An ideal gas system undergoes an isothermal process, then the work done during the process is

Let an ideal gas goes isothermally from its initial state ($P,V$) to the final state ($P_2,V_2$), then work done

                  $W = {\int }_{V_1}^{V_2}PdV$     Taking $V = nRT$or $P = \frac{nRT}{V}$

Then, $W = {\int }_{V_1}^{V_2}\frac{nRT}{V}dV = nRT{\int }_{V_1}^{V_2}\frac{1}{V}dV = nRT\ln \left(\frac{V_2}{V_1}\right)$