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 **system**s, 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 **system**s 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**