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 UU, which is a thermodynamic potential, in terms of the change in entropy SS, volume VV, and the number of particles NN.

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

dU=dQ+dW=TdSPdV\begin{aligned} \dd{U} &= \dd{Q} + \dd{W} = T \dd{S} - P \dd{V} \end{aligned}

The term TdST \dd{S} comes from the second law of thermodynamics, and represents the transfer of thermal energy, while PdVP \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 NN is variable), then each such particle costs an amount μ\mu of energy, where μ\mu is known as the chemical potential:

dU=TdSPdV+μdN\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 μi\mu_i. In that case, we sum over all species ii:

dU=TdSPdV+iμidNi\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.