Page 175 - Physical Chemistry

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Chapter 5 The degree superscript in (5.22) indicates the standard pressure of 1 bar. The sub-
Standard Thermodynamic script zero indicates a temperature of absolute zero. As we shall see, absolute zero is
Functions of Reaction
unattainable, so we use the limit in (5.22). Helium remains a liquid as T goes to zero
at 1 bar. All other elements are solids in this limit. Since elements are never inter-
converted in chemical reactions, we are free to make the arbitrary assignment (5.22)
for each element.
To find the conventional S° m,T for an element at any T, we use (5.22) and the
constant-P equation S T 2 (C /T) dT [Eq. (3.28)], including also the S of any
T 1 P
phase changes that occur between absolute zero and T.
How do we find the conventional entropy of a compound? We saw that U or H
values for reactions are readily measured as q or q for the reactions, and these H val-
V
P
ues then allow us to set up a table of conventional enthalpies (or enthalpies of formation)
for compounds. However, S for a chemical reaction is not so easily measured. We have
S q /T for constant temperature. However, a chemical reaction is an irreversible
rev
process, and measurement of the isothermal irreversible heat of a reaction does not give
S for the reaction. As we shall see in Chapter 13, one can carry out a chemical reac-
tion reversibly in an electrochemical cell and use measurements on such cells to find S
values for reactions. Unfortunately, the number of reactions that can be carried out in an
electrochemical cell is too limited to enable us to set up a complete table of conventional
entropies of compounds, so we have a problem.

The Third Law of Thermodynamics
The solution to our problem is provided by the third law of thermodynamics. About
1900, T. W. Richards measured G° as a function of temperature for several chemical
reactions carried out reversibly in electrochemical cells. Walther Nernst pointed out
that Richards’s data indicated that the slope of the G°-versus-T curve for a reaction
goes to zero as T goes to absolute zero. Therefore in 1907 Nernst postulated that for
any change

lim 10 ¢G>0T2 0 (5.23)
P
TS0
From (4.51), we have (
G/
T) S; hence (
G/
T)
(G G )/
T
G /
T
2
P
1
P
2

G /
T S S S. Thus (5.23) implies that
1
2
1
lim ¢S 0 (5.24)
TS0
Nernst believed (5.24) to be valid for any process. However, later experimental work
by Simon and others showed (5.24) to hold only for changes involving substances in
internal equilibrium. Thus (5.24) does not hold for a transition involving a supercooled
liquid, which is not in internal equilibrium. (See also Sec. 21.9.)
We therefore adopt as the Nernst–Simon statement of the third law of thermo-
dynamics:
For any isothermal process that involves only substances in internal equilibrium,
the entropy change goes to zero as T goes to zero:
lim ¢S 0 (5.25)*
TS0
The Nernst–Simon statement is often restricted to pure substances but in fact it is valid for
mixtures. (See J. A. Beattie and I. Oppenheim, Principles of Thermodynamics, Elsevier,
1979, secs. 11.18, 11.19, and 11.24.) The ideal-gas entropy-of-mixing formula (3.33) gives
a nonzero isothermal S of mixing that is independent of T and so seems to contradict the
third law. The term ideal gas as used so far in this book means a classical ideal gas, which
is a gas with no intermolecular interactions and with the molecules obeying classical
mechanics. For such gases, PV nRT and the mixing formula (3.33) are obeyed. In reality,