Let ABCD is a cyclic process where, AB is a general process (reversible or irreversible) and other processes are reversible. Now ABCD is divided into a number of small cycles. Let us take AEFD as one of these small cycles and its efficiency is ɳ = – dQ_{2} / d Q were dQ is heat added to isothermal (T = C) process AE and aQ_{2} is the rejection at T_{2} in the isothermal process FD.

We know that

ɳ _{general} < ɳ _{nev}

or 1- dQ_{2} / dQ < [ 1- dQ/ dQ] _{nev}

or dQ_{2} / dQ> [ dQ_{2} / dQ] _{nev}

or dQ / dQ_{2} < [ dQ / dQ_{2}] _{nev}

since [ dQ /dQ_{2}] _{nev} = T/ T_{2}

Thus dQ / dQ_{2} < T / T_{2} for any general process AB, (reversible or Irreversible)

For a reversible process dS = dQ _{rev} / T = dQ_{2} / T_{2}

Hence for any process dQ/ T< dS

Thus in any cyclic process

Since entropy is a point function and in a cyclic process initial and final point is the same,

thus

Equation (8.11) is called the inequality of Clausius which supplies information about

the reversibility of a cycle.

Equation (8.11) provides the following two equations

(i) = 0 for a reversible cycle.

(ii) = 0 for a irreversible cycle (Possible).

Thus the following equation does not exist i.e.,