A system of physical chemistry - Index of

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A system of physical chemistry - Index of

ENTROPY AND THERMODYNAMIC PROBABILITY 15

complexions is a maximum, i.e. greater than the number of complexions

in any other distribution of the constituents of the system. A maximum

number of complexions is identical with the idea of maximum disorder

or maximum molecular chaos.

In all spontaneous processes the thermodynamical probability tends

to reach a maximum. But on purely thermodynamical grounds we know

that in spontaneous processes the entropy of a system tends towards

a maximum value consistent with the total energy of the system. It

follows therefore that there must be some close relation between the

thermodynamical probability of a state and the We can express this by writing—

S = F {w\

entropy of the state.

where S is the entropy of the system in any state, not necessarily the

equilibrium state, w the thermodynamic probability of the same state,

and F is some function still to be determined. To determine the nature

of F, let us suppose that we have two independent systems, each one

in a definite state, the entropy of the first being denoted by Si, the

probability of the state or arrangement of the first system being W]^, the

entropy of the second system being S2, and the probability of the state

of the second system being w>i. We then have the relations—

Si = FK)

S2 = FM-

The total entropy S of the two systems taken together is the sum of

the separate entropies. That is—

S = Si + S2 = Y{w^ + Y{w.^.

Since the particular state or arrangement of the first system can be

realised by selecting any one of the Wx complexions (contained in or

characteristic of that arrangement or state) and similarly for the second

system, it follows that the state or arrangement of the combined

system can be realised by selecting any one of the Wx complexions of

the first and combining them with the w^2 complexions of the second.

That is, the compound arrangement is obtained by selecting any one

of the Wx and w-i complexions. That is the probability iv of the

compound state is Wx x jf/g.

But for the compound system we have the relation : S = F(ze/).

Hence from the above we get : S = F(z£/i . w.^. But we have already

seen that S = ^{w-^ + ^{w^)- Hence, Y{wx . w^ = Y{wx) + ^ip^-

The only function which will satisfy this relation is the logarithmic

one, i.e. log xy = log .t + log y.

Hence, the connection between the thermodynamic probability and

the entropy of a system is given by the relation :—

Entropy oc log^ Probability

or S = /^ loge W,

where /^ is a constant independent of the chemical nature of the system

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