Fundamental thermodynamic relation
The fundamental thermodynamic relation combines the first two laws of thermodynamics, and gives the change of the internal energy , which is a thermodynamic potential, in terms of the change in entropy , volume , and the number of particles .
Starting from the first law of thermodynamics, we write an infinitesimal change in energy as follows, where is the temperature and is the pressure:
The term comes from the second law of thermodynamics, and represents the transfer of thermal energy, while represents physical work.
However, we are missing a term, namely matter transfer. If particles can enter/leave the system (i.e. the population is variable), then each such particle costs an amount of energy, where is known as the chemical potential:
To generalize even further, there may be multiple species of particle, which each have a chemical potential . In that case, we sum over all species :
- H. Gould, J. Tobochnik, Statistical and thermal physics, 2nd edition, Princeton.