Problems and Solutions on Thermodynamics and Statistical Mechanics

Statistical Physics 399

through unit area perpendicular to the 2-direction in unit time is nG/4.
In a collision the momentum transfer in the y-direction is mAu,. Since a
collision occurs over a distance - 1, the molecular mean free path, for an
order of magnitude calculation we can take

au, av,

aY aY

Av, = Ay- - I-.

Thus the shearing force across the unit area or the viscous force is

nmCAv, 1 au,

4 dY

f= = -nmGl-.

Hence the coefficient of viscosity is

v=-- - md-~,

nmZl

4 40

where u is the molecular collision cross section. q is seen to be independent
of pressure at constant temperature.

2203

Consider a dilute gas whose molecules of mass m have mean velocity

of magnitude 5. Suppose that the average velocity in the x-direction u,

increases monotonically with z, so that u, = uz(z) with Iu,~ << 5 and all

gradients small. There are n molecules per unit volume and their mean free

path is 1 where 1 >> d (molecular diameter) and 1 << L (linear dimension of

enclosing vessel).

(a) The viscosity q is defined as the proportionality constant between

the velocity gradient and the stress in the 2-direction on an imaginary plane

whose normal points in the z-direction. Find an approximate expression

for q in terms of the parameters given.

(b) If the scattering of molecules is treated like that of hard spheres,

what is the temperature dependence of q? The pressure dependence? As-

sume a Maxwellian distribution in both cases.

(c) If the molecular scattering cross section u o( E&>, where E,, is

the center-of-mass energy of two colliding particles, what is the temperature

dependence of q? Again assume a Maxwellian distribution.

(d) Estimate q for air at atmospheric pressure (10 dyn/cm2) and room

(Princeton)

temperature. State clearly your assumptions.