A system of physical chemistry - Index of

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A system of physical chemistry - Index of

APPENDIX I 163

and we see that any one particular component, say u, enters into this

expression in a manner similar to the way the error x enters into the

expression for the normal law of errors, viz. \e-'^^dx.

We can employ a geometrical method of representing this law.

Imagine that u, v, w, are chosen to be the Cartesian co-ordinates of a

point in a three dimensional diagram, which we will refer to as the

velocity diagram. The origin O of the co-ordinates in this diagram

represents absolute rest, while any other point P represents a velocity

whose magnitude is given by the length OP and whose direction is the

direction of OP. Let us write c for the length OP, i.e.—

gives the actual speed corresponding to the point P, Now suppose we

represent, as it were, every molecule by a point in this diagram, i.e.

€ach point represents the velocity— not the — position of some molecule,

so that we have therefore N points marked in the diagram these ; points

will, of course, move about with lapse of time, because of the changing

velocity of the corresponding molecule. The expression (2) states the

volume distribution of these points at a givert instant. Thus, taking

dudvdw to be an element of volume of the velocity diagram the

" density " of the points around the point representing zero speed, i.e.

representing the condition of rest, is—

Around any point representing a speed c (with no restriction as to

of radius c

direction), i.e. around a point lying anywhere on a sphere

with its centre representing rest, the density of the points is—

N J^T/Sn^c'^ . e-y""!^^- . . . . (3)

So the density of the swarm of points diminishes in this exponential

manner as we recede from the point representing rest. The reader is

warned against drawing the erroneous conclusion that there are at any

moment more molecules in the gas at rest, or very nearly so, than there

are those possessing any other assigned speed or near to it. The fallacy

involved in such an inference will be pointed out presently.

An element of volume in the velocity diagram represents what is

called an "extension-in-velocity," just as an element of volume in actual

space occupied by a gas is an " extension-in-position ". Thus we may

refer to expression (3)

as the " "

density-in-velocity of the molecules

about any velocity of the c.

magnitude

We can readily pass to the second manner of formulating Maxwell's

Law, and in so doing clear away the possibility of misconception referred

to a few lines back. Suppose we wish to find the number of molecules

whose speeds {i.e. merely velocity magnitudes) are at any moment

between the limits c and c + dc, but with no limits placed on their

directions of motion. We must obviously find the number of points in

our velocity diagram which lie between spherical surfaces of radii c and

£ + dc. The volume of the elementary region of this diagram is ^irc-dc,

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