A system of physical chemistry - Index of

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A system of physical chemistry - Index of

APPENDIX 1 i6i

bility of their occurrence is very small indeed. There are, however,

no privileged directions, all directions of motion being equally likely.

This rough statement will serve to introduce the mathematical law of

the distribution of velocities, first formulated by Maxwell and often

stated in the words: "the law of distribution of the velocities among

the molecules of a gas in the steady state is the law of errors ".

As a matter of fact there are two ways of stating Maxwell's Law,

one of which assumes a velocity to be measured by its components

parallel to the Cartesian axes, the other way assumes that a velocity

is to be measured by its magnitude and direction on the plan explained

above (the unit sphere device). In the former case the law of the

distribution is for each cojnponeni similar to the normal law of errors,

but in the latter case the law of distribution of the speeds is not the

normal law of errors. Let us take the first method of formulation.

Considering an enormous number of molecules, N,

that in the steady state those of them which at one instant have veloci-

it can be shown

ties whose X components lie between narrow ii limits, say and u + du,

are in number equal to—

NA e-^'''^du

where A and y are constants to be determined shortly. This number

constitutes a fraction A^-v^Vw of the whole. It should be carefully

observed that the group of velocities here dealt with are not neces-

sarily close to each other in magnitude and direction : it is only their

X components which are close to each other in magnitude ; no restriction

is so far imposed on the Yand Z components.

Of this limited number there will be at the same instant a certain

fraction which have their Y components between v and v + dv, this

fraction being in fact Ae~y^'''dv so that the number whose X and Y

;

components are between u and u + du, and v and v + dv respectively,

is—

NA-

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