Equation (4.1) refers to first law for a control mass during a cycle. Now we will discuss about a control mass under a change of state. If a system undergoes a change of state, net heat transfer to system Q minus network transferred from the system W will be stored in the system. This stored energy is called total internal energy. Mathematically

Q- W = ∆E … (4.2)

where E is the total internal energy

Let us consider a system undergoing a change of state 1 to 2 by process A, and returns from state 2 to state 1 by process B. This cycle is shown in Fig. 4.3 on a pressure-volume coordinate system (any other intensive and extensive property can be used as a coordinate system).

From equation (4.1) we can write

Considering processes A and B together we get

Now we consider another cycle in which the control mass changes from state 1 to state 2 by process C and returns back to state 1 by process B. For this cycle we can write,

Since A and B are two arbitrary p rocesses between states 1 and 2, the quantity (dQ – dW) is the same for all processes between states 1 and 2. Therefore (dQ- dW) depends on end states (here 1 and 2) and not by the path it traces. So this is a point function.

Therefore it is an exact differentials. This property is the energy of the mass and call total internal energy. Thus mathematically

dQ – dW = dE

dQ =dE+ dW

Integrating between states 1 and 2, we get

Q_{l} _{-2} = (E_{2}- E_{1}) + W_{l -2} … (4.4(a))

where E_{1} and E_{2} are the value of the total internal energy of the control mass in state 1 and 2 respectively. Q_{1 – 2 }is the heat supplied during 1 to 2 and W _{1-2} is the work output from the system between states 1 – 2.