Categories: Physics, Thermodynamics.

Fundamental thermodynamic relation

The fundamental thermodynamic relation combines the first two laws of thermodynamics, and gives the change of the internal energy \(U\), which is a thermodynamic potential, in terms of the change in entropy \(S\), volume \(V\), and the number of particles \(N\).

Starting from the first law of thermodynamics, we write an infinitesimal change in energy \(\dd{U}\) as follows, where \(T\) is the temperature and \(P\) is the pressure:

\[\begin{aligned} \dd{U} &= \dd{Q} + \dd{W} = T \dd{S} - P \dd{V} \end{aligned}\]

The term \(T \dd{S}\) comes from the second law of thermodynamics, and represents the transfer of thermal energy, while \(P \dd{V}\) represents physical work.

However, we are missing a term, namely matter transfer. If particles can enter/leave the system (i.e. the population \(N\) is variable), then each such particle costs an amount \(\mu\) of energy, where \(\mu\) is known as the chemical potential:

\[\begin{aligned} \dd{U} = T \dd{S} - P \dd{V} + \mu \dd{N} \end{aligned}\]

To generalize even further, there may be multiple species of particle, which each have a chemical potential \(\mu_i\). In that case, we sum over all species \(i\):

\[\begin{aligned} \boxed{ \dd{U} = T \dd{S} - P \dd{V} + \sum_{i}^{} \mu_i \dd{N_i} } \end{aligned}\]


  1. H. Gould, J. Tobochnik, Statistical and thermal physics, 2nd edition, Princeton.

© Marcus R.A. Newman, a.k.a. "Prefetch". Available under CC BY-SA 4.0.