Problems and Solutions on Thermodynamics and Statistical Mechanics

Statistical Physics 397

own estimates of collision cross-sections and molecular mean free paths.
You may restrict your attention to pressures near atmospheric, temper-
atures near room temperature and dimensions of the order of centime-
ters or meters. Do not concern yourself with heat transfer by convection.
(k = 1.38 x loT1' erg/K).
(UC, Berkeley)

Solution:
dT
dx
Assume that a temperature gradient - exists in the gas and molecules

drift from region of higher temperature to that of lower temperature. The
number crossing unit area perpendicular to the drift in unit time is nis/4.
Each molecule, on the average, makes a collision in travelling a distance 1,

the mean free path, and transfers an energy c,AT - c,lx. The heat flow

dT

U+l.

1 dT dT

4 dx dx

per unit area per unit time is therefore q = -nisc,,l- = K-, where

1

4 4u

K = -nlGc, -

is the thermal conductivity of the gas. Taking air as an example, with

M = 29 x 1.67 x lopz7 kg, u = 10-20m2, c, = 5k/2, T = 300 K, we have

K = 0.44 J/mK.

2201

A propagating sound wave causes periodic temperature variations in a

gas. Thermal conductivity acts to remove these variations but it is generally

claimed that the waves are adiabatic, that is, thermal conductivity is too

slow.

The coefficient of thermal conductivity for an ideal gas from kinetic

theory is k M 1.23CU?il where C, is the heat capacity per unit volume, is is

the mean thermal speed, and 1 is the mean free path.

What fraction of the temperature variation AT will be conducted away

vs X and what is the condition on X for thermal conductivity to be ineffec-

tive?

( wis c 0 nsin)