Problems and Solutions on Thermodynamics and Statistical Mechanics

58 Problems 8 Sdutio~ on Thermodpamics 8 Statistical Mechanics

At every temperature point, there exists a large heat rcservoir. Let
the water come into contact with them successively from low temperature
to high temperature, to make the process of thermal contact quasi-static.
Then AS = 0 at every step and consequently for the entire process.

1062
Two finite, identical, solid bodies of constant total heat capacity per
body, C, are used as heat sources to drive heat engine. Their initial tem-
peratures are TI and T2 respectively. Find the maximum work obtainable
from the system.

Soh t ion :
As energy is conserved, the work obtainable is W = C(T1 + T2 - 2q),
where Tf is the final temperature of the system. From the second law of
thermodynamics, we have

(MIT)

Tf Tr

Tl TZ

AS = Cln - + Cln - > 0 , so that Tf > m.

Hence W,,, = C(T1 + T2 - 2m).

1063

A rigid box containing one mole of air at temperature 2’0 (in K) is

initially in thermal contact with an “infinite’ heat-capacity reservoir” at

the same temperature TO. The box is removed from the reservoir and a

cyclic engine is used to take some heat from the reservoir and put some

into the air in the box. What is the minimum amount of work from To

to TI? Express W in terms of TO, TI and the gas constant R, and state

units. Ignore vibrational degrees-of-freedom in the air molecules and the

heat capacity of the container. Would inclusion of vibrational degrees-of-

freedom increase or reduce the value of W?

( Columbia)

Solution:

“infinite heat-capacity reservoir” , we get

As AQ + W = C,(T1 - To), where AQ is the heat absorbed from the

0 I AS = ASsource + Asair = -AQ/To + C,, ln(Tl/To).