Problems and Solutions on Thermodynamics and Statistical Mechanics

Thermodynamics 61

1066
Consider an arbitrary heat engine which operates between two reser-
voirs, each of which has the same finite temperature-independent heat ca-
pacity c. The reservoirs have initial temperatures TI and T2, where T2 > TI,
and the engine operates until both reservoirs have the same final tempera-
ture T3.
(a) Give the argument which shows that T3 > m.
(b) What is the maximum amount of work obtainable from the engine?
(UC, Berkeley)
Solution:
(a) The increase in entropy of the total system is

Thus T: 2 T1T2, or T3 2 a.

(b) The maximum amount of work can be obtained using a reversible

heat engine, for which AS = 0.

Wmax = C(TI + T2 - 2T31nin) = c(T1 + T2 - 2m) = ~(fi-.

1067

(a) What is the efficiency for a reversible engine operating around

the indicated cycle, where T is temperature in K and S is the entropy in

joules/K?

T

300 - - - - - - - - -

~oo~;--------rL~ S

Fig. 1.22.

(b) A mass M of a liquid at a temperature TI is mixed with an equal

mass of the same liquid at a temperature T2. The system is thermally

insulated. If cp is the specific heat of the liquid, find the total entropy

change. Show that the result is always positive.

(UC, Berkeley)