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}

## References

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

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