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Entropy Change in an Irreversible Process

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,

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.,

>0  impossible and also violates second law.