A system of physical chemistry - Index of

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A system of physical chemistry - Index of

1 68 A SYSTEM OF PHYSICAL CHEMISTRY

Now it can be shown that for a gas mixture consisting of N^ molecules

of type I, N2 molecules of type 2, etc., in a steady state, and

therefore with the same mean energy of motion {i.e. the same tempera-

ture) throughout, the number of molecules of type i, of type 2, etc.,

within the defined limits of kinetic energy are given by expression (9),

with N replaced by Nj, N2, etc., in succession. Briefly, the distribution

of kinetic energy ignores differences in molecular mass.

Hitherto we have considered the gas to be free from all external

forces such as gravity, and subject only to the forces arising from collision

with each other or with the comparatively fixed molecules in the solid

walls of the containing vessel. If we introduce external forces of a

conservative type the previous formulae must be modified as follows :—

Denote the co-ordinates of a point in the gas as X, Y, Z the ; potential

energy of a molecule m of the gas at this point is a function of

X, Y, Z ; call it x^YZ or simply x-

We no longer have a uniform density of distribution m position for

the molecules. The density, in fact, diminishes as we move to places

of greater potential energy. A good illustration of this is the progressive

decrease in the density of the atmosphere as the altitude increases.

It can be shown that the molecular density is proportional to—

e - x/*T_

Hence, the number of molecules in an element of volume dx, dy, dz at

the point XYZ is—

jte - '>il^'^

dxdydz

where ft is a constant. As a matter of fact « is the molecular density

in a small volume surrounding a point where the potential energy, x, is

zero.

The distribution in kinetic energy of the molecules in this element

of volume is just as before. Hence we can write in full—

..¥.he-(^ + x)lk^dE.dxdydz . . (10)

as the number of molecules in an element of volume dxdydz, where the

potential energy of a molecule is ^5 and whose kinetic energies lie be-

tween E and E + dE.

The extension of this result to the case of a gaseous mixture is easily

given. Let the potential energies of each type of molecule in the external

field of force at the point XYZ be xi> X2) XS' ^'^- > ^^^^ '^^

numbers of each type of molecule in the volume element dxdydz, at

this point limited as in (10), are—

and so on.

__ifL. .EKe- i^ + xiVkTdEdxdydz]

n/^^'T=^ V

. EK e

. . (11)

- 2^2

(^ + I

X2)lkr^Edxdydz

Vtt/^^T^

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