Physical Chemistry Third Edition

3.1 The Second Law of Thermodynamics and the Carnot Heat Engine 109

ACarnot heat pumpis a Carnot heat engine that is driven backwards by another

engine. It removes heat from the cool reservoir and exhausts heat into the hot reservoir.

Figure 3.3 represents a Carnot heat pump cycle, which is the reverse of the cycle of

Figure 3.2. The steps are numbered with a prime (′) and are numbered in the order in

which they occur. Since we are considering the same Carnot engine run backwards,

q′ 4 −q 1 (3.1-7)

and

q′ 2 −q 3 (3.1-8)

V

T

Th

(a)

Step 3:

Adiabatic

compression

Step 2:

Isothermal expansion

Step 1:

Adiabatic

expansion

Step 4: Isothermal compression

Tc

Figure 3.3 The Path of the State

Point in theV–TPlane during a Carnot

Heat Pump Cycle.

Because the cycles are reversible, the amount of work done on the system in the

reverse (heat pump) cycle,w′, is equal to the amount of work done on the surroundings

in the forward (engine) cycle:

w′cycle−wcyclewsurr (3.1-9)

For a heat pump, the output is the heat delivered to the hot reservoir and the input is the

work put into the heat pump. The ratio of the output to the input is called thecoefficient

of performanceof the heat pump and is denoted byηhp. The coefficient of performance

equals the reciprocal of the Carnot efficiency, because the input and output are reversed

in their roles as well as their signs.

ηhp

∣

∣q′

4

∣

∣

w′cycle

−

q′ 4

−q′ 2 −q′ 4



q 1

q 1 +q 3



1

1 +q 3 /q 1



1

ηc

(3.1-10)

The Carnot efficiency is always smaller than unity, so the Carnot heat pump coefficient

of performance is always greater than unity. The amount of heat delivered to the hot

reservoir is always greater than the work put into the heat pump because some heat has

been transferred from the cold reservoir to the hot reservoir. There is no violation of

the Clausius statement of the second law because the heat pump is driven by another

engine. A real heat pump must have a lower coefficient of performance than a reversible

heat pump but can easily have a coefficient of performance greater than unity.

No reversible heat engine can have a greater efficiency than the Carnot engine. We

prove this by assuming the opposite of what we want to prove and then show that

this assumption leads to a contradiction with experimental fact and therefore must be

incorrect. Assume that a reversible heat engine does exist with a greater efficiency than

a Carnot engine. We call this engine a “superengine” and label its quantities with the

letter s and label the quantities for the original Carnot engine by the letter c. By our

assumption,ηs>ηc, so that

1 +

q 3 (s)

q 1 (s)

> 1 +

q 3 (c)

q 1 (c)

(3.1-11)

Now use the superengine to drive the Carnot engine as a heat pump between the same

two heat reservoirs that are used by the superengine. If there is no friction all of the

work done by the engine is transmitted to the heat pump:

w(s)−w′(c) (3.1-12)