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Chapter 3 entropy. When the entropy of the isolated system is maximized, things cease happen-
The Second Law of Thermodynamics ing on a macroscopic scale, because any further processes can only decrease S, which
would violate the second law. By definition, the isolated system has reached equilib-
rium when processes cease occurring. Therefore (Fig. 3.11):
S
Thermodynamic equilibrium in an isolated system is reached when the system’s
Isolated
system entropy is maximized.
Thermodynamic equilibrium in nonisolated systems is discussed in Chapter 4.
Thermodynamics says nothing about the rate at which equilibrium is attained. An
Equilibrium isolated mixture of H and O at room temperature will remain unchanged in the
reached 2 2
absence of a catalyst. However, the system is not in a state of true thermodynamic
equilibrium. When a catalyst is introduced, the gases react to produce H O, with an
2
increase in entropy. Likewise, diamond is thermodynamically unstable with respect to
conversion to graphite at room temperature, but the rate of conversion is zero, so no
one need worry about loss of her engagement ring. (“Diamonds are forever.”) It can
Time even be said that pure hydrogen is in a sense thermodynamically unstable at room tem-
perature, since fusion of the hydrogen nuclei to helium nuclei is accompanied by an
Figure 3.11 increase in S univ . Of course, the rate of nuclear fusion is zero at room temperature, and
we can completely ignore the possibility of this process.
The entropy of an isolated system
is maximized at equilibrium.

3.6 THE THERMODYNAMIC TEMPERATURE SCALE
In developing thermodynamics, we have so far used the ideal-gas temperature scale,
which is based on the properties of a particular kind of substance, an ideal gas. The
state functions P, V, U, and H are not defined in terms of any particular kind of sub-
stance, and it is desirable that a fundamental property like temperature be defined in a
more general way than in terms of ideal gases. Lord Kelvin pointed out that the sec-
ond law of thermodynamics can be used to define a thermodynamic temperature scale
that is independent of the properties of any kind of substance.
We showed in Sec. 3.2 that, for a Carnot cycle between temperatures t and t ,
H
C
the efficiency e rev is independent of the nature of the system (the working substance)
and depends only on the temperatures: e rev 1 q /q f(t , t ), where t sym-
H
C
C
H
bolizes any temperature scale whatever. It follows that the heat ratio q /q (which
H
C
equals 1 e ) is independent of the nature of the system that undergoes the Carnot
rev
cycle. We have
q >q 1 f1t , t 2 g1t , t 2 (3.41)
H
H
C
C
H
C
where the function g (defined as 1 f ) depends on the choice of temperature scale
but is independent of the nature of the system. By considering two Carnot engines
working with one reservoir in common, one can show that Carnot’s principle (3.6)
(which is a consequence of the second law) requires that g have the form
g1t , t 2 f1t 2>f1t 2 (3.42)
C
H
H
C
where f (phi) is some function. The proof of (3.42) is outlined in Prob. 3.25. Equa-
tion (3.41) becomes
q >q f1t 2>f1t 2 (3.43)
H
H
C
C
We now use (3.43) to define a temperature scale in terms of the Carnot-cycle
ratio q /q . To do so, we choose a specific function for f. The simplest choice
H
C
for f is “take the first power.” This choice gives the thermodynamic temperature