From 6ce0bb9a8f9fd7d169cbb414a9537d68c5290aae Mon Sep 17 00:00:00 2001 From: Prefetch Date: Fri, 14 Oct 2022 23:25:28 +0200 Subject: Initial commit after migration from Hugo --- .../fundamental-thermodynamic-relation/index.md | 53 ++++++++++++++++++++++ 1 file changed, 53 insertions(+) create mode 100644 source/know/concept/fundamental-thermodynamic-relation/index.md (limited to 'source/know/concept/fundamental-thermodynamic-relation') diff --git a/source/know/concept/fundamental-thermodynamic-relation/index.md b/source/know/concept/fundamental-thermodynamic-relation/index.md new file mode 100644 index 0000000..a251fcd --- /dev/null +++ b/source/know/concept/fundamental-thermodynamic-relation/index.md @@ -0,0 +1,53 @@ +--- +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. -- cgit v1.2.3