Problems and Solutions on Thermodynamics and Statistical Mechanics

56 Problems d Sdutiom on Thermodynamics d Statistical Mechanics

2AT,... , Ti + NAT = Tf. Assuming that the heat capacity of the material,
C, is temperature independent, calculate the entropy change of the total
system, material plus reservoirs. What is the entropy change in the limit
N + 00 for fixed Tf - z?
( wis co nsin)

Solution:

reservoir at temperature Ti + (t + 1)AT.

equilibrium, the change of entropy of the material is

Consider the material at temperature Ti + tAT in contact with the

When they come to thermal

Ti+(t+l)AT CdT ~- T + (t + 1)AT

• Cln
  T z+tAT '

AS1 = 1

Ti+tAT

The change in entropy of the heat reservoir is

CAT

z+(t+l)AT'

AS2 = -

The total change in entropy is

• AT ).
  Ti + (t + 1)AT

Therefore, after the material of initial temperature Ti has had contacts with

the series of reservoirs, the total change of entropy of the whole system is

N-1 N-1

AS= CASt=C (ln' ll -

t=O t=O 2?, + (t + 1)AT

When N -+ 00, or AT --t 0, the above sum can be written as an integration,

so that