The Kinetic Theory of Gases and the Ideal Gas
Equation

ASSUME:

Gases are composed of

- Numerous

- Elastic molecules

- of Negligible Size compared to Bulk Container

- whose Thermal Motion is 'Random'

Consider a rectangular box length ** l**,
area of ends

The time between
collisions with the left wall is the distance of travel between wall collisions
divided by the speed (magnitude of the velocity) of the particle, *u*.

The frequency of collisions with that wall in collisions per second is

According to Newton, force is the time rate of change of the momentum

The momentum change
upon collision is the momentum after the collision minus the momentum
before
the collision To hit the left wall the initial velocity must have been
*-u*, so:

The average force on the left end wall is the force per collision times the frequency of colllsions

The pressure,*
p*, (not to be confused with momentum, ) exerted by this single molecule
constrained to move in one horizontal direction (one dimension) is the
average force per unit area

where *V***
=***A ^{.}l* is the volume of the rectangular box.

The pressure exerted by *N* molecules moving in the same way as
a single molecule is simply

where *<u ^{2}>*
is the mean square speed

(If the elastic molecules collide but still remain travelling only in
the left-right dimension the pressure is unaffected by that collision)

If the molecues are free to move in three dimensions, they will hit
walls in one of the three dimensions one third as often. The pressure then
of a gas sample of *N* molecules in 3-D is

We __define __the
Temperature, *T*, as a measure of the thermal (random) motion of the
gas particles. The only kind of motion this model of a gas can have is
kinetic energy. This energy is

Thus, we can replace
*m*<*u ^{2 }*>/3

Here *n = N/N _{A}*
is the number of moles of gas molecules and

The empirical observation that one mole of gas in a 22.4 liter vessel
at 273K exerts a pressure of 1.00 atmosphere allows the determination of
the constant, *R*

The kinetic energy
for *n*=1 mol of gas particles at temperature *T* is

Comparison of eqns (8) and (10) give the relation

So, from equations (12) and (13), the total kinetic energy of a mole of gas is

The root mean square
velocity, (<*u *^{2}>)^{1/2}, a measure of the
average speed of the molecules may be directly derived from the ideal gas
constant, the molecular weight of the gas and the temperature

Note that *R*
is in units of energy per mole per kelvin and *M*=*N _{A} m*
is the mass of one mole of molecules (the molecular mass)

The gas constant, *R*, must be expressed in appropriate units for
the correct determination of numerical values. In the ideal gas equation
of state, *pV=nRT*, the logical chioce of units is (latmmol^{-1}
K^{-1}). But latm is a non-standard unit of work or energy, so
for equation (15) we convert the units of *R*** **to obtain

^{*} Note: The average force exerted by a single molecule is
associated with a __time__ average and is denoted with an overbar. The
mean square speed is associated with an __ensemble__ average and is
denoted by 'pointy braces', < >. The average over molecules in the
sample (or ensemble) is an important part of chemistry since there are
so many molecules (10^{23}) in every macroscopic object.