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^