From equation (8.10a) we can write for any process

dQ < dS

dS > dQ / T

For clarification of equation (8.12), let us consider Fig. 8.11, where a and reversible process and c is an irreversible process. T

For (a) & (b) cycle

(for a & c irreversible cycle) … (8.14)

From equation (8.13) & (8.14) we can write

(as b is reversible)

(as entropy change is same in band c)

So for any irreversible process dS > dQ / T and for a reversible process dS = dQ _{rev} / T

Therefore, for a general process, we can write

dS > dQ / T

S _{f} – S_{t} > ^{f} ʃ _{1} dQ / T

Note that= sign holds good for reversible process and> sign for irreversible process.

**PRINCIPLE OF INCREASE OF ENTROPY**

For any process we Can write for the total mass

dS > dQ / T

If the system is isolated, dQ= 0

(dS) _{isol} =0

For reversible process (dS) _{isol} = 0

S = const.

For irreversible process (dS) _{isol }> 0

Thus it is proved that the entropy of an isolated system always increases, it can never decrease. The last two lines mentioned above is called the principle of increase of entropy or simply entropy principle and this is the general quantitative statement of 2nd law.

We can form an isolated system by putting system and surroundings together. The universe is an isolated system.

Universe = System+ surroundings

dS _{univ} >0 for all possible processes

dS _{system} + dS _{surr }> 0. Fig. 8.12

The entropy may decrease locally at some region within the isolated system, but it will be compensated by a greater increase of entropy somewhere within the system. So that the net effect of an irreversible process will be an increase of entropy of the whole system.