Let process 1 – 2 is achieved through reversible path R_{1} and the cycle is completed by process 2 – 1 through reversible path R_{2.}

Applying Clausius’ theorem to paths R_{1} & R_{2}, we get

We have taken arbitrarily any two reversible paths and we have come upto an equation which shows that the expression ^{2}ʃ_{1} _{R} d Q / T is independent of reversible path connecting states 1 and 2. Thus there exists a property d Q_{R}/ T which is called entropy (S).

Thus entropy is a point function.

^{2}ʃ_{1} _{R} d Q / T = S_{2} – S_{1}

When two states 1 – 2 are very near, then

Q_{R}/ T / dS

Where dS is the change of entropy. ds is an exact differential because S is a point function and a thermodynamic property. dQR heat transfer is reversible. (Rà7 Reversible).

Entropy is taken from Greek Word ‘tropee’ meaning transformation and introduced by Clausius. Entropy per unit mass is called specific entropy. So mathematically specific entropy s = S / m J/ kgK.

In fig 8.5

R = R =Reversible path

IR = Irreversible path

IR is replaced by R in view point of entropy which depends on end points. We integrate in the reversible path to get entropy for irreversible path.

Integration is performed only on a reversible path.