An ideal gas exists in a state with pressure $$P_0$$, volume $$V_0$$. It is isothermally expanded to 4 times of its initial volume $$(V_0)$$, then isobarically compressed to its original volume. Finally the system is heated isochorically to bring it to its initial state. The amount of heat exchanged in this process is:

Initial state:

For one complete cycle,

So net heat exchanged equals net work done:

$$Q=W$$

Now calculate work in each process.

1. Isothermal expansion from $$V_0\ ​to\ 4V_0$$

For isothermal process,

$$W_1=nRT\ln\frac{4V_0}{V_0}$$

Since initially

$$nRT=P_0V_0$$

So

$$W_1=P_0V_0\ln4$$

1. Isobaric compression

After isothermal expansion pressure becomes

$$P=\frac{P_0V_0}{4V_0}=\frac{P_0}{4}$$

Compression occurs at constant pressure $$\frac{P_0}{4}\ from\ 4V_0​\ to\ V_0​$$.

Work done:

1. Isochoric heating

$$W_3=0$$

Net work:

$$W=P_0V_0\ln4-\frac{3}{4}P_0V_0$$