---
title: "Fundamental thermodynamic relation"
date: 2021-07-07
categories:
- Physics
- Thermodynamics
layout: "concept"
---

The **fundamental thermodynamic relation** combines the first two
[laws of thermodynamics](/know/concept/laws-of-thermodynamics/),
and gives the change of the internal energy $U$,
which is a [thermodynamic potential](/know/concept/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.