Compression and expansion of gases

Two fundamental questions often arise [1] Why the change in enthalpy is usually expressed in an expansion process as dE = dU + PdV and not as dE = dU + PdV + VdP and [2] Is true isothermal compression really possible?

Briefly, In the isothermal compression of a gas, there is work done on the system to decrease the volume and increase the pressure. Doing work on the gas increases the internal energy and will tend to increase the temperature. For isothermal expansion, the energy supplied to the system does work on the surroundings.

1.PdV vs VdP  

∫ PdV is the specific work for a closed system, as in a piston and cylinder. During isothermal expansion, both P and V change along an isotherm with a constant PV product. The gas will expand and perform work on the surroundings as long as appropriate heat is added to keep PV constant

Significance of VdP 

∫ VdP is the specific work for an open system. This work, Vdp, is used for open flow systems like a turbine or a pump in which there is a “dp”, i.e. change in pressure. There are no changes in the control volume.

 An open system can exchange both energy and matter with its surroundings. A closed system, on the other hand, can exchange only energy with its surroundings, not matter.

2. Real compression processes operate between adiabatic and isothermal compression. Actual compression processes are polytropic because the gas being compressed is not at constant entropy as in the adiabatic process, or at a constant temperature as in the isothermal processes.

We are quite familiar with constant temperature and constant pressure isothermal phase transition like liquid to vapor.

Isothermal compression is not so common. Isothermal compression is not possible in a true sense as the compression process happens so fast you will require infinite numbers of cold sources of heat sinks to keep the temperature constant. And as a consequence, the gas expands as the temperature does not remain constant resulting in a drop in compressor efficiency.