KINETIC DERIVATION OF THE GAS EQUATION AND COLLISION FREQUENCY
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RALPH HULME University College, Gold Coast, British West Africa
Tm
nmc2,for an elementary derivation of P V = ideal gas, usually proceeds from a consideration of molecules moving freely in a cubic container. But the practical apparatus, customarily used to verify Boyle's law, approximates a sphere in shape so that the following derivation has been evolved. I t is somewhat similar to one given by T. Boyer in the School Science Review, 21, No. 81, p. 747 (1939). Consider one perfectly elastic molecule of mass, ml, and of negligible volume, moving with velocity, u,,inside a sphere of radius, r (Fig. 1). Suppose that when the molecule strikes the sphere a t B its path makes an angle with the radius 0B. Because the molecule is perfectly elastic1 ABO = OBC = BCO = ..... = p, and BC = CD = ...... = 2r cos (3, which is the distance between collisions. The time taken by the molecule to travel this distance will be 27 ros 6 (1) UI
Pressure is the rate of change of momentum per unit area, so that the pressure on the sphere due to one molecule will be
If we now consider n noninterfering molecule^,^ the total pressure, P, will be
For a single gas where all the molecules have the same mass, m, we may express this result more simply in terms of the root mean square velocity, c, where by definition
~1
1
= -
n
Substitution then gives
Hence the number of collisions/sec. will be
THE COLLISION FREQUENCY On collision with the sphere, the momentum change along a radius will be from mlul cos 13 towards B to mlul cos (3 away from B, i.e., 2mlul cos 8, while the momentum of the molecule perpendicular to the radius will remain unaltered. Therefore, change of momentum of one molecule per second will be
This result is perfectly general and true for all values of a. ' A statistical truth. A subela~tiocollision involves energy loss in the form of heat. But on a, molecular scale such "heat" would augment the kinetic energy of other molecules, making them superelastic. The over-all picture is one of perfect elasticity.
Another advantage of this derivation is that i t will yield the correct result, na/4V, for the average number of collisions per second per unit area made by molecules with the walls of the containing vessel. It is a curious limitation of the proof derived for the cube that it will give the correct formula for the pressure, but an incorrect value, na/GV, for the average number of molecules colliding with unit area of the wall per second. From (I) the average time taken by one molecule to travel from one collision to the next, irrespective of the angle of incidence, 8, will be =2 ?LI
where cos (3 is the mean value of cos (3 between 0 and
It is necessary to average cos p in this case because all angles are possible and the average time is directly proportional to the angle (unlike the momentum change per second). Note that averaging velocity would in no way average p, or vice-versa, hecause 0 and u are quite independent of each other. To a first approximation this mean value is
*/2.
1;'.
COS
6 dB
SO"/' dB
Fig".
1
VOLUME 34, NO. 9, SEPTEMBER, 1957
Fi-
2
" 2
=
2
3
¶In fact the molecule^ do interfere-they collide. But theso collisions are perfectly elastic and can therefore only alter the angle 4. Because our result is independent of 0 we may neglect collisions between molecules. 459
More precisely, in obtaining an average value of cos 3 ! we must allow for the fact that not all values of cos 13 are equally likely. For molecules with any particular
velocity, the number making impact with the walls a t B (Fig. 2) will depend both on the number in the solid angle dw (= 2 s sin p dp), and also on the rate a t which these molecules approach the tangent plane a t B, i.e.: upon u, cos 8. Hence
For n noninterfering molecules the collision frequency becomes
This may be expressed more simply as nzi ~-
4v
where the average velocity,
Consequently, the average time between collisions is Z r G = 2 3111
161
The selection of c or a in these derivations is largely a matter of convenience. They are, of course, related:
Thus the average number of collisions made by one molecule on the walls of the sphere per second is 3% -
4r
Hence the average number of collisions/sec./unit area made by one molecule will he
ACKNOWLEDGMENT
I wish to record my thanks to a former colleague, Mr. Bryan Higman, for making helpful suggestions regarding this derivation.
JOURNAL OF CHEMICAL EDUCATION
