Page 87 - Essentials of physical chemistry

The Kinetic Molecular Theory of Gases 49

Gas at T
1cm
Z
λ 1cm
2λ
λ

Gas at T–δ

Y

X
FIGURE 3.9 Schematic showing momentum transfer between layers of gas under a thermal gradient.

use cgs units here because viscosity was derived historically in poise, which is a cgs unit and
because we want to emphasize the small domain of the process we are considering. The question is
how much vertical interaction is there between the two postage-stamp size sheets of gas as they
move to the right? We are only concerned with the few molecules in the upper layer that have
components toward the lower layer and the few molecules in the lower layer that have vertical
components moving up to the upper layer. The diagram shows that only 1=4 of the possible
directions will result in a transfer of molecules with differing velocities from one layer to the
other. Since the atoms=molecules have the same mass, the difference in speeds leads to a transfer of
momentum and that is a force, which leads to a small viscous drag f z .
We can itemize this momentum transfer between the two layers, which are separated by 2l using
the formal definition of the speeds due to the vertical gradient as v z ¼ (dv=dz)(z þ l) at the upper slit
and v z ¼ (dv=dz)(z l) at the lower slit. By assuming 1 cm 1 cm layers, we can write the
momentum transfer though a (1 cm 1 cm) window between the layers. For the moment, ignore
the fact that we do not know the speed gradient (dv=dz) and write out the two momenta in the upper
and lower layers:

1 dv
(z þ l),
4 dz
mv #¼ n*hvi

1 dv
(z l) ,
4 dz
mv "¼ n*hvi
________________________

2 dv dv 2
(l) ¼ f z ¼ h x (1 cm ):
4 dz dz
m(Dv) ¼ n*hvi visc
Thus, using our hypothetical ‘‘box with slits’’ we see that we can cancel the unknown gradient
(dv=dz) to obtain an expression for the viscosity of the gas as

1 1 1 mhvi
h visc ¼ n*mhvil ¼ n*mhvi p ffiffiffi ¼ p ffiffiffi ¼ h visc :
2 2 2pd n* 2 2pd 2
2