Thermodynamics and Chemistry

CHAPTER 4 THE SECOND LAW

4.6 APPLICATIONS 125

zero, the entropy change of the experimental system is dS, the heat transferred between the
Carnot engine and the experimental system is∂q, and the heat transferred between the heat
reservoir and the Carnot engine is given by∂q^0 DTres∂q=Tb(Eq.4.4.1). The heat transfer
between the heat reservoir and Carnot engine is reversible, so the entropy change of the heat
reservoir is

dSresD

∂q^0

Tres

D

∂q

Tb

(4.5.1)

The entropy change of the supersystem is the sum of the entropy changes of its parts:

dSssDdSCdSresDdS

∂q

Tb

(4.5.2)

The process within the supersystem is adiabatic and includes an irreversible change within
the experimental system, so according to the conclusions of Sec.4.5.1, dSssis positive.
Equation4.5.2then shows that dS, the infinitesimal entropy change during the irreversible
change of the experimental system, must be greater than∂q=Tb:

dS >

∂q

Tb

(4.5.3)

(irreversible change, closed system)

This relation includes the case of an irreversibleadiabaticchange, because it shows that if
∂qis zero, dSis greater than zero.
By integrating both sides of Eq.4.5.3between the initial and final states of the irre-
versible process, we obtain a relation for the finite entropy change corresponding to many
infinitesimal cycles of the Carnot engine:

ÅS >

Z

∂q

Tb

(4.5.4)

(irreversible process, closed system)

4.6 Applications

The lengthy derivation in Secs.4.3–4.5is based on the Kelvin–Planck statement describing
the impossibility of converting completely into work the energy transferred into the sys-
tem by heat from a single heat reservoir. The derivation has now given us all parts of the
mathematical statement of the second law shown in the box on page 102. The mathemati-
cal statement includes an equality, dSD∂q=Tb, that applies to an infinitesimalreversible
change, and an inequality, dS > ∂q=Tb, that applies to an infinitesimalirreversiblechange.
It is convenient to combine the equality and inequality in a single relation that is a general
mathematical statement of the second law:

dS

∂q

Tb

(4.6.1)

(irrevrev, closed system)

The inequality refers to an irreversible change and the equality to a reversible change, as
indicated by the notation irrevrev in the conditions of validity. The integrated form of this