Problems and Solutions on Thermodynamics and Statistical Mechanics

Thermodynamics 149

(EL

Compare it with wave equation ~ = v2V2p', we have v2 =

a2
at2
(Note: An assumption has been made here that the pressing of the fluid
created by the disturbances is adiabatic for which S = const. Generally
speaking, such approximation is reasonable as the heat conductivity is neg-
ligible.)

(b) At T = 0 K, for a system of spin 1/2 Fermioii gas we have

2 NPo

P=iv

1150

Gas, in equilibrium at pressure po and mass density PO, is confined to

a cylinder of length L and cross sectional area A. The right hand end of

the cylinder is closed and fixed. At the left hand end there is a friction-

less and massless movable piston. In equilibrium the external force that

must be exerted on the piston is of course fo = poA. However, suppose

a small additional force is supplied by an external agency: the harmonic

force f(t) = fo cos(wt). This produces small motions of the piston and thus

small amplitude disturbances in the gas. Let c be the speed of sound in the

gas; neglect viscosity. Let v(t) be the velocity of the piston. Compute v(t).

(CUSPEA)

piston

/

...... ,. *.. *.

X

-L-

Fig. 1.4F.

Solution:

whose origin is the equilibrium point (as shown in Fig. 1.46).

Consider the gas as an ideal fluid. We choose a coordinate system

Let the