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.
Is my reasoning valid and will the work required to compress the adiabatic system be more than the isothermal system?