The work done during an isothermal process is:

Explanation:When work done by the system is taken as positive. Then Work done during an isothermal process is GIVEN by –\({W_{1 - 2}} = {p_1}{V_1}\ln \left( {\frac{{{V_2}}}{{{V_1}}}} \right) = {p_2}{V_2}\ln \left( {\frac{{{V_2}}}{{{V_1}}}} \right)\;\;\;\left( \because {{p_1}{V_1} = {p_2}{V_2}} \right)\)∵ p1V1 = p2V2 during an isothermal process.\( \Rightarrow \frac{{{V_2}}}{{{V_1}}} = \frac{{{p_1}}}{{{p_2}}}\)\(\therefore {W_{1 - 2}} = {p_1}{V_1}\ln \left( {\frac{{{p_1}}}{{{p_2}}}} \right) = {p_2}{V_2}\ln \left( {\frac{{{p_1}}}{{{p_2}}}} \right)\)∵ p1V1 = mRT1 & p2V2 = mRT2\(\therefore {W_{1 - 2}} = mR{T_1}\ln \left( {\frac{{{V_2}}}{{{V_1}}}} \right) = mR{T_2}\ln \left( {\frac{{{V_2}}}{{{V_1}}}} \right) = mR{T_1}\ln \left( {\frac{{{p_1}}}{{{p_2}}}} \right) = mR{T_2}\ln \left( {\frac{{{p_1}}}{{{p_2}}}} \right)\)ProcessWork DoneConstant Pressure (Isobaric / Isopiestic)W1-2 = p(V2 – V1)CONSTANT Volume (ISOCHORIC)W1-2 = 0PolytropicFor Adiabatic (N = γ = 1.4)\({W_{1 - 2}} = \frac{{{p_1}{V_1} - {p_2}{V_2}}}{{n - 1}} = \frac{{{p_1}{V_1}}}{{n - 1}}\left[ {1 - {{\left( {\frac{{{p_2}}}{{{p_1}}}} \right)}^{\frac{{n - 1}}{n}}}} \right]\)