How does an adiabatic or isothermal system affect the work required to compress a gas? Lets assume I have two piston cylinders both containing the same ideal gas which are then both compressed by the same volume. One is compressed adiabatically and the other isothermally.
My thought process:
As volume decreases, the adiabatic system will have an increase in pressure, temperature and therefore an increase in internal energy as well. As work is done on this system (in the form of compression), the internal energy will continue to increase.
The isothermal system however, will have an increase in pressure but not temperature meaning that internal energy is relatively unchanged.
Therefore, the adiabatic system will require more work because as it is compressed the internal energy increases as well effectively increasing the work needed to keep compressing. Whereas the work required for the isothermal system will be less because there is no energy increase within the system.
Question:
Is my reasoning valid and will the work required to compress the adiabatic system be more than the isothermal system?
 A: A P-V diagram is a good way to show what happens as the area under the graph is the work done and the gradient for an isothermal change is always larger (less negative) than that for an adiabatic change (the gradients are negative).

Compression from a volume $V_i$ to a volume $V_f$ results in a higher final pressure for an adiabatic change $P_{fa}$ than for an isothermal change$P_{fi}$ and hence more work is done during the adiabatic change.
Because the temperature has increased after the adiabatic change the internal energy of the system after the adiabatic change is greater than that after the isothermal change for which there is no change.
A: You are correct that, for the same volume change, the work will be greater for the adiabatic system than the isothermal system.  As you said, at each volume change, the adiabatic system will have a higher pressure than the isothermal system, because, in the adiabatic system, the temperature will also be higher.  This all applies to reversible compressions in each case.
