1. Maxwell Relations
Maxwell relations are a “family” of relations between
derivatives of thermal parameters. Their list is quite long.
We will show how to derive some we most often use.
The procedure used for deriving them is always the same –
first, you take a differential of U, or the differential of one
of the “thermodynamic functions” that are obtained by
performing a Legendre transformation of U. We have no
time in this course to discuss in detail what a Legendre
Transformation is – but the transformed functions, as you
can see below, have rather simple forms:
H ≡ U + pV – this is the enthalpy we already know;
F ≡ U – TS – is called the Helmholtz Free Energy;
G ≡ U + pV – TS – is called the Gibbs Function.
The list is much longer, but these three are the most often
used ones.

2. The Maxwell relation we obtain by processing U :
We begin with writing the First Law in the form:
It clearly suggests a functional relation:
Therefore, one can write:
From the two differential
forms it follows that:
Now, we simply apply the
Euler Criterion and we get
a Maxwell relation:

3. Now, let’s do the same trick using the Gibss Function:
By comparing
the two dG
forms, we get:
By applying the Euler Criterion to the dG form obtained,
we get another useful Maxwell relation:
We will
soon use
these two!

4. Heat capacities expressed in terms of entropy and enthalpy
All essential parameters describing thermal systems are
interrelated through the state equations. Therefore, one
parameter can be expressed as a function of the other
parameters in different ways. For instance, entropy can be
though of either as a function of T and V: S=S(T,V),
or as a function of T and p: S=S(T,p).
In a constant
V process
dV=0, hence:
meaning
that:

5. A similar trick can be done to obtain the heat capacity Cp
expressed in terms of entropy:
In a p=const.
process dp=0,
and then:
meaning
that:
In term of enthalpy H: we have
already shown earlier that:

6. Heat capacity – summary:
BUT
!!!!

7. Joule-Thompson Process again:
We further processed
that derivative, but is
was a very tedious job.
Earlier we have shown
(File 4, slide 5) that:
Here is a much faster method.
1. Let’s use the cyclic
chain rule:
2. Take the Gibbs function and the enthalpy:
So:
On the
preceding
slide we
showed
this is Cp

8. Continiued from Slide 7:
Use the
standard
coefficient:
to obtain
a compact
form:
Putting everyting together, we obtain:
which is exactly the same expression as that we have
obtained earlier using a much more time-consuming method.