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author | Prefetch | 2024-07-21 17:52:57 +0200 |
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committer | Prefetch | 2024-07-21 17:52:57 +0200 |
commit | 96447d884e02012a4ed9146dc6c00d186a201038 (patch) | |
tree | de03b43cc8e4b64a5aff6f7aa69c06f2085020c9 /source/know/concept/fundamental-thermodynamic-relation | |
parent | 075683cdf4588fe16f41d9f7b46b9720b42b2553 (diff) |
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diff --git a/source/know/concept/fundamental-thermodynamic-relation/index.md b/source/know/concept/fundamental-thermodynamic-relation/index.md deleted file mode 100644 index 0d945fa..0000000 --- a/source/know/concept/fundamental-thermodynamic-relation/index.md +++ /dev/null @@ -1,54 +0,0 @@ ---- -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. |