# Problem with derivation of kinetic gas theory

While studying kinetic theory of gas I studied that work done, $W =PdV$ and also internal energy, $U=\frac{3}{2}nRT$ so change in internal energy is equal to the work done by gas so $PdV=\frac{3}{2}nRT$ but we have $PV=nRT$

You have some things confused. Work done by gas should be $W = \int_{V_i}^{V_f} P dV$ and the change in internal energy is $\Delta U = \frac{3}{2}nR\Delta T$ for monoatomic gas. The thermodynamic process where the change in internal energy is only equal to the work done is adiabatic process. You can refer to Adiabatic Process to see the prove that the change in internal energy is indeed $\Delta U = \frac{3}{2}nR\Delta T$ for monoatomic gas.