Physical Chemistry Third Edition

3.3 The Calculation of Entropy Changes 121

No violations of the second law of thermodynamics have ever been observed in

a properly done experiment, so there is no reason to doubt its applicability. If it is

universally applicable, the ultimate fate of the universe will be to approach a state of

thermodynamic equilibrium in which every object in the universe will be at the same

temperature. There will be no energy flow from stars to planets, and no life or any

other macroscopic processes will be possible. This“heat death” of the universewill of

course not occur for a very long time, but is unavoidable if the second law is universally

valid.

Some people have speculated that the second law might not be universally valid,

but might just be a statement of what nearly always occurs. If so, perhaps under some

circumstances violations of the second law could be observed (possibly if the universe

begins to contract instead of expand). This idea is unsupported speculation, and we

have every reason to apply the second law of thermodynamics to any process in any

macroscopic system.^3

PROBLEMS

Section 3.2: The Mathematical Statement of the Second
Law: Entropy

3.12 Calculate the entropy change for each of the four steps in
the Carnot cycle of Problem 3.2, and show that these
entropy changes sum to zero.

3.3 The Calculation of Entropy Changes

For a process in a closed system that begins at an equilibrium or metastable state and

ends at an equilibrium state, the entropy change of the process is given by the line

integral on a reversible path from the initial state to the final state.

∆SSfinal−Sinitial

∫

c

dS

∫

c

dqrev

T

(3.3-1)

Since entropy is a state function, we can calculate∆Sfor a process that is not reversible

so long as it has equilibrium or metastable initial and final states by calculating on a

reversible path with the same initial and final states.

Entropy Changes of Isothermal Reversible Processes

in Closed Systems

Since the process is reversible we integrate along the actual path of the process, and

sinceTis constant we can factor 1/Tout of the integral:

∆S

∫

c

dqrev

T



1

T

∫

c

dqrev

qrev

T

(reversible isothermal

process, closed system)

(3.3-2)

(^3) See S. Frautschi, “Entropy in an Expanding Universe,”Science, 217 , 592 (1982).