A system of physical chemistry - Index of

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A system of physical chemistry - Index of

Hence we derive—

APPENDIX I 167

.

2 - - 3 R ^

-wE = RT, or E = - - . T.

'

3

2 nv

But nv is the number of molecules in the ; gram-molecule so is

R/^Z'

the gas constant per molecule, usually denoted by k. Thus we obtain,

as the connection between the temperature of the gas and its mean

molecular motion—

E = |/^T

(8)

We have so far dealt with a gas composed of similar molecules, but

if we consider a gas mixture containing molecules of different masses,

little alteration is required in the formulae. It can be shown that among

each group of molecules of one type, the relative distribution in speed

and direction is similar to that already outlined for a simple gas, and is

unaffected by the presence of other types of molecules. This is a

statistical generalisation of the well-known law of Dalton concerning

gas mixtures. Thus, each group of molecules has a certain r-m-s-speed,

but the value of this varies from group to group ; however, a very simple

relation connects them. If we denote types of molecules by suffixes,

I, 2, 3, etc., then— ;«i^i^ = m.2^>^ = m-^i^ — etc.

where /«i, m.2, m-^, etc., are the masses of each type of molecule. In

terms of average kinetic energies per group of molecules this is—

Ej = E2 = E3 = etc.

Referring to (8) we see that this dynamical conclusion is the statistical

statement of the equality of temperature which obtains throughout the

gas mixture when a steady state has been attained. On account of this

relation it becomes more convenient to state the distribution law in

terms of limits of kinetic energy. This is easily done. Thus, revert-

ing to the case of a simple gas for a moment, we introduce the following

changes into (4) :—

Write

and we obtain after a few steps —

E = \mr

E = \mc^

dE = mc . dc

4-N^^-^3 E^ . . - 3=/-^E

as the number of molecules with kinetic energies between E and

E + ^E at an instant.

We may avail ourselves of (8) and obtain instead of this expression

the following one :—

f^ .

EK

g - ^/^^ . ^E . . . (9)

as the number of molecules limited in the manner mentioned.

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