Page 104 - Physical Chemistry

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Most newer power plants recirculate most of their cooling water after cooling it in a Section 3.3
cooling tower, thereby reducing fish kills by 95%. Some power plants avoid the use of Entropy
cooling water by using air cooling, but this is much more expensive than water cooling.
The analysis of this section applies only to heat engines, which are engines that
convert heat to work. Not all engines are heat engines. For example, in an engine that
uses a battery to drive a motor, the energy of a chemical reaction is converted in the
battery to electrical energy, which in turn is converted to mechanical energy. Thus,
chemical energy is converted to work, and this is not a heat engine. The human body
converts chemical energy to work and is not a heat engine.

3.3 ENTROPY
For any closed system that undergoes a Carnot cycle, Eq. (3.16) shows that the inte-
gral of dq /T around the cycle is zero. The subscript rev reminds us of the reversible
rev
nature of a Carnot cycle.
We now extend this result to an arbitrary reversible cycle, removing the constraint
that heat be exchanged with the surroundings only at T and T . This will then show
C
H
that dq /T is the differential of a state function (Sec. 2.10).
rev
The curve in Fig. 3.5a depicts an arbitrary reversible cyclic process. We draw re-
versible adiabats (shown as dashed lines) that divide the cycle into adjacent strips
(Fig. 3.5b). Consider one such strip, bounded by curves ab and cd at the top and bot-
tom. We draw the reversible isotherm mn such that the area under the zigzag curve
amnb equals the area under the smooth curve ab. Since these areas give the negative
of the reversible work w done on the system in each process, we have w amnb w ,
ab
where ab is the process along the smooth curve and amnb is the zigzag process along
the two adiabats and the isotherm. U is independent of the path from a to b, so
U amnb U . From U q w, it follows that q amnb q . Since am and nb are
ab
ab
adiabats, we have q amnb q . Hence q mn q . Similarly, we draw the reversible
ab
mn
isotherm rs such that q q . Since mn and rs are reversible isotherms and ns and
rs
cd
rm are reversible adiabats, we could use these four curves to carry out a Carnot cycle;
Eq. (3.16) then gives q /T mn q /T 0, and
sr
mn
sr
q ab q dc
0 (3.17)
T mn T sr
We can do exactly the same with every other strip in Fig. 3.5b to get an equation sim-
ilar to (3.17) for each strip.
Now consider the limit as we draw the adiabats closer and closer together in
Fig. 3.5b, ultimately dividing the cycle into an infinite number of infinitesimally nar-
row strips, in each of which we draw the zigzags at the top and bottom. As the adia-
bat bd comes closer to the adiabat ac, point b on the smooth curve comes closer to
point a and, in the limit, the temperature T at b differs only infinitesimally from that
b
at a. Let T ab denote this essentially constant temperature. Moreover, T mn in (3.17)
(which lies between T and T ) becomes essentially the same as T . The same thing
b
a
ab
happens at the bottom of the strip. Also, the heats transferred become infinitesimal
quantities in the limit. Thus (3.17) becomes in this limit Figure 3.5
dq ab dq dc An arbitrary reversible cycle and
0 (3.18) its relation to Carnot cycles.
T ab T dc
The same thing happens in every other strip when we take the limit, and an equation
similar to (3.18) holds for each infinitesimal strip. We now add all the equations like
(3.18) for each strip. Each term in the sum will be infinitesimal and of the form dq/T,
where dq is the heat transfer along an infinitesimal portion of the arbitrary reversible