---
title: "Fundamental thermodynamic relation"
sort_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.