Isothermal Expansion and Compression
Work Done and Heat Transferred
In an isothermal process, the temperature is constant. Applying the first law of thermodynamics to this closed process
For an ideal gas, the internal energy is a function of temperature only, and since the temperature is constant, then dU is zero and
dQ = dW = PdV
using the ideal gas law and integrating between the start and end of the process
This equation tells us that if we do some work on a gas to compress it, the same amount of energy will appear as heat transferred from the gas as it is compressed.
The Entropy change comes from the equation which incorparates the first and second laws. The energy balance is the first law, and the heat transfer is expressed as an entropy change which is a statement of the second law.
dU = TdS - PdV
dU is zero because the process is isothermal and the working fluid is an ideal gas, so that
TdS = PdV
substituting for the pressure from the ideal gas law for the pressure
and finally integrating between the start and end of the process
Isothermal compression is shown above on P-V and T-S diagrams. Note that as the gas is compressed heat is given out and that as it expands heat is absorbed.