Problems and Solutions on Thermodynamics and Statistical Mechanics

400 Problems d Solutions on Thermcdpizm'cs €4 Statistical Mechanics

Solution:
(a) The number of molecules passing through a unit area perpendicular
to the z-direction per unit time is n5/4. Each particle makes a collision after
travelling a distance of the order of magnitude of 1 in which a momentum

in the 2-direction is transferred of the amount m(u, + Az- - uz) =

mAz-. For an approximate estimate, we take Az - I, so that the

viscous force and the viscosity are respectively

auz

aZ

a UZ

aZ

1 a uz

4 aZ

r = -nvml--- ,

1

q = -mhl

4

mi7

nu 40

as V cx fi, q cc @ and is independent of pressure.

(b) AS 1 = - , q = -. The hard sphere model gives u = const., then

1

(c) If u cc E:,,, cx T2, then q cx T-3/2 and is independent of pressure.

(d) For room temperature, we can approximately take the molecular

weight of air to be 30, V to be the speed of sound and u - m2. Then

q w 1.3 x kg/ms.

2204

Electrical conductivity. Derive an approximate expression for the

electrical conductivity, u, of a degenerate electron gas of density n

electrons/cm3 in terms of an effective collision time, 7, between the elec-

trons.

(MIT)

Solution:

For a degenerate electron gas, the velocity is uniform on the Fermi

surface. Then the net current in any direction is zero. Under the effect

of an electrical field E,, the electrons move as a whole in the z-direction,

forming an electrical current. We have

du, duz

dt dt

E,e = m- , Av, w 7- ,