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author | Prefetch | 2022-10-20 18:25:31 +0200 |
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committer | Prefetch | 2022-10-20 18:25:31 +0200 |
commit | 16555851b6514a736c5c9d8e73de7da7fc9b6288 (patch) | |
tree | 76b8bfd30f8941d0d85365990bcdbc5d0643cabc /source/know/concept/fundamental-thermodynamic-relation | |
parent | e5b9bce79b68a68ddd2e51daa16d2fea73b84fdb (diff) |
Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'
Diffstat (limited to 'source/know/concept/fundamental-thermodynamic-relation')
-rw-r--r-- | source/know/concept/fundamental-thermodynamic-relation/index.md | 22 |
1 files changed, 11 insertions, 11 deletions
diff --git a/source/know/concept/fundamental-thermodynamic-relation/index.md b/source/know/concept/fundamental-thermodynamic-relation/index.md index 392e0b3..0d945fa 100644 --- a/source/know/concept/fundamental-thermodynamic-relation/index.md +++ b/source/know/concept/fundamental-thermodynamic-relation/index.md @@ -10,35 +10,35 @@ 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 $U$, +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 $N$. +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: +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, +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. +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**: +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 $i$: +which each have a chemical potential $$\mu_i$$. +In that case, we sum over all species $$i$$: $$\begin{aligned} \boxed{ |