.
Isotope
Copper-63
Copper-65
Isotope
Percentage or
Abundance
69.09%
30.91%
Mass of a single atom
1.045x10-22grams
1.078x10-22grams
Mass of Avogadro’s Number
of Atoms, relative gram
molecular mass, Ar in grams
62.93grams
64.94grams
The Mass of 100 Cu atoms= 69.09x62.93+30.91x64.94=6353amu.
The Average Mass of a Single Cu Atom=63.53amu.
Average Mass of 1 mole or Avogadro’s number of
Cu atoms=0.6909x62.93+0.3091x64.94=63.53grams
63.53grams of Cu has the two isotopes in correct proportions or abundances and the total
number of atoms is Avogadro’s number or one mole of atoms.
We still need to find the number of particles in a mole. There are at least 2 ways to do
this. The first is to count electrons in an electric circuit where we produce a metallic
element from a solution of its ions. We then determine how much mass formed for a
given total charge. The mass of the element gives the number of moles of the element
and the number of electrons used is just the current x time/charge on a single electron.
We also need to know how many electrons were transferred to form each metal atom.
Was it 1 or 2 electrons? If we know the total current and total charge for the experiment
how many electrons total?
The most advanced method for finding Avogadro’s number uses a polished silicon
ball (Si) and uses x-ray diffraction to determine the spacing of the atoms in the ball.
From the spacing of the silicon atoms, the total number of atoms in the ball is determined.
The ball is fabricated to mass at 1kg. The mass of a mole of Si is 28.09g. This is not
very precise but for our purposes illustrates our point. From the number of atoms in a
1kg ball of silicon using x-ray diffraction we can then determine the number of atoms in
28.09g of silicon. For more information look at the following link discussing the
determination of Avogadro’s number: http://en.wikipedia.org/wiki/Avogadro_constant
You might want to consider whether the relative average masses given in the periodic
table are universal. What I mean is: Are the % for each of the isotopes of an element the
same throughout the world and the universe?
Some Relationships Between Avogadro’s Number, the Number of Atoms,
The Mole and Molar Mass
If m is the mass in grams of an element, M is the average molar mass of the element, and
n is the number of moles then
m=Mxn
The answer is in grams.
(mass=Average molar mass x number of moles)
Example: 3 moles of oxygen gas have what total mass?
Answer: M=32g/mol n=3 therefore m=96g
Also if NA is Avogadro’s number and N is the total number of atoms then the total
number of atoms in a sample is
N=nxNA
(Number of atoms=number of molesx Avogadro’s number)
Example: How many atoms are there in 5 moles of He gas.
Answer: N=5x6.02x1023=30.1x1023 atoms
Example: How many moles is 3.01x1023 atoms of He?
Answer: n=N/NA=3.01x1023/6.02x1023=0.5moles
Simple Atomic Structure
The nucleus is at the center of the atom. The nucleus has neutrons plus protons in
general. The electrons circle the nucleus in discrete energy levels. Niels Bohr was the
first to propose a quantum theory of the atom. His model was named the Bohr model.
His model only works for simple atoms such as hydrogen and helium.
Niels Bohr proposed, in 1913, what is now called the Bohr model of the atom. He
suggested that electrons could only have certain classical motions:
1. The electrons can only travel in certain orbits: at a certain discrete set of distances
from the nucleus with specific energies.
2. The electrons of an atom revolve around the nucleus in orbits. These orbits are
associated with definite energies and are also called energy shells or energy
levels. Thus, the electrons do not continuously lose energy as they travel in a
particular orbit. They can only gain and lose energy by jumping from one allowed
orbit to another, absorbing or emitting electromagnetic radiation with a frequency
ν determined by the energy difference of the levels according to the Planck
relation:
Where h is Planck's constant.
3. The frequency of the radiation emitted at an orbit of period T is as it would be in
classical mechanics; it is the reciprocal of the classical orbit period:
The diagram below shows the Bohr atom. Unfortunately it only explains the
simplest of atoms such as hydrogen. The large circles labeled 1, 2, 3 are the
orbits of the electron. These circles represent the path of an electron. For a single
electron in hydrogen the normal state is for the electron to be in the lowest orbit
closest to the nucleus, n=1. This is a quantum mechanical model. The electron
can only exist in these orbits. It cannot exist between the orbits. Also when an
electron is in an n=1 orbit it can remain there forever and not lose energy. Things
slow-down in our world but electrons in an atom at the lowest level always keep
their energy and they do not crash into the nucleus. Finally, electrons can lose or
gain energy only by going from one orbit to another. If the electron goes from an
inner to an outer orbit it gains energy but if it goes in the opposite direction from
an outer orbit to an inner orbit it loses energy. It can gain energy by absorbing
light or lose energy by giving off light. The light in a fluorescent lamp comes
from or is emitted by electrons losing energy.
Summary of Allowed Combinations of Quantum Numbers
Number of
Number of
Electrons
Total Number of
Subshell Orbitals in Needed to Fill
Electrons in
n l m
Notation the Subshell
Subshell
Subshell
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
1 0 0
1s
1
2
2
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
2 0 0
2s
1
2
2 1 1,0,-1
2p
3
6
8
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
3 0 0
3s
1
2
3 1 1,0,-1
3p
3
6
3 2 2,1,0,-1,3d
5
10
18
2
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
4 0 0
4s
1
2
4 1 1,0,-1
4p
3
6
2,1,0,-1,4 2
4d
5
10
2
3,2,1,0,4 3
4f
7
14
32
1,-2,-3
The Relative Energies of Atomic Orbitals
Because of the force of attraction between objects of opposite charge, the most important
factor influencing the energy of an orbital is its size and therefore the value of the
principal quantum number, n. For an atom that contains only one electron, there is no
difference between the energies of the different subshells within a shell. The 3s, 3p, and
3d orbitals, for example, have the same energy in a hydrogen atom. The Bohr model,
which specified the energies of orbits in terms of nothing more than the distance between
the electron and the nucleus, therefore works for this atom.
The hydrogen atom is unusual, however. As soon as an atom contains more than one
electron, the different subshells no longer have the same energy. Within a given shell, the
s orbitals always have the lowest energy. The energy of the subshells gradually becomes
larger as the value of the angular quantum number becomes larger.
Relative energies: s < p < d < f
As a result, two factors control the energy of an orbital for most atoms: the size of the
orbital and its shape, as shown in the figure below.
A very simple device can be constructed to estimate the relative energies of atomic
orbitals. The allowed combinations of the n and l quantum numbers are organized in a
table, as shown in the figure below and arrows are drawn at 45 degree angles pointing
toward the bottom left corner of the table.
The order of increasing energy of the orbitals is then read off by following these arrows,
starting at the top of the first line and then proceeding on to the second, third, fourth lines,
and so on. This diagram predicts the following order of increasing energy for atomic
orbitals.
1s < 2s < 2p < 3s < 3p <4s < 3d <4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p < 7s < 5f < 6d <
7p