PPT 107 PHYSICAL CHEMISTRY
Semester 2
CHAPTER 1
THERMODYNAMICS
CHAPTER 1
THERMODYNAMICS
CONTENT:
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
Physical Chemistry
Thermodynamics
Temperature
The Mole
Ideal Gases
Differential Calculus
Equations of State
Integral Calculus
1.1 Physical Chemistry
• What is Physical Chemistry?
– Physical chemistry is the study of the underlying physical principles
that govern the properties and behavior of chemical systems.
• What is Chemical Systems?
– A chemical system can be studied from either a microscopic or a
macroscopic viewpoint.
The microscopic viewpoint is based on the
concept of molecules.
Chemical
Systems
The macroscopic viewpoint studies large-scale
properties of matter without explicit use of the
molecule concept.
The first half of this book uses mainly a macroscopic viewpoint; the second half uses mainly a
microscopic viewpoint.
• "microscopic" implies detail at the atomic or
subatomic levels which cannot be seen directly
(even with a microscope!).
• The macroscopic world is the one we can know
by direct observations of physical properties
such as mass, volume, etc.
4 Branches of
Physical
Chemistry
Thermodynamics
Quantum
Chemistry
Statistical
Mechanics
Kinetics
Thermodynamics is a
macroscopic
science that studies the
interrelationships of the
various equilibrium
properties
of a system and the
changes in equilibrium
properties in processes.
Molecules and the
electrons and nuclei
that compose them do
not obey classical
mechanics. Instead,
their motions are
governed by the laws of
quantum mechanics.
Application of quantum
mechanics to atomic
structure, molecular
bonding,
and spectroscopy gives
us quantum chemistry.
The molecular and
macroscopic levels are
related to each other by
the branch of science
called statistical
mechanics. Statistical
mechanics gives insight
into why the laws of
thermodynamics hold
and allows calculation
of macroscopic
thermodynamic
properties from
molecular properties.
Kinetics is the study of
rate processes such as
chemical reactions,
diffusion, and
the flow of charge in an
electrochemical cell.
Kinetics uses relevant
portions of
thermodynamics,
quantum chemistry,
and statistical
mechanics.
1.2 Thermodynamics
THERMODYNAMIC SYSTEM
An important concept in thermodynamics is the thermodynamic system. A thermodynamic
system is one that interacts and exchanges energy with the area around it (transformation
of energy). A system could be as simple as a block of metal or as complex as a
compartment fire. Outside the system are its surroundings. The system and its
surroundings comprise the universe.
Systems:
A region of the universe that we direct our attention to.
Surroundings:
Everything outside a system is called surroundings.
Boundary:
The boundary or wall separates a system from its
surroundings.
UNIVERSE
For example, to study the vapor
pressure of water as a function of
temperature, we might put a
sealed container of water (with
any air evacuated) in a constanttemperature bath and connect a
manometer to the container to
measure the pressure. Here, the
system consists of the liquid water
and the water vapor in the
container, and the surroundings
are the constant-temperature bath
and the mercury in the
manometer.
For example we might consider a burning fuel
package as the system and the compartment as the
surroundings. On a larger scale we might consider
the building containing the fire as the system and the
exterior environment as the surroundings.
A key property in
thermodynamics is temperature,
and thermodynamics is
sometimes defined as the study
of the relation of temperature to
the macroscopic properties of
matter.
Energy transfer is studied in three types
of systems:
Open systems
Open systems can exchange both matter and energy with an outside system. They are portions of larger
systems and in intimate contact with the larger system. Your body is an open system.
Closed systems
Closed systems exchange energy but not matter with an outside system. Though they are typically
portions of larger systems, they are not in complete contact. The Earth is essentially a closed system; it
obtains lots of energy from the Sun but the exchange of matter with the outside is almost zero.
Isolated systems
Isolated systems can exchange neither energy nor matter with an outside system. While they may be
portions of larger systems, they do not communicate with the outside in any way. The physical universe
is an isolated system; a closed thermos bottle is essentially an isolated system (though its insulation is
not perfect).
Heat can be transferred between open systems and between closed systems, but not between
isolated systems.
Example
Example
For example, in figure above, the system
of liquid water plus water vapor in the
sealed container is closed (since no
matter can enter or leave) but not
isolated (since it can be warmed or
cooled by the surrounding bath and can
be compressed or expanded by the
mercury).
A thermodynamic system is either open or closed and is either isolated or non-isolated.
Most commonly, we shall deal with closed systems
WALLS
A system may be separated from its surroundings by various kinds
of walls.
1. A wall can be either rigid or nonrigid (movable).
2. A wall may be permeable or impermeable.
Impermeable means that it allows no matter to pass through it.
3. A wall may be adiabatic or nonadiabatic.
An adiabatic wall is one that does not conduct heat at all,
whereas a nonadiabatic wall does conduct heat.
In Fig. 1.2, the system is
separated from the bath by
the container walls
EQUILIBRIUM
• An isolated system is in equilibrium when its macroscopic
properties remain constant with time.
• A nonisolated system is in equilibrium when the
following two conditions hold:
– The system’s macroscopic properties remain constant with
time;
– removal of the system from contact with its surroundings
causes no change in the properties of the system.
•
If condition (a) holds but (b) does not hold, the system is in a
steady state.
Types of Equilibrium:
1. Mechanical equilibrium
•
No unbalanced forces act on or within the system; hence the system undergoes no
acceleration, and there is no turbulence within the system.
2. Material equilibrium
•
No net chemical reactions are occurring in the system, nor is there any net transfer of
matter from one part of the system to another or between the system and its
surroundings; the concentrations of the chemical species in the various parts of the
system are constant in time.
3. Thermal equilibrium between a system and its surroundings
•
There must be no change in the properties of the system or surroundings when they are
separated by a thermally conducting wall.
Likewise, we can insert a thermally conducting wall between two parts of a system to test
whether the parts are in thermal equilibrium with each other. For thermodynamic
equilibrium, all three kinds of equilibrium must be present.
THERMODYNAMIC PROPERTIES
-used to characterize a system in equilibrium
extensive
Is one whose value is equal to the
sum of its values for the parts of
the system. Thus, if we divide a
system into parts, the mass of the
system is the sum of the masses
of the parts; mass is an extensive
property. So is volume.
Phase
intensive
Is one whose value does not
depend on the size of the
system, provided the system
remains of macroscopic. Density
and pressure are examples of
intensive properties. We can
take a drop of water or a
swimming pool full of water, and
both systems
will have the same density.
A phase is a region of space (a thermodynamic system), throughout which all
physical properties of a material are essentially uniform. Examples of physical
properties include density, index of refraction, and chemical composition
• Extensive Parameters:
– Parameters which values for the composite system are
the sum of the values for each of the subsystems. These
parameters are non-local in the sense that they refer to
– the entire system.
Examples are: Volume, internal energy, mass, length.
• Intensive Parameters:
– These parameters are identical for each subsystem into
which we might subdivide our system.
– Examples are: Pressure, temperature, and density.
• Homogenous system :
– A system is homogenous when it has some chemical
composition throughout.
– e.g. mixture of gases or true solution of solid in liquid.
• Heterogenous system :
– Two or more different phases which are homogenous
but separated by a boundary.
– e.g. Ice in water.
1.3 Temperature
• To determine whether or not thermal equilibrium
exists between systems.
• By definition, two systems in thermal equilibrium
with each other have the same temperature; two
systems not in thermal equilibrium have different
temperatures.
• Symbolized by θ (theta).
The Zeroth Law
Two systems that are each found to be in
thermal equilibrium with a third system
will be found to be in thermal
equilibrium with each other.
It is so called
because only after the first, second, and third
laws of thermodynamics had been formulated
was it realized that the zeroth law is needed for
the development of thermodynamics.
Moreover, a statement of the zeroth law
logically precedes the other three. The
zeroth law allows us to assert the existence of
temperature as a state function.
1.4 The Mole
Relative Atomic Mass, Ar
• The ratio of the average mass of an atom of an element
to the mass of some chosen standard.
• The Relative Atomic Mass of a chemical element gives us
an idea of how heavy it feels (the force it makes when
gravity pulls on it).
•
The relative masses of atoms are measured using an
instrument called a mass spectrometer.
•
Look at the periodic table, the number at the bottom of
the symbol is the Relative Atomic Mass (Ar ):
Relative Molecular Mass, Mr
•
•
•
Most atoms exist in molecules.
To work out the Relative Molecular Mass, simply add up the
Relative Atomic Masses of each atom in the molecule:
A relative molecular mass can be calculated easily by adding
together the relative atomic masses of the constituent atoms.
For example, ethanol, CH3CH2OH, has a Mr of 46 (Try it!).
Gram molecular mass
• Molecular mass expressed in grams is
numerically equal to gram molecular mass of the
substance.
• Molecular mass of O2 = 32Gram
Calculation of Molecular Mass
• Molecular mass is equal to sum of the atomic
masses of all atoms present in one molecule of
the substance.
• Example:
– H2OMass of H atom = 18g
– NaCl = 58.44g
The statement that the molecular weight of H2O is 18.015 means that a water molecule
has on the average a mass that is 18.015/12 times the mass of a 12C atom.
Why unitless?
Find out!
Remember that relative atomic
mass/relative molecular mass is a ratio
and has no units while gram molecular
mass and gram atomic mass are
expressed in grams.
Mole Concept and Avogadro’s Number
• It is convenient to consider the number of atoms needed to make 12g of
carbon and for this number to be given a name - one mole of carbon
atoms.
• Avogadro's number and the mole are very important to the
understanding of atomic structure.
• The Mole is like a dozen. You can have a dozen guitars, a dozen roosters,
or a dozen rocks. If you have 12 of anything then you would have what we
call a dozen. The concept of the mole is just like the concept of a dozen.
• You can have a mole of anything. The number associated with a
mole is Avogadro's number. Avogadro's number is
602,000,000,000,000,000,000,000 (6.02 x 1023).
• A mole of marbles would spread over the surface of the earth, and
produce a layer about 50 miles thick. A mole of sand, spread over the
United States, would produce a layer 3 inches deep. A mole of dollars
could not be spent at the rate of a billion dollars a day over a trillion years.
This shows you just how big a mole is.
• Probably the only thing you will ever have a mole of is atoms or
molecules. One mole of magnesium atoms (6.02 X 1023 ) magnesium
atoms weigh 24.3 grams. 6.02 X 1023 carbon atoms weigh a total of 12.0
grams. 6.02 X1023 molecules of CO2 gas only weigh a total of 44.0 grams.
• The actual number of atoms that is needed to give the relative atomic
mass expressed in grams is called Avogadro's number.
Avogadro's number = 6.02 x 1023
1 Mole C atom = 6.02 x 1023 C atoms = 12g
1 Mole Mg atom = 6.02 x 1023 Mg atoms = 24.3g
Example
• How many atoms are there in 24g carbon?
24g of carbon = 24/12 = 2 moles
1 mole of atoms = 6.02 x 1023
Therefore 2 moles of carbon contains:
2 x 6.02 x 1023 atoms = 1.204 x 1024 atoms
Try This!
• How many atoms and moles of silicon are in a
sample of silicon that has a mass of 5.23g?
• Answers = 0.186 mol Silicon; and
= 1.12 x 1023 atoms
•
Molar Mass, M
• The mole is just a number; it can be used for atoms, molecules, ions,
electrons, or anything else we wish to refer to.
• Because we know the formula of water is H2O, for example, then we
can say one mole of water molecules contains one mole of oxygen
atoms and two moles of hydrogen atoms.
• One mole of hydrogen atoms has a mass of 1.008 g and 1 mol of
oxygen atoms has a mass of 16.00 g, so 1 mol of water has a mass of
(2 x 1.008 g) + 16.00 g = 18.02 g. The molar mass of water is 18.02
g/mol.
1 mol of oxygen atoms has a mass of 16.00 g  Molar mass of O = 16 g/mol
Molar mass of H2O = 2 mol of H + 1 mol of O
= (2x1.008 g/mol of H) + (16 g/mol of O)
= 18.02 g/mol
M = mass = m
mole n
1.5 Ideal Gases
Ideal Gas Law
An ideal gas is defined as one in which all collisions between atoms or molecules are
perfectly elastic and in which there are no intermolecular attractive forces. One can
visualize it as a collection of perfectly hard spheres which collide but which
otherwise do not interact with each other. In such a gas, all the internal energy is in
the form of kinetic energy and any change in internal energy is accompanied by a
change in temperature. An ideal gas can be characterized by three state variables:
absolute pressure (P), volume (V), and absolute temperature (T). The relationship
between them may be deduced from kinetic theory and is called the
Where:
n = number of moles
R = universal gas constant = 8.3145 J/mol K
N = number of molecules
k = Boltzmann constant = 1.38066 x 10-23 J/K = 8.617385 x 10-5 eV/K
k = R/NA
NA = Avogadro's number = 6.0221 x 1023
The Ideal Gas Law
PV = nRT
P = Pressure (in kPa)
T = Temperature (in K)
V = Volume (in L)
n = moles
R = 8.3145 kPa • L
mol • K
R is constant. If we are given three of P, V, n, or
T, we can solve for the unknown value.
R = 82.06 cm 3.atm
mol . K
or
For the Volume-Pressure relationship:
Boyle’s Law
• n1 = n2 and T1 = T2 therefore the n's and T's
cancel in the above expression resulting in the
following simplification:
• P1V1 = P2V2
or PV = constant
(mathematical expression of Boyle's Law)
For the Volume-Temperature relationship:
Charles's Law
• n1 = n2 and P1 = P2 therefore the n's and the P's
cancel in the original expression resulting in the
following simplification:
• V1T2 = V2T1
or V / T = constant
(mathematical expression of Charles's Law)
For the Pressure-Temperature Relationship:
Gay-Lussac's Law
• n1 = n2 and V1 = V2 therefore the n's and the V's
cancel in the above original expression:
• P1T2 = P2T1
or P / T = constant
(mathematical expression of Gay Lussac's Law)
For the Volume-Mole relationship:
Avagadro's Law
• P1 = P2 and T1 = T2 therefore the P's and T's
cancel in the above original expression:
• V1n2 = V2n1
or V / n = constant
(mathematical expression of Avagadro's Law)
Boyle’s Law
•
At constant temperature, the volume of a given quantity of gas is inversely
proportional to its pressure : V 1/P
So at constant temperature, if the volume of a gas is doubled, its pressure is halved.
OR
At constant temperature for a given quantity of gas, the product of its volume and its
pressure is a constant : PV = constant, PV = k
•
At constant temperature for a given quantity of gas : PiVi = PfVf
where Pi is the initial (original) pressure, Vi is its initial (original) volume, Pf is its final
pressure, Vf is its final volume
Pi and Pf must be in the same units of measurement (eg, both in atmospheres), Vi
and Vf must be in the same units of measurement (eg, both in litres).
All gases approximate Boyle's Law at high temperatures and low pressures. A
hypothetical gas which obeys Boyle's Law at all temperatures and pressures is called
an Ideal Gas. A Real Gas is one which approaches Boyle's Law behaviour as the
temperature is raised or the pressure lowered.
•
•
Boyle’s Law
P1V1=P2V2
Charles Law
•
•
•
•
•
At constant pressure, the volume of a given quantity of gas is directly proportional to the
absolute temperature : V T (in Kelvin)
So at constant pressure, if the temperature (K) is doubled, the volume of gas is also doubled.
OR
At constant pressure for a given quantity of gas, the ratio of its volume and the absolute
temperature is a constant : V/T = constant, V/T = k
At constant pressure for a given quantity of gas : Vi/Ti = Vf/Tf
where Vi is the initial (original) volume, Ti is its initial (original) temperature (in Kelvin), Vf is its
final volme, Tf is its final tempeature (in Kelvin)
Vi and Vf must be in the same units of measurement (eg, both in litres), Ti and Tf must be in
Kelvin NOT celsius.
temperature in kelvin = temperature in celsius + 273 (approximately)
All gases approximate Charles' Law at high temperatures and low pressures. A hypothetical gas
which obeys Charles' Law at all temperatures and pressures is called an Ideal Gas. A Real Gas is
one which approaches Charles' Law as the temperature is raised or the pressure lowered.
As a Real Gas is cooled at constant pressure from a point well above its condensation point, its
volume begins to increase linearly. As the temperature approaches the gases condensation
point, the line begins to curve (usually downward) so there is a marked deviation from Ideal
Gas behaviour close to the condensation point. Once the gas condenses to a liquid it is no
longer a gas and so does not obey Charles' Law at all.
Absolute zero (0K, -273oC approximately) is the temperature at which the volume of a gas
would become zero if it did not condense and if it behaved ideally down to that temperature.
Charles Law
V1/V2=T1/T2
P1V1/T1=P2V2/T2
Pressure and Volume Units
VOLUME
PRESSURE
P (Pressure) = F (Force)
A (Area)
ATMOSPHERE
In SI:
2
1 Pa (Pascal) = 1 N/m
1 atm = 760 torr = 1.01325 x 10 5 Pa
Chemists use:
2
1 torr = 133.322 Pa or 2
= 133.322 N/m or
= 133.322 kg/ms
5
1 L = 1 dm3 = 1000 cm 3
1 bar =10 Pa = 0.986923 atm = 750 torr
Example 1.1: Density of an Ideal Gas
Page 16
• Find the density of F2 gas at 20.0°C and 188 torr.
Exercise
• Find the molar mass of a gas whose density is
1.80 g/L at 25.0°C and 880 torr.
• (Answer: 38.0 g/mol.)
1.6 Differential Calculus
Functions and Limits
To say that the variable y is a function of the variable x means that for
any given value of x there is specified a value of y; we write y=f(x).
Dependent variable
Function f
lim y= f (x)
x a
Limit of the function f(x)
as x approaches the value of a
Independent variable
What is a limit?
•
•
A limit is a certain value to which a function approaches. Finding a limit means finding
what value y is as x approaches a certain number. You would typical say that the limit of a
certain function is <a number> as x approaches <some x coordinate>. For example,
imagine a curve such that as x approaches infinity, that curve may come closer and closer
to y=0 while never actually getting there. So, how do we algebraically find that limit? One
way to find the limit is by the SUBSTITUTION METHOD.
For example, the limit of the following graph is 0 as x approaches infinity, because the
graph approaches 0:
y = f(x)
Approaches 0
x
Approaches ∞
Examples
Sample A: Find the limit of f(x) = 4x, as x approaches 3 or
Steps:
1) Replace x for 3.
2) Simplify.
f(x) = 4x becomes f(3) = 4(3) = 12.
So, the limit of f(x) = 4x as x approaches 3 is 12; or
Lim (4x)
x 3
Examples
Sample B: Find the limit:
2
lim
(x
+
5x
–
3)
x 1
Follow the same steps,
2
x + 5x – 3 = 1 2+ 5(1) – 3
= 3
2
So, the limit of x + 5x – 3 as x approaches 1 is 3.
Slope
•
•
•
What is slope?
If you have ever walked up or down a hill, then you have already experienced a real life example of slope
By definition, the slope is the measure of the steepness of a line.
Example: How to find the slope
Examples: How to find the slope when points are given
Let (x1,y1) = (4, 9) and (x2,y2) = (2, 1)
Slope, m = (y1 − y2) = (9 − 1)
(4 − 2 )
(x1 − x2)
=
8
2
=
4
If we write the equation of the straight
line in the form y=mx+b, it follows from
this definition that the line’s slope
equals m.The intercept of the line on
the y axis equals b, since y=b when x=0.
Positive
slope
The Derivative
Derivatives
•
The derivative tells us the rate of change of one quantity compared to another at a
particular instant or point (so we call it "instantaneous rate of change").
•
Let y f (x). Let the independent variable change its value from x to x+ h; this will change
y from f (x) to f (x +h). The average rate of change of y with x over this interval equals the
change in y divided by the change in x and is
•
The instantaneous rate of change of y with x is the limit of this average rate of change
taken as the change in x goes to zero. The instantaneous rate of change is called the
derivative of the function f and is symbolized by f :
•
Wherever a quantity is always changing in value, we can use calculus (differentiation and
integration) to model its behaviour.
•
The derivative of a function with respect to the variable is defined as
•
but may also be calculated more symmetrically as
•
the second derivative may be defined as
•
and calculated more symmetrically as
The Partial Derivative
Partial derivatives are defined as derivatives of a function of multiple variables when all
but the variable of interest are held fixed during the differentiation.
Example - Function of 2 variables
Here is a function of 2 variables, x and y:
F(x,y) = y + 6 sin x + 5y2
•
The derivative is carried out in the same way as ordinary differentiation with this
constraint. For example, given the polynomial in variables x and y,
•
the partial derivative with respect to x is written
•
and the partial derivative with respect to y is written
Partial Differentiation with respect to x
• "Partial derivative with respect to x" means
"regard all other letters as constants, and just
differentiate the x parts".
• In our example (and likewise for every 2variable function), this means that (in effect)
we should turn around our graph and look at
it from the far end of the y-axis. We are
looking at the x-z plane only.
Now for the partial derivative of
•
•
We see a sine curve at the bottom
and this comes from the 6 sin x part
of our function F(x,y) = y + 6 sin x +
5y2. The y parts are regarded as
constants.
(The sine curve at the top of the
graph is just where the software is
cutting off the surface - it could have
been made it straight.)
F(x,y) = y + 6 sin x + 5y2
with respect to x:
The derivative of the 6 sin x part is 6 cos x. The
derivative of the y-parts is zero since they are
regarded as constants.
Notice that we use the symbol "∂" to denote "partial
differentiation", rather than "d" which we use for
normal differentiation.
Partial Differentiation with respect to y
• "Partial derivative with respect to y" means
"regard all other letters as constants, just
differentiate the y parts".
• As we did above, we turn around our graph and
look at it from the far end of the x-axis. So we see
(and consider things from) the y-z plane only.
• We see a parabola. This comes from
the y2 and y terms in F(x,y) = y + 6 sin x + 5y2. The
"6 sin x" part is now regarded as a constant.
Now for the partial derivative of
F(x,y) = y + 6 sin x + 5y2
with respect to y.
The derivative of the y-parts with respect to y is 1 + 10y. The derivative of the 6 sin x part is
zero since it is regarded as a constant when we are differentiating with respect to y.
If now both x and y undergo infinitesimal changes, the infinitesimal
change in z is the sum of the infinitesimal changes due to dx and dy:
In this equation, dz is called the total differential of z(x, y). This
equation is often used in thermodynamics. An analogous equation holds
for the total differential of a function of more than two variables. For
example, if z=z(r, s, t), then:
1.7 Equations of State
Please Read Topic 1.7 (Equations of
State) in page 22 to 25.
•
Equations of State:
An equation of state is a relation between P, V, and T (for a pure material). For a mixture, it must also
involve the composition of the mixture (usually in mole fractions).
•
Liquids:
V
dV
V (T , P)
V
T
V
P
dT
P
dP
T
Define a thermal expansion coefficient and an isothermal compressibility by
•
V
T
1
V
1
V
•
•
•
•
P
V
P
T
These can be assumed constant for liquids, as long as we are not near the critical point.
The equation of state can then be written as
dV
V
V
ln 2
V1
dT
T2
dP
T1
P2
P1
Values of ҡ and β (β=α) can be found in many handbooks.
1.8 Integral Calculus
Definition:
A function F(x) is the antiderivative of a function ƒ(x) if for all x in the domain of ƒ,
F'(x) = ƒ(x)
ƒ(x) dx = F(x) + C, where C is a constant.
• Example 1: Evaluate
• Use formula (4):
• and get this:
Logarithms
• Integration of 1/x gives the natural logarithm ln
x.
End of Chapter 1
Understanding, rather than mindless
memorization, is the key to learning
physical chemistry…