A system of physical chemistry - Index of

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A system of physical chemistry - Index of

1 62 A SYSTEM OF PHYSICAL CHEMISTRY

Hence N = NA3( J^f.

Hence A = Jyfi^.

We can also calculate y in terms of the total kinetic energy of the

molecular motion. For the fraction of the whole, i.e. fractional number

of the whole number of molecules, which is represented by —

have each kinetic energy \m{u'^ + z/^ + ze;"'') where m is the mass of a

molecule.

Hence the total kinetic energy is—

mK^m •+CO

du

+CO r+00 _y(«2-t-t)2-(.z£,!8)

dv dw . {u- + v'^ + . e

iv"-) dudzdw

-oo J — oo

Leaving aside the constant factor for the moment, we see that the triple

integral is the sum of three triple integrals, one of which, for instance,

'+00

tih-y''"du

Now it is known that— r+

+=»

e-y'^dv

+00

ti •^

. e-y^'dic = I Jirlf

e-y'^'^dw.

and the two remaining integral factors in the above chosen integral havt

already been evaluated. Hence the selected member of the three triple

integrals has the value -^ J^^ly^, and the other two have each the same

value. Hence the total kinetic energy is—

|NA^w jTT^Iy", i.e. fNw/y ; since A = v/y/''".

Now we could conceive the molecules all moving with one uniform

velocity as a rigid body, for instance, and that velocity such as to give

the same kinetic energy as that due to the gaseous motion. Denote

the magnitude of this hypothetical velocity by c, and we have—

-^Nwf^ = |Nz«/y,

or y = 3/2^^

This particular speed c is not the true average speed, nor is it the

maximum probability speed referred to above (its relation with these

will be it is given presently)

in fact a ;

speed whose square is the average

of the squares of the molecular speeds at one instant, and on that

account is called the root-mean-square or r-m-s-speed. Assuming that

this is r-m-s-speed known, we formulate Maxwell's Law in one way

thus :—

Of a great number N of gaseous molecules in a steady state with an

r-m-s-speed c, the number whose velocity components at one instant

lie between the limits u io u + du, v to v + dv, w to w + dw, is—

N .

• e-^(^+^''+^^)l^^-dudvdw . . . (2)

j'g^

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