Form the Clausius inequality, ∫ δQ / T ≤ 0
As the entropy is a property of the system, therefore the cyclic integral of a property is zero and the above equation can also be written as:
∫ δQ / T ≤ ∫ dS
SQ / T ≤ dS
dS ≥ δQ / T
for isolated system, SQ = 0
(dS)iso ≥ 0
For a reversible process: (dS) iso = 0
S = constant
For an irreversible process: dS) iso > 0 or entropy increases.
Thus it may be concluded that entropy of an isolated system can never decrease. It always increases with every irreversible process and remains constant when the process is reversible. This is known as principle of entropy increase or in other words the entropy of principle.