Problems and Solutions on Thermodynamics and Statistical Mechanics

Statistical Physics 387

and M is the ratio of the flow velocity v to the velocity of sound c at
temperature T.

(a) Derive the above expression.

(b) Derive a corresponding expression for Po -, and calculate the value

P

Po

P

of M for a condition where - = lo4.

Po

P

(c) Calculate the value of T for - = lo4 and TO = 300 K.

(d) Find the maximum value of u in the limit T -+ 0.

(UC, Berkeley)

Solution:
(a) Consider the process of a small volume of gas consisting of N
molecules passing through a small hole. When it enters the hole it carries
internal energy Nc,To and the bulk of the gas does work on it to the
amount of poV0 = NkTo. When the volume of gas leaves the hole, its
internal energy is Nc,T and it does work pV = NkT on the external gas.
Its kinetic energy is now Nmv2/2. Thus we have for each molecule of the
volume cPTo = c,T + mv2/2, where

cP = C, + k = -^7 k , 7 = cp/c, f

7-1

Noting that the velocity of sound is c = E, we have

5 TO M2
3' T 3
For air, 7 = - and - = 1 + -.

(b) From the adiabatic relation,

we have