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Can a process on an ideal gas be both adiabatic and isothermal?

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Solution

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,ā€‹W, such that ΔQ=ΔU+ΔWā€‹. In an adiabatic process, Q = 0 and in an isothermal process, change in temperature, T = 0. Therefore,
ΔQ=ΔU+ΔWΔQ=nCvΔT+ΔW0=nCv(0)+ΔWΔW=0,
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.

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