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+---
+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.