A system of physical chemistry - Index of

A system of physical chemistry - Index of

4

A SYSTEM OF PHYSICAL CHEMISTRY

take advantage of the fact that the number of the molecules involved

in any system with which we are concerned in physics or chemistry is

so enormous that we are justified in dealing with these aggregations of

molecules in a statistical manner, by introducing the principle of probability

or chance into the mechanics of the process considered.

It is not proposed to attempt to give a systematic account of what

may be called the principles of statistical mechanics. We are concerned

mainly with one such principle, known as the principle of equipartition

of kinetic energy among degrees of freedom. We shall state

and apply this principle later. For the present it is necessary to

familiarise ourselves with the idea of probability.

Probability.

In a purely algebraic sense probability may be defined as follows :

If an event can occur in a ways and fail in b ways, each of these ways

being equally likely, then the chance or probability of its occurring is

a/((3 + b), and the chance or probability of its failing to occur is

bl{a + The b). sum of these two terms is necessarily unity, for the sum

of the two probabilities covers all eventualities, i.e. the event must

either happen or fail, and the sum represents certainty. It follows that

mathematical probability is a fractional quantity which may be small or

large, but can never exceed unity, ^

i.e. certainty. We may illustrate

the idea by one or two examples. Suppose we have equal numbers of

black and white balls inside a bag, the bag being well shaken so as to

destroy any possible regularity or ordered arrangement of the what is the probability or chance that, say, a white ball will

balls,

be drawn

from the bag ? It is evident that the chance of drawing a white is the

same as that of drawing a black. In other words, the is probability one

half, for here a = b when a is the number of white and b the number of

It is evident that in

black balls, and ajia + b) = o-\$ = bl{a -f b).

the limit, if b becomes very small compared with a, the probability

of drawing a white increases almost to a certainty, i.e. the fraction

al{a + b)

is nearly unity. We are here considering the probability of

a single event occurring. Let us now consider the probability that two

independent events may occur simultaneously. The probability in such

a case is easily shown ^ to be the product of the probabilities of the

separate events. That is, if the probability of the first event is P^, and

that of the second is P2, then the probability P of both events occurring

simultaneously is P = PjPo- Thus, if we have two bags, each con-

taining a white balls and b black ones, the chance of drawing a single

white from one bag is Pj, where Pi = a\{a -\- b), and the chance

of drawing, say, a black ball from the other is bag P2, where P2 =

bl{a 4- b). The chance of drawing a white ball from the first bag

^ Whilst this is true of mathematical probability we shall find later that there is

a quantity to which the term "thermodynamic probability" has been given, this

quantity being in general a large integral number.

^C/., for example, Hall and Knight's Algebra.

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