Problems and Solutions on Thermodynamics and Statistical Mechanics

Thermodynamics 73

Solution:
Let C, and C, be the principal molar specific heats.
(a) From pV = NkT and TdS = dU + pdV, we find

C, - C, = T (%),- T (g)v =p (g) = Nk.

P

Hence C, - C, = k.

pV = NkT, we have

(b) For an adiabatic process, TdS = 0 and hence C,dT = -pdV.From

pdV + Vdp = NkdT = (C, - C,)dT ,

giving 7pdV + Vdp = 0, i.e.,

pV7 = const.

1076

The difference between the speficif heat at constant pressure and the

specific heat at constant volume is nearly equal for all simple gases. What

is the approximate numerical value of cp - c,? What is the physical reason

for the difference between cp and c,? Calculate the difference for an ideal

gas.

Solution:

( wis cons in)

c, - c, = 1 m [. ( g), - T ( 3v]

where m is the mass of the gas. From the functional relationship

we can find

(%),= (%)v + (z)T (%),

Utilizing Maxwell’s relation (g)T = (g)v, the above formula be-

comes

VTa2

(*I

m P