Problems and Solutions on Thermodynamics and Statistical Mechanics

50 Problem d Sdutiom on Thermodyamica d Statistical Mechanics

process, we have dS = pdV/T,pV = NkT. Hence,

S, - S = / dS = -dV P = Nkln Vl ___ +v2 > ().

Vl

Thus the freely expanding process of the gas is irreversible.

1054
A thermally conducting, uniform and homogeneous bar of length L,
cross section A, density p and specific heat at constant pressure cp is brought
to a nonuniform temperature distribution by contact at one end with a hot
reservoir at a temperature TH and at the other end with a cold reservoir
at a temperature T,. The bar is removed from the reservoirs, thermally
insulated and kept at constant pressure. Show that the change in entropy
of the bar is

where C,, = cppAL,

(SVNY, Buflulo)

Solution:

As the temperature gradient in the bar is (T'-T,)/L, the temperature

at the cross section at a distance x from the end at T, can be expressed by

T, = T, + (TH - T,)x/L. As the bar is adiabatically removed, we have

Tf = (TH + T,)/2.

from which we obtain

But cp = T(aS/aT),

rL

Jo

Tf =

L

AS =cppA dx

where C,, = c,pAL.