1000 Solved Problems in Modern Physics

4.3 Solutions 263

mu−

(

d

dz

mu

)

rcosθ

Let dnbe the number of molecules with velocity betweenνandν+dν

per unit volume. The number of molecules with velocityνandν+dνin the

volume element dνis dndν. Molecules within the volume element undergo

collisions and are scattered in various directions.

Fig. 4.2Transport of
momentum of gas molecules

Number of collisions that occur in dVin time dtwill be^12 νλdt. The fac-

tor^12 is introduced to avoid counting each collision twice, since the collision

between molecules 1 and 2 and that between 2 and 1 is same.

Each collision results in two new paths for the scattered molecules. Hence

the number of molecules that are scattered in various directions from this vol-

ume element dVin time dtwill be 2×^12 νλdt×dndVorνλdtdndV.

Now the solid angle subtended by dAof the reference plane at dV is

dAcosθ/r^2.

Assuming the scattering to be isotropic the number of molecules moving

downward toward dAis

ν

λ

dtdndVdAcosθ

4 πr^2

or

νdtdn(2πr^2 sinθdθdr)dAcosθ

λ. 4 πr^2

or

νdtdndAsinθcosθ

2 λ

Transport of momentum downward from molecules in the upper hemi-

sphere through dAin time dtis