# How is Isothermal compression work less than isentropic compression work?

I know that isothermal compression work(input) is lesser than the isentropic/adiabatic compression work(input) for steady flow compressors. But does this also hold for positive displacement compressors (like piston-cylinder compressors)? I am asking as the area under the PV graph for the Isothermal compression process is greater than the adiabatic compression process.

[I am guessing this has to do with dV being negative here but can someone confirm this?]

I am asking as the area under the PV graph for the Isothermal compression process is greater than the adiabatic compression process.

The problem with your figures is neither involves the same volume change. Since you are interested in comparing the work done by both process, and work is the integral $$\int_1^2pdV$$ between the initial and final volume, you should probably compare them for the same volume change starting with the same initial pressure. See Fig 1 below.

Fig 1 compares an isothermal compression to an adiabatic compression that begin at the same pressure for the same volume change. Note that the magnitude of the work (area under the curve) is greater for the adiabatic process than the isothermal process. But since the work is done on the system, the work is negative work.

Normally when we talk about more or less work being done we're generally referring to the work done by the system (expansion work), i.e., the magnitude of positive work. So in terms of the amount of work done by the system, more work is done for the isothermal compression in the sense that it is less negative (and therefore more positive) than the work done for the adiabatic compression.

Hope this helps.

• So for a steady flow system (VdP work) to compare the work done, we have to bring the system to the same pressure level and compare? Jul 21, 2022 at 3:57
• I’m not sure what you’re asking Jul 21, 2022 at 17:43
• If we have to compare the work done in a steady flow system (the work done is VdP), say, a steady flow compressor (axial flow compressor), do we bring the process line to the same 'pressure' level? (instead of the same volume level, as done in your answer for a positive displacement compressor) Jul 21, 2022 at 17:58
• Compare the work done to what? Jul 21, 2022 at 18:21
• Work done for 2 different compression processes (Isothermal vs Adiabatic) or compression with cooling and without cooling. But instead of work done in positive displacement compressor, I am wondering for a steady flow compression process, say steady flow isotherm and steady flow adiabatic. Jul 22, 2022 at 4:33