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.