Page 128 - Essentials of physical chemistry

90 Essentials of Physical Chemistry

However, we could take the second derivative in the reverse order by taking the first derivative
with respect to P first and then with respect to S as shown here:

qH
¼ V
qP
S
so that

2
q H qV
:
qS qP ¼ qS P
But a state variable has the property that it does not matter which variable is changed first, so the
two second derivatives must be the same.
qT qV
Thus, we obtain the Maxwell relationship for H as ¼ . You can prove the other
qP S qS P
three equations easily in the homework problem set.
What is the meaning of these eight equations? They are mathematical truisms, which we can use
for problem solving. For instance, while the basic equation for the Helmholtz free energy, A, is not
often useful, the fourth Maxwell equation proves to be very useful. For instance, while we see that

qS qP qS
it is very difficult to imagine what could possibly mean in terms of
¼
qV qT qV
T V T
something we can measure in the laboratory, but it is very convenient to realize it is equal to the
pressure change with respect to temperature when the volume is constant, which is easily measured.
Thus, the Maxwell relationships are very valuable in resolving strange dependencies among the four
state variables H, U, G, and A as related to laboratory variables (P, V, T). Perhaps we have violated
the formal presentation of the historical development of thermodynamics, but for the students we
have just saved you countless hours of frustrating reading, and NOW we are ready to do some real
thermodynamics (after we get past the third law). It might be a good idea to write the eight equations
over a few times till you master the patterns therein. Knowing those eight equations is the key to
thermodynamics.

THIRD LAW OF THERMODYNAMICS

So far, we are confident that entropy exists and can be described quantitatively. On the other hand,
we have only talked about DS so far. The mention of increasing disorder as a liquid vaporizes into a
gas or even as a solid melts into a liquid helps us to realize that entropy is related to disorder
somehow. This leads up to the idea that there is an absolute value for the entropy of a substance at a
given temperature. Once again, the history of the third law is shared by several scientists where
credit is given for later consolidation of ideas developed earlier by others. W. Nernst (1864–1941) is
generally given the main credit for his work in 1905 called the Nernst heat theorem for which he
received the Nobel Prize in 1920. Although Nernst can be said to have founded the field of physical
chemistry, his work translated into what is now analytical chemistry. Many students associate his
name with the ‘‘Nernst equation’’ of electrochemistry. Much of what is now analytical chemistry
was formerly the field of research in physical chemistry, especially in electrochemistry, but a split
occurred later in the 1930s when physical chemists were lured into the fields of spectroscopy opened
up by the development of quantum mechanics. However, the mathematical basis for the third law
was already put in place by Boltzmann in the 1880s and we will mainly use the Boltzmann statistical
approach to entropy. In words, we favor a simple statement of the law:

The entropy of a pure, perfectly crystalline substance is zero at 08K.