Page 103 - Physical Chemistry

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Chapter 3 Hence (3.9) becomes
The Second Law of Thermodynamics
dq 0 Carnot cycle, perf. gas (3.12)
T

We have
dq 2 dq 3 dq 4 dq 1 dq (3.13)
T T T T T
1 2 3 4
Since processes 2 → 3 and 4 → 1 are adiabatic with dq 0, the second and fourth inte-
grals on the right side of (3.13) are zero. For the isothermal process 1 → 2, we have T
2
2
T . Since T is constant, it can be taken outside the integral: T 1 dq T H 1 dq
H
1
1
4
q /T . Similarly, T 1 dq q /T . Equation (3.12) becomes
H
H
C
3
C
dq q H q C 0 Carnot cycle, perf. gas (3.14)
T T H T C
We now find e , the maximum possible efficiency for the conversion of heat to
rev
work. Equations (3.3) and (3.14) give e 1 q /q and q /q T /T . Hence
C H C H C H
T C T T C
H
e rev 1 Carnot cycle (3.15)
T H T H
We derived (3.15) using a perfect gas as the working substance, but since we earlier
proved that e rev is independent of the working substance, Eq. (3.15) must hold for any
working substance undergoing a Carnot cycle. Moreover, since the equations e rev 1
q /q and e rev 1 T /T hold for any working substance, we must have q /q
H
C
H
C
C
H
T /T or q /T q /T 0 for any working substance. Therefore
C
H
H
C
C
H
dq q C q H 0 Carnot cycle (3.16)
T T C T H
Equation (3.16) holds for any closed system undergoing a Carnot cycle. We shall use
(3.16) to derive the state function entropy in Sec. 3.3.
Note from (3.15) that the smaller T is and the larger T is, the closer e ap-
C H rev
proaches 1, which represents complete conversion of the heat input into work output.
Of course, a reversible heat engine is an idealization of real heat engines, which in-
volve some irreversibility in their operation. The efficiency (3.15) is an upper limit to
the efficiency of real heat engines [Eq. (3.7)].
Most of our electric power is produced by steam engines (more accurately, steam
turbines) that drive conducting wires through magnetic fields, thereby generating elec-
tric currents. A modern steam power plant might have the boiler at 550°C (with the
pressure correspondingly high) and the condenser at 40°C. If it operates on a Carnot
cycle, then e 1 (313 K)/(823 K) 62%. The actual cycle of a steam engine is
rev
not a Carnot cycle because of irreversibility and because heat is transferred at temper-
atures between T and T , as well as at T and T . These factors make the actual effi-
H C H C
ciency less than 62%. The efficiency of a modern steam power plant is typically about
40%. (For comparison, James Watt’s steam engines of the late 1700s had an efficiency
of roughly 15%.) River water is commonly used as the cold reservoir for power plants.
A 1000-MW power plant uses roughly 2 million L of cooling water per minute
(Prob. 3.29). About 10% of the river flow in the United States is used by power plants
for cooling. A cogeneration plant uses some of the waste heat of electricity generation
for purposes such as space heating, thereby increasing the overall efficiency.
14
The annual intake of 10 gallons of cooling water from rivers, lakes, and coastal
waters by United States electric-power plants and industrial plants kills billions of fish.