So if the density of the gas is ρ (= N / V ; N = number of molecules in a volume of V), then the number of molecules moving out of the hole per unit time will be given by, ф.

Thomas Graham

For a hole with area A_o;

[   Substituting N/V = p / kT from the perfect gas equation.   ]

Graham's Law of effusion (1846): The rate of effusion is inversely proportional to its molar mass.

FLUX

There are three other types of translational motion brought about by "the uneven distribution of properties". They are diffusion, thermal conductivity and viscosity. The rate of movement of such properties is known as flux, ξ, and is directly proportional to the property gradient.

Such translational motion is governed by two factors. The random translational motion, ф, and the collision between molecules. That is a fraction of the molecules move forward while some are slowed down because they have collided with other molecules.

Let us derive an expression for the frequency of collision. From tutorial 2 we have derived the collision frequency, the number of collisions made by a molecule per unit time, to be   πd�cp / kT. So the mean free path, λ, the distance the molecule travelled before hitting a molecule is;

λ c / (πd�cp / kT) kT / πd�p.

The proportionality constant was determined to be 2‾^�. So λ = kT / (2^�πd�p).

The equation for flux can then be expressed as;

ξ ф x λ x {property factor} x {property gradient}

Let us illustrate this with the diffusion of gases

DIFFUSION

Diffusion is the translational motion brought about by a difference in concentration. For example when we release a smoke at one end of a tube it will eventually spread until it is distributed evenly throughout the tube.

Consider the gas moving for a distance of 2λ. The change in density, dN / dV = (dN / dx) x (dx / dV), where dV = unity. So;

Of course this is not an exact expression as many assumptions were made in its derivation. It was found that a better fit would be;      ξ = - (⅓ĉλ) � (dN / dx)}

THERMAL CONDUCTIVITY

Thermal conductivity is the translational motion brought about by a difference in energy of the molecules. If you heat a gas at one end of the tube eventually the gas at the other end of the tube will get hot.

Now pV = force � distance = energy = kT. However there is also a partition of energy for the different motions - translational, rotational, and vibrational - in the molecule. So the relevant term to use here will be the heat capacity of the molecule at constant volume per molecule, C_v,m.

A better fit would be;      ξ = - (⅓ĉλC_v,mρ) � (dT / dx)}

In general   C_v,m = � {3 + ξ(rotational) + ξ(vibrational)} k. So more monoatomic gases (noble gases) there will be no vibrational or rotational motion so C_v,m = 3k / 2.

For linear molecules at normal temperature ξ(rotational)=2 for linear molecules and ξ(rotational) = 3 for non-linear molecules. ξ(vibrational) is only important at high temperatures.

VISCOSITY

Viscosity is the translational motion brought about by a difference in momentum of the molecules.

A better fit would be;      ξ = - (⅓ĉλmρ) � (dv_x / dx)}

SUMMARY

PROPERTY	GRADIENT	COEFFICIENT	UNITS
Diffusion	- dN / dx	of diffusion, D = (⅓ĉλ)	m� s‾�
Energy	- dT / dx	of thermal conductivity, k = (⅓ĉλ) � (Cv,mρ)	JK‾� m‾� s‾�
Momentum	- dvx / dx	of viscosity, ŋ = (⅓ĉλ) � mρ	kg m‾� s‾�

The classical method for determining the viscosity coefficient of a gas is to measure the volume flow of the gas through a cylindrical tube of radius r and length L, under a pressure of p₁ with the outlet at a pressure of p₂. According to Poiseulle's equation;

where p_o is the pressure at which the volume was measured, usually the same as p₂.