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----
-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.