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 = – dQ2 / d Q were dQ is heat added to isothermal (T = C) process AE and aQ2 is the rejection at T2 in the isothermal process FD.
We know that
ɳ general < ɳ nev
or 1- dQ2 / dQ < [ 1- dQ/ dQ] nev
or dQ2 / dQ> [ dQ2 / dQ] nev
or dQ / dQ2 < [ dQ / dQ2] nev
since [ dQ /dQ2] nev = T/ T2
Thus dQ / dQ2 < T / T2 for any general process AB, (reversible or Irreversible)
For a reversible process dS = dQ rev / T = dQ2 / T2
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,
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
Thus the following equation does not exist i.e.,