The Significance of pV=nRT

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 A, with a single molecule travelling left and right the length of the box by virtue of collisions with the end walls.

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 <u2> is the mean square speed* of the molecules in the box

(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<u2 >/3 ( = (2/3)Ekinetic ) with a constant (call it R ) times the temperature, T

Here n = N/NA is the number of moles of gas molecules and NA is Avogadro's number.

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=NA 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 (1023) in every macroscopic object.



PJ Brucat // University of Florida