Lecture Notes for Chapter 5
In this chapter we will take the ideas of the second law and apply them further. The first
thing that is addressed is the incorporation of the concepts of entropy into internal energy.
Simply it amounts to taking the relationship between entropy and heat and that between
work, pressure and volume changes. The first law says:
dU  dw  dq
The second law says:
dqrev  TdS
Our definition of expansion-type work tells us:
dw  PdV
putting all of this together we have:
dU  TdS  PdV
Chapter 5.1
This last equation suggests that perhaps we should consider U as a function of S and V
(remember before we used T and V but now we have S which is a function of S and it
will turn out that the relationships between U and S are useful). In other words, S is just
a function of T, V and P, so if we know any two of the four (T, V, P, or S) we know the
other one. We can write our usual expansion of dU, now in terms of dS and dV instead
of dT and dV:
 U 
 U 
dU    dS    dV
 S  V
 V  S
But this has the same form as the equation above for dU:
dU  TdS  PdV
By comparison, we can see that:
 U 

 T
 S  V
 U 

  P
 V  S
The book then goes on to derive a series of such relationships which are referred to as the
Maxwell equations. While these are very important relationships to the theoretical
physical chemist, they do not have a very obvious conceptual value (they are very useful
in deriving other things) for us at this point and I will not require that you do more than
know that they exist.
Chapter 5.2
Instead we will go on to trying to develop relationships for the Gibbs energy in analogy
to those derived above for the internal energy. Recall that G=H-TS. Thus we can write
down in general for dG:
dG  dH  TdS  SdT
but recall that
dH  dU  PdV  VdP
This comes from the definition of H, H=U+PV. We found a little bit ago that:
dU  TdS  PdV
If we combine all of this together:
dG  TdS  PdV  PdV  VdP  TdS  SdT
dG  VdP  SdT
This again suggests that G should be a function of P and T and we can write:
 G 
 G 
dG    dP    dT
 P  T
 T  P
comparing this to the equation above, we see that
 G 
  V
 P  T
 G 
   S
 T  P
With these two relationships, we can determine what will happen to the Gibbs free
energy when either the pressure is adjusted or the temperature is changed. This will be
very important when we start to try and work through how the phase of a substance
changes with pressure and temperature. What we will ask is, at any given temperature
and pressure, which phase has the lowest Gibbs free energy? These equations tell us how
that energy should change with temperature and pressure and thus allow us to predict
which phase (gas, liquid or solid) will have the lowest free energy. We can generalize
this to chemical reactions as well.
For now, we can just use the above equations in a more general sense. If we wanted to
determine the change in free energy with pressure of an ideal gas, we could just substitute
V by nRT/P and then integrate to obtain
Pf
Pf
nRT
G  
 nRT ln
P
Pi
Pi
For most liquids, which have only very small Volume changes at moderate pressures, we
can say the V is roughly constant and we simply find that G = V(Pf - Pi). See example
5.2 in the book for a specific application.
Similarly, changes in temperature can be dealt with. However, here S is rarely
independent of T, so we substitute in (G-H)/T for S (remember that G = H - TS) and then
do a series of algebraic manipulations and obtain:
GH
 G 

  S  
T
 T  P

   G 
H

    2
T
 T  T   P
This is the general form of the Gibbs-Helmholtz equation, but in its present form it is a
bit difficult to see what it is good for. It can be rearranged (see Example 5.3) to obtain:
 G / T 

 H
  1 / T   P
This is useful experimentally, because if one were able to measure the Gibbs free energy
as a function of Temperature and then to plot G/T vs. 1/T, one should get a line with a
slope equal to the enthalpy. As we will see later, G is closely related to T times the log of
the relative concentrations of reactants and products. Usually one measures these
concentrations and makes plots closely related to the G/T vs. 1/T plot to determine H.
Chapter 5.3
Ok, most of what we have done so far in chapter 5 is just a setup for things that we will
use in chapters 6 and 7. However, the next point is key, as it opens the door to using the
free energy in a very general way for many different kinds of processes. We will start be
making another definition. I am afraid that the large number of different names for terms
or expressions is sort of a historical artifact which cannot be avoided, but if you buckle
down and learn the names, you will find both the book and the literature much easier to
read. This term is called the chemical potential. I am afraid it has the same symbol as
does the Joule-Thompson coefficient, but the two are completely unrelated.
 G 
 
 n  T , P
This is called the chemical potential. What is really is is the change in Gibbs free energy
per mole of substance or (for a pure substance) just the molar Gibbs free energy. We
give it a special name because later we will apply it to mixtures and talk about the
chemical potentials of individual components. However, remember that the chemical
potential of some compound is just it molar Gibbs free energy.
This is useful, because now we can add up all the chemical potentials for the individual
components of a system, multiply each by its number of moles and end up with the total
Gibbs free energy. More on this later. For now, lets think about what this tells us about
simple cases. Remember what happened to the Gibbs free energy when we changed the
pressure of an ideal gas:
Pf
Pf
nRT
G  
dP  nRT ln
P
Pi
Pi
Rewriting this, we have:
Pf
G( Pf )  G( Pi )  nRT ln
Pi
We can put this in terms of the molar free energy (the chemical potential) since it is for a
pure substance by simply dividing by n:
Pf
G ( Pf )  G ( Pi )  RT ln
Pi
 ( Pf )   ( Pi )  RT ln
Pf
Pi
Now, where things become more useful is if we define (as we have done before) a
standard state for the ideal gas (P = 1 bar) and define the chemical potential at any other
pressure as the sum of the chemical potential at the standard pressure plus the change in
the chemical potential (in other words, just let PI be 1 bar, which we call P0, and then
always define the chemical potential at any other pressure, P, relative to the chemical
potential at P0:
P
 ( P)   ( P )  RT ln 
P
or
P
P
This equation you will use about a hundred times in the next two months, so make sure
you understand it. It simply says that we can define a molar free energy for any
substance (in this case an ideal gas, but we will generalize as time goes on and the
equation will always look the same just with appropriate fudge factors built in) as just the
molar free energy at the standard condition plus some term that depends on the
temperature and the log of the relative amount of the substance (expressed as a pressure
in the case of a gas or later a concentration in the case of a solute in a liquid). This is a
very general concept which we will see over and over again. Let's give it just a bit more
thought. Lets take the familiar Gibbs free energy equation and manipulate it a bit and see
what we get:
G  H  TS
But for an ideal gas at constant temperature:
V
P
S  nR ln F   nR ln F
VI
PI
     RT ln
PF
PI
if PI = 1 bar or P0 then
P
S   R ln 
P
So for an ideal gas, the chemical potential at some particular pressure is just the chemical
potential at the standard pressure plus the change in entropy associated with changing to
the new pressure. In general, we will try to split things into terms that are properties of
the molecules in question (e.g., the chemical potential at a standard pressure or
concentration), and terms that simply have to do with changing the amount of room or
total number of states that are available to the molecules. The first of these have usually
a large enthalpic component, depending on molecular structure and interactions, while
the second term depends primarily on the change in entropy that occurs upon changing
the concentration or pressure.
S   R ln
Ok, so let's consider the first situation. What happens if the gas is not an ideal gas? How
do we come up with the chemical potential now? Well, we just … well kinda…well, to
be honest… we arbitrarily throw in a fudge factor. What we do is to replace the pressure
with an "effective" pressure. In other words, the pressure that comes from applying the
ideal gas law. We call this effective pressure a fugacity. Sounds impressive anyway. In
fact the definition of fugacity, f, is that it is the number that makes the equation
f
P
work for a nonideal gas. Then we have a fudge factor that we give the auspicious name
"fugacity coefficient" and label as  that relates the real pressure to our effective pressure:
f  P
     RT ln
so
P
P
It turns out we do this a great deal in physical chemistry. And we get away with it
because in equations like the one above, the fudge factor just becomes a offset which
often does not effect the change in free energy of chemical potential of the system very
much.
Don't concern yourselves too much with the calculation of fugacity given at the end of
the chapter. Just understand what it is and how it relates to the true pressure.
     RT ln   RT ln