Page 105 - Physical Chemistry

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Chapter 3 cycle and T is the temperature at which this heat transfer occurs. The sum of the
The Second Law of Thermodynamics infinitesimals is a line integral around the cycle, and we get

dq rev 0 (3.19)
T
The subscript rev reminds us that the cycle under consideration is reversible. If it is ir-
reversible, we can’t relate it to Carnot cycles and (3.19) need not hold. Apart from the
reversibility requirement, the cycle in (3.19) is arbitrary, and (3.19) is the desired gen-
eralization of (3.16).
Since the integral of dq /T around any reversible cycle is zero, it follows
rev
2
(Sec. 2.10) that the value of the line integral dq /T is independent of the path be-
1
rev
tween states 1 and 2 and depends only on the initial and final states. Hence dq /T is
rev
the differential of a state function. This state function is called the entropy S:
dq rev
dS K closed syst., rev. proc. (3.20)*
T
The entropy change on going from state 1 to state 2 equals the integral of (3.20):
1
¢S S S 2 dq rev closed syst., rev. proc. (3.21)*
2
1 T
Throughout this chapter we have been considering only closed systems; q is undefined
for an open system.
If a system goes from state 1 to state 2 by an irreversible process, the intermedi-
ate states it passes through may not be states of thermodynamic equilibrium and the
entropies, temperatures, etc., of intermediate states may be undefined. However, since
S is a state function, it doesn’t matter how the system went from state 1 to state 2; S
is the same for any process (reversible or irreversible) that connects states 1 and 2. But
it is only for a reversible process that the integral of dq/T gives the entropy change.
Calculation of S in irreversible processes is considered in the next section.
Clausius discovered the state function S in 1854 and called it the transformation
content (Verwandlungsinhalt). Later, he renamed it entropy, from the Greek word trope,
meaning “transformation,” since S is related to the transformation of heat to work.
Entropy is an extensive state function. To see this, imagine a system in equilib-
rium to be divided into two parts. Each part, of course, is at the same temperature T.
Let parts 1 and 2 receive heats dq and dq , respectively, in a reversible process. From
2
1
(3.20), the entropy changes for the parts are dS dq /T and dS dq /T. But the
1
2
2
1
entropy change dS for the whole system is
dS dq>T 1dq dq 2>T dq >T dq >T dS dS 2 (3.22)
1
1
2
2
1
Integration gives S S S . Therefore S S S , and S is extensive.
2
1
2
1
For a pure substance, the molar entropy is S S/n.
m
The commonly used units of S in (3.20) are J/K or cal/K. The corresponding units
of S are J/(mol K) or cal/(mol K).
m
The path from the postulation of the second law to the existence of S has been a
long one, so let us review the chain of reasoning that led to entropy.
1. Experience shows that complete conversion of heat to work in a cyclic process is
impossible. This assertion is the Kelvin–Planck statement of the second law.
2. From statement 1, we proved that the efficiency of any heat engine that operates
on a (reversible) Carnot cycle is independent of the nature of the working sub-
stance but depends only on the temperatures of the reservoirs: e rev w/q
H
1 q /q f (t , t ).
C
H
C
H