Consider a system which changes its state from state 1 to state 2 by following the path L, and

returns from state 2 to state 1 by following the path M (Fig.2.1).

So the system undergoes a cycle.

Writing the First law for path L

$Q_L =ΔE_L +W_L$. . . . . . . . . . . (i)

and for path M

$Q_M = ΔE_M +W_M$. . . . . . . . . . (ii) but ∫ dW = ∫ dQ

$W_L +W_M = Q_L + Q_M$

$Q_L–W_L = W_M–Q_M$. . . . . . . . . . . . . . . . (iii)

From equations (i), (ii) and (iii), it yields

$ΔE_L = – ΔE_M$. . . . . . . . . . . . . . . . (iv)

Similarly, had the system returned from state 2 to state 1 by following the path N instead of path M

$ΔE_L = – ΔE_N$. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (v)

From equations (iv) and (v),

$ΔE_M = ΔE_N$. . . . . . . . . . . . . . . . (vi)

Thus, it is seen that the change in energy between two states of a system is the same, whatever path the system may

follow in undergoing that change of state.

If some arbitrary value of energy is assigned to state 2, the value of energy at state 1 is fixed independent of the path the system follows.

Therefore, energy has a definite value for every state of the system.

Hence, it is a point function and a property of the system.