Problems and Solutions on Thermodynamics and Statistical Mechanics

390 Problems tY Solutions on "hermidynamics 6' Statistical Mechanics

2196

An insulated box of volume 2V is divided into two equal parts by a
thin, heat-conducting partition. One side contains a gas of hard-sphere
molecules at atmospheric pressure and T = 293 K.

(a) Show that the number of molecules striking the partition per unit

area and unit time is nG/4.

(b) A small round hole of radius r is opened in the partition, small
enough so that thermal equilibrium between the two sides is maintained
via heat conduction through the partition. Calculate the pressure and
temperature as functions of time in both halves of the box.

(c) Suppose the partition is a non-conductor of heat. Discuss briefly

and qualitatively any deviations from the time-dependence of temperature

and pressure found in part (b).

(UC, Berkeley)

Solution:

(a) The Maxwell distribution is given by

Among the molecules which strike on unit area of the partition in unit

time, the number in the velocity interval v - v + dv is nu, fdv,dvydv,.

Integrating, we get the number of molecules striking the partition per unit

area per unit time:

(b) Take the gas as an ideal gas whose internal energy is only dependent

on temperature. As the box is insulated, the temperature of the gas is

constant. Then we need only obtain the molecular number densities as

functions of time in the two parts. Let n1,n2 be the molecular number

densities of the left and right parts respectively at time t, V the volume of