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 **system**s 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