Physical Chemistry Third Edition

120 3 The Second and Third Laws of Thermodynamics: Entropy

Entropy Changes for Nonadiabatic Processes

Consider a closed system and its surroundings arranged as in Figure 3.8. We call

the system plus its surroundings the “combination” and assume that the combination

is isolated from the rest of the universe and can undergo only adiabatic processes.

The entropy of the combination is the sum of the entropy of the system and the entropy

of the surroundings:

ScomS+Ssurr (3.2-23a)

dScomdS+dSsurr (3.2-23b)

where a symbol without a subscript refers to the system. Since the combination can

undergo only adiabatic processes,

dScomdS+dSsurr≥ 0 (3.2-24)

Rest of

universe

Surroundings

System

Adiabatic

insulation

Figure 3.8 A System and Its Sur-
roundings (a “Combination”) Adiabat-
ically Insulated from the Rest of the
Universe.

In order to focus on the system, we rewrite Eq. (3.2-24) in the form

dS≥−dSsurr (3.2-25)

It is not necessary thatdSbe positive. However, ifdSis negative, thendSsurrmust

be positive and large enough that the sumdS+dSsurris not negative. Since we wish

to focus our attention on the system, we make the simplest possible assumption about

the surroundings. We assume that the heat capacity and thermal conductivity of the

surroundings are so large that all processes in the surroundings can be considered to

be reversible. This means that

dSsurr

dqsurr

Tsurr

−

dq

Tsurr

(3.2-26)

The second equality comes from the fact that any heat transferred to the surroundings

must come from the system. Equation (3.2-25) is now

dS≥

dq

Tsurr

(3.2-27)

where the equality applies to reversible processes and the inequality applies to irre-

versible processes. The temperature of the surroundings, not the temperature of the

system, occurs in Eq. (3.2-27). This equation is the complete mathematical statement

of the second law and has been derived from a physical statement of the second law.

It is also possible to take the mathematical statement as a postulate and to derive the

physical statements from it.^2

Exercise 3.4

Show from the mathematical statement of the second law that heat cannot flow from a cooler

object to a hotter object if nothing else happens.Hint: Consider two objects that form a combi-

nation that is isolated from the rest of the universe and are at different temperatures.

(^2) Kirkwood and Oppenheim,op. cit.