Problems and Solutions on Thermodynamics and Statistical Mechanics

80 Problems d Solutio~ on Thermodpamics d Statiaticd Mechanics

1083

The tension of a rubber band in equilibrium is given by

where t = tension, T = absolute temperature, z = length of the band, lo
= length of the band when t = 0, A = constant.
When z is the constant length lo, the thermal capacity cx(z,T) is
observed to be a constant K.

(a) ,, Find as functions of T and x:

8E

(1) (a,) where E = internal energy, (2)

T

E(z, T), (5) S(z,T), where S = entropy.

(b) The band is stretched adiabatically from z = lo to z = 1.510. Its

(CUSPEA)

initial temperature was TO. What is its final temperature?

Solution:

Then as

(a) From the theory of thermodynamics, we know dE = TdS + tdz.

cx=T(%) X ,

we have

Generally, E = E(z, T), and we have

(g) T dz.

i.e., dE = c,dT +

On the other hand,

dS = -(dE- 1 tdx) = -dT+ CX

T T