According to the first law of thermodynamics, change in internal energy, U is equal to the difference between heat supplied to the gas, Q and the work done on the gas,W, such that . In an adiabatic process, Q = 0 and in an isothermal process, change in temperature, T = 0. Therefore,
where Cv is the heat capacity at constant volume.
This shows that if the process is adiabatic as well as isothermal, no work will be done. So, a process on an ideal gas cannot be both adiabatic and isothermal.