Physical Chemistry Third Edition

108 3 The Second and Third Laws of Thermodynamics: Entropy

V 1 V 4 V 2

V

P

P 1

P 2

P 4

P 3

V 3

(b)

Step 1:

Isothermal expansion

Step 2:

Adiabatic expansion

Step 3:

Isothermal

compression

Step 4:

Adiabatic

compression

V 1 V 4 V 2 V 3

V

T

Tc

Th

(a)

Step 2:

Adiabatic expansion

Step 3:

Isothermal

compression

Step 4:

Adiabatic

compression

Step 1: Isothermal expansion

Figure 3.2 The Path of the State Point during a Carnot Cycle.(a) In theV–Tplane. (b) In theV–Pplane.

the cylinder in contact with the cold reservoir. This step ends at a volume such that

the fourth step, a reversible adiabatic compression, ends with the piston at top dead

center and the system at the temperature of the hot reservoir. The engine is now ready

to repeat the cycle.

Figure 3.2a shows the path that the state point of the system follows as the engine

undergoes one cycle, usingVandTas the state variables. The state at the beginning

of each step is labeled with the same number as the step. Figure 3.2b shows the same

cycle usingVandPas the state variables.

Since the second and fourth steps of the Carnot cycle are adiabatic,

q 2 q 4  0 (3.1-2)

For the entire cycle,

qcycleq 1 +q 2 +q 3 +q 4 q 1 +q 3 (3.1-3)

SinceUis a state function and because the cycle begins and ends at the same state,

∆Ucycle 0 (3.1-4)

From the first law of thermodynamics,

wcycle∆Ucycle−qcycleqcycle−q 1 −q 3 (3.1-5)

The efficiency,ηCarnot, of the Carnot engine is the work done on the surroundings

divided by the heat input from the hot reservoir. The heat exhausted at the cold reservoir

is wasted and is not included in the efficiency calculation.

ηCarnot

wsurr

q 1



−wcycle

q 1



qcycle

q 1



q 1 +q 3

q 1

 1 +

q 3

q 1

(3.1-6)

From the Kelvin statement of the second law, the efficiency must be less than unity, so

thatq 3 must be negative. It is not possible to run a Carnot engine without exhausting

some heat to a cool reservoir.