Physical Chemistry Third Edition

110 3 The Second and Third Laws of Thermodynamics: Entropy

From Eq. (3.1-10) the amount of heat put into the hot reservoir by the Carnot heat

pump is equal to

−q′ 4 (c)

w′(c)

1 +q 3 (c)/q 1 (c)

The amount of heat removed from the hot reservoir by the superengine is

q 1 (s)−

w(s)

1 +q 3 (s)/q 1 (s)

<

w′(c)

1 +q 3 (c)/q 1 (c)

Therefore

q 1 (s)<−q′ 4 (c)

A greater amount of heat has been put into the hot reservoir by the Carnot heat pump

than was removed from this reservoir by the superengine. This conclusion is contrary

to the Clausius statement of the second law. The only source of this contradiction is

our assumption that a superengine exists with a greater efficiency than that of a Carnot

engine, so the efficiency of the second reversible engine cannot be larger than that of

a Carnot engine.

The second reversible heat engine also cannot have a smaller efficiency than the

first Carnot engine. If it did its coefficient of performance as a heat pump, which is

the reciprocal of its efficiency as a heat engine, would be larger than that of a Carnot

heat pump, and the second law could be violated by using the first engine to drive the

second engine as a heat pump. We have shown thatthe efficiency of a reversible heat

engine operating with two heat reservoirs does not depend on the nature of the working

fluid or on the details of its design, but depends only on the temperatures of the heat

reservoirs.

Exercise 3.1

Carry out the proof that a reversible engine cannot have a smaller efficiency than a Carnot engine

if it uses the same heat reservoirs.

A heat engine operating irreversibly can have a lower efficiency than a Carnot

engine. If a heat engine operates irreversibly its coefficient of performance as a heat

pump will not necessarily be the reciprocal of its engine efficiency since each step

cannot necessarily be reversed. Therefore, driving the irreversible engine backward as

a heat pump by a Carnot engine would not necessarily violate the second law if it has

a lower efficiency than a Carnot engine.

The Thermodynamic Temperature and the Zeroth Law

of Thermodynamics

The zeroth law of thermodynamics states a fact that many had tacitly accepted during

the development of thermodynamics, and it gradually became apparent that it required

a formal statement. It is stated:If two objects, A and B, are at thermal equilibrium

with each other and if B is at thermal equilibrium with a third object, C, then A is also

at thermal equilibrium with C.This law is considered to be basic to the other laws

of thermodynamics, so it is called the zeroth law of thermodynamics, although it was