A system of physical chemistry - Index of

4

A SYSTEM OF PHYSICAL CHEMISTRY

This, as already pointed out, would be quite impossible. Instead, we

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 **system**atic 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.