Constant pressure: Isobaric processes, W 1 − 2 = ∫ P d V = P ( V 2 − V 1 ) {\displaystyle W_{1-2}=\int PdV=P(V_{2}-V_{1})} Constant volume: Isochoric processes, W 1 − 2 = ∫ P d V = 0 {\displaystyle W_{1-2}=\int PdV=0} Constant temperature: Isothermal processes, W 1 − 2 = ∫ P d V , {\displaystyle W_{1-2}=\int PdV,} where P varies with V via P V = P 1 V 1 = C {\displaystyle \quad PV=P_{1}V_{1}=C} , so W 1 − 2 = P 1 V 1 ln ⁡ V 1 V 2 {\displaystyle W_{1-2}=P_{1}V_{1}\ln {\frac {V_{1}}{V_{2}}}} Polytropic processes, W 1 − 2 = P 1 V 1 − P 2 V 2 n − 1 {\displaystyle W_{1-2}={\frac {P_{1}V_{1}-P_{2}V_{2}}{n-1}}}