From 16555851b6514a736c5c9d8e73de7da7fc9b6288 Mon Sep 17 00:00:00 2001 From: Prefetch Date: Thu, 20 Oct 2022 18:25:31 +0200 Subject: Migrate from 'jekyll-katex' to 'kramdown-math-sskatex' --- source/know/concept/maxwell-relations/index.md | 30 +++++++++++++------------- 1 file changed, 15 insertions(+), 15 deletions(-) (limited to 'source/know/concept/maxwell-relations') diff --git a/source/know/concept/maxwell-relations/index.md b/source/know/concept/maxwell-relations/index.md index aa51b06..892ced1 100644 --- a/source/know/concept/maxwell-relations/index.md +++ b/source/know/concept/maxwell-relations/index.md @@ -14,15 +14,15 @@ for well-behaved functions (sometimes known as the *Schwarz theorem*), applied to the [thermodynamic potentials](/know/concept/thermodynamic-potential/). We start by proving the general "recipe". -Given that the differential element of some $z$ is defined in terms of -two constant quantities $A$ and $B$ and two independent variables $x$ and $y$: +Given that the differential element of some $$z$$ is defined in terms of +two constant quantities $$A$$ and $$B$$ and two independent variables $$x$$ and $$y$$: $$\begin{aligned} \dd{z} \equiv A \dd{x} + B \dd{y} \end{aligned}$$ -Then the quantities $A$ and $B$ can be extracted -by dividing by $\dd{x}$ and $\dd{y}$ respectively: +Then the quantities $$A$$ and $$B$$ can be extracted +by dividing by $$\dd{x}$$ and $$\dd{y}$$ respectively: $$\begin{aligned} A = \Big( \pdv{z}{x} \Big)_y @@ -30,7 +30,7 @@ $$\begin{aligned} B = \Big( \pdv{z}{y} \Big)_x \end{aligned}$$ -By differentiating $A$ and $B$, +By differentiating $$A$$ and $$B$$, and using that the order of differentiation is irrelevant, we find: $$\begin{aligned} @@ -55,11 +55,11 @@ $$\begin{aligned} \end{aligned}$$ The following quantities are useful to rewrite some of the Maxwell relations: -the iso-$P$ thermal expansion coefficient $\alpha$, -the iso-$T$ combressibility $\kappa_T$, -the iso-$S$ combressibility $\kappa_S$, -the iso-$V$ heat capacity $C_V$, -and the iso-$P$ heat capacity $C_P$: +the iso-$$P$$ thermal expansion coefficient $$\alpha$$, +the iso-$$T$$ combressibility $$\kappa_T$$, +the iso-$$S$$ combressibility $$\kappa_S$$, +the iso-$$V$$ heat capacity $$C_V$$, +and the iso-$$P$$ heat capacity $$C_P$$: $$\begin{gathered} \alpha \equiv \frac{1}{V} \Big( \pdv{V}{T} \Big)_{P,N} @@ -77,7 +77,7 @@ $$\begin{gathered} ## Internal energy The following Maxwell relations can be derived -from the internal energy $U(S, V, N)$: +from the internal energy $$U(S, V, N)$$: $$\begin{gathered} \mpdv{U}{V}{S} = @@ -119,7 +119,7 @@ $$\begin{gathered} ## Enthalpy The following Maxwell relations can be derived -from the enthalpy $H(S, P, N)$: +from the enthalpy $$H(S, P, N)$$: $$\begin{gathered} \mpdv{H}{P}{S} = @@ -161,7 +161,7 @@ $$\begin{gathered} ## Helmholtz free energy The following Maxwell relations can be derived -from the Helmholtz free energy $F(T, V, N)$: +from the Helmholtz free energy $$F(T, V, N)$$: $$\begin{gathered} - \mpdv{F}{V}{T} = @@ -203,7 +203,7 @@ $$\begin{gathered} ## Gibbs free energy The following Maxwell relations can be derived -from the Gibbs free energy $G(T, P, N)$: +from the Gibbs free energy $$G(T, P, N)$$: $$\begin{gathered} \mpdv{G}{T}{P} = @@ -245,7 +245,7 @@ $$\begin{gathered} ## Landau potential The following Maxwell relations can be derived -from the Gibbs free energy $\Omega(T, V, \mu)$: +from the Gibbs free energy $$\Omega(T, V, \mu)$$: $$\begin{gathered} - \mpdv{\Omega}{V}{T} = -- cgit v1.2.3