A system of physical chemistry - Index of

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A system of physical chemistry - Index of

PROBABILITY 5

and simultaneously a black ball from the second bag is P where P =

ablia + b)'^. This simple relation between the probabilities of two

separate events and the probability that both will occur simultaneously

is of great importance, and will be made use of later. Let us now

consider one or two cases which possess a more distinctly chemical

character.

The first which we shall take is the elementary deduction of the law

of mass action given in Vol. I., Chap. III. Suppose we have a gaseous

system containing «„ molecules of the substance A, and «6 molecules

of the substance B. The probability of a collision between a single

molecule of A and a single molecule of B is the product of the fractional

concentration of each, for a collision is analogous to two events

happening simultaneously, which we have just seen depends upon the

product of two single chances, each of which is represented by the

fractional concentration of A and B respectively.

We might regard the problem in the following way. Suppose that

V is the molecular volume, i.e, the actual volume " occupied "

by or

allotted to any single molecule of A or B in the mixture. Let V be the

total volume. Then, V = (w^ + nb)v. Suppose for the moment that

there is only one molecule of A present. Then the chance that this

molecule would occupy a given volume v dX a. given instant of time

would be the ratio of this volume to the total volume, i.e. the ratio z>/V

or i/(«a + n^. Since there are ?ia molecules of the substance A actually

"

present, the chance that any one of them occupies a certain " position

or space v is given by the ratio nal{na + «6). This term is likewise the

fractional concentration of the substance A. The chance that any

B molecule occupies the same position is given by the expression

nbl{fta + ^b)-

If a- molecule of A and a molecule of B occupy the

same position together, this is equivalent to a collision, and hence the

chance of a collision is the product of the fractional concentrations.

If the reaction required say two molecules of A to meet one molecule

of B simultaneously the chance of this occurring is .

, ,

\na + «&/ «a + fib

which finally takes the form : rate of collision = • kC^a Cb, for the total

volume is proportional to the total number of molecules present.

These simple probability ideas may also be used to account for the

influence exerted upon the collision frequency by the fact that in actual

gaseous systems the molecules possess volume. Thus if r is the radius

of a molecule, and / the average distance between two molecules, then

when a molecule moves over a distance / it sweeps out a cylinder the

cross-section of which is Trr'-^ and the /. length The volume of this cylinder

is therefore Trr^/. Each molecule has, on the average, a free space

allotted to it which is a cube of volume P. Hence, as far as the radius

affects the question, the chance of one molecule encountering another

is Trr-ljl^ or trr'^iP. If the average velocity is u, the time of a journey

between two successive collisions is Iju. Hence the number of encounters

per second is («//) x chance of collision = uttr'-lB. This

Ub

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