Page 23 - Instant notes

Molecular behavior in perfect gases 9

The molecular origin of pressure

In the kinetic theory of gases, the pressure which a gas exerts is attributed to collisions of
the gas molecules with the walls of the vessel within which they are contained. A
molecule colliding with the wall of the vessel will change its direction of travel, with a
corresponding change in its momentum (the product of the mass and velocity of the
particle). The force from the walls is equal to the rate of change of momentum, and so the
faster and heavier and more dense the gas molecules, the greater the force will be. The
equation resulting from mathematical treatment of this model may be written as:

where n is the number of moles of gas in a volume V (i.e. the density). This equation may
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be rearranged to a similar form to that of the ideal gas law: PV=n Mc /3.
Substituting for c, yields PV=n m(3RT/m)/3=nRT, i.e. the ideal gas law.
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Alternatively, we may recognize that the value ½Mc represents the rms kinetic energy
of the gas, E kinetic, and rewrite the equation to obtain the kinetic equation for gases:

Effusion

Effusion is the escape of a gas through an orifice. The rate of escape of the gas is directly
related to c:

where ρ is the density of the gas. For two gases at the same temperature and pressure, for
example nitrogen and hydrogen, it follows that the ratio of the velocities is given by:

This is Graham’s law of effusion.

Mean free path

Gas particles undergo collisions with other gas particles in addition to colliding with the
walls. The mean distance travelled by a gas molecule between these random collisions is
referred to as the mean free path, λ. If two molecules are regarded as hard spheres of
radii r A and r B, then they will collide if they come within a distance d of one another