Question

If ρ is the density, <ν> the mean speed and λ the mean free path of the gas molecules, then show that the coefficient of viscosity is given by η = 1/3ρ <ν>λ.

Answer 1

Consider a volume element dV = 2πr²sinθdθdr located on a layer at a height z = r cos θ. If mu is the momentum of a molecule at the XY-plane at z = 0, then its value at dV will be mu + ((d/dz)mu) r cos θ (Fig. 4.2). At an identical layer below the reference plane dA, the momentum would be

mu − ((d/dz)mu)rcosθ

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.

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/r². 

Assuming the scattering to be isotropic the number of molecules moving downward toward dA is

Transport of momentum downward from molecules in the upper hemisphere through dA in time dt is

The factor e^−r/λ is included to ensure that the molecule in traversing the distance r toward dA does not get scattered and prevented from reaching dA. 

Similarly, transport of momentum upward, from molecules in the lower hemisphere through dA in time dt is