4.3 Solutions
263
mu −
d
dz
mu
r cos θ
Let dn be the number of molecules with velocity between ν and ν + dν
per unit volume. The number of molecules with velocity ν and ν + dν in the
volume element dν is dndν. Molecules within the volume element undergo
collisions and are scattered in various directions.
Fig. 4.2 Transport of
momentum of gas molecules
Number of collisions that occur in dV in time dt will be
1
2
ν
λ
dt. The factor
1
2
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 volume element dV in time dt will be 2 ×
1
2
ν
λ
dt × dndV or
ν
λ
dtdndV .
Now the solid angle subtended by dA of the reference plane at dV is
dA cos θ/r
2 .
Assuming the scattering to be isotropic the number of molecules moving
downward toward d A is
ν
λ
dtdndV dA cos θ
4πr 2
or
νdtdn(2πr
2 sin θdθdr )dA cos θ
λ.4πr 2
or
νdtdndA sin θ cos θ
2λ
Transport of momentum downward from molecules in the upper hemisphere through dA in time dt is