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3 The second and third laws
3A Entropy
Answers to discussion questions
D3A.2 Everyday experience indicates that the direction of spontaneous change in an
isolated system is accompanied by the dispersal of the total energy of the sys-
tem. For example, for a gas expanding freely and spontaneously into a vacuum,
the process is accompanied by a dispersal of energy and matter. For a perfect
gas this entropy change of such an expansion is derived in Section 3A.2(a) on
page 80 as ∆S = nR ln(Vf/Vi). �e entropy change is clearly positive if Vf is
greater than Vi.
�emolecular interpretation of this thermodynamic result is based on the iden-
ti�cation of entropy with molecular disorder. An increase in disorder results
from the chaotic dispersal of matter and energy and the only changes that can
take place within an isolated system (the universe) are those in which this kind
of dispersal occurs. �is interpretation of entropy in terms of dispersal and
disorder allows for a direct connection of the thermodynamic entropy to the
statistical entropy through the Boltzmann formula S = k lnW , where W is
the number of microstates, the number of ways in which the molecules of the
system can be arranged while keeping the total energy constant.
�e concept of the number of microstates makes quantitative the more ill-
de�ned qualitative concepts of ‘disorder’ and ‘the dispersal of matter and en-
ergy’ used above to give a physical feel for the concept of entropy. A more
‘disorderly’ distribution of energy and matter corresponds to a greater number
of microstates associated with the same total energy.
Solutions to exercises
E3A.1(b) For the process to be spontaneous it must be irreversible and obey the Clausius
inequality [3A.12–86] implying that ∆Stot = ∆S + ∆Ssur > 0. In this case,
∆Stot = 105 JK−1 + (−95 JK−1) = +10 JK−1, the total entropy increases and
thus the process is spontaneous .
E3A.2(b) �e thermodynamic de�nition of entropy is [3A.1a–80], dS = dqrev/T or for
a �nite change at constant temperature ∆S = qrev/T . �e transfer of heat is
speci�ed as being reversible, which can o�en be assumed for a large enough
metal block, therefore qrev = 250 kJ.