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Diffstat (limited to 'source/know/concept/thermodynamic-potential/index.md')
| -rw-r--r-- | source/know/concept/thermodynamic-potential/index.md | 76 |
1 files changed, 38 insertions, 38 deletions
diff --git a/source/know/concept/thermodynamic-potential/index.md b/source/know/concept/thermodynamic-potential/index.md index 3e211f7..ece1551 100644 --- a/source/know/concept/thermodynamic-potential/index.md +++ b/source/know/concept/thermodynamic-potential/index.md @@ -16,8 +16,8 @@ Which potential (of many) decides the equilibrium states for a given system? That depends which variables are assumed to already be in automatic equilibrium. Such variables are known as the **natural variables** of that potential. For example, if a system can freely exchange heat with its surroundings, -and is consequently assumed to be at the same temperature $T = T_{\mathrm{sur}}$, -then $T$ must be a natural variable. +and is consequently assumed to be at the same temperature $$T = T_{\mathrm{sur}}$$, +then $$T$$ must be a natural variable. The link from natural variables to potentials is established by [thermodynamic ensembles](/know/category/thermodynamic-ensembles/). @@ -34,7 +34,7 @@ by [Legendre transformation](/know/concept/legendre-transform/). ## Internal energy -The **internal energy** $U$ represents +The **internal energy** $$U$$ represents the capacity to do both mechanical and non-mechanical work, and to release heat. It is simply the integral @@ -46,9 +46,9 @@ $$\begin{aligned} } \end{aligned}$$ -It is a function of the entropy $S$, volume $V$, and particle count $N$: +It is a function of the entropy $$S$$, volume $$V$$, and particle count $$N$$: these are its natural variables. -An infinitesimal change $\dd{U}$ is as follows: +An infinitesimal change $$\dd{U}$$ is as follows: $$\begin{aligned} \boxed{ @@ -57,9 +57,9 @@ $$\begin{aligned} \end{aligned}$$ The non-natural variables are -temperature $T$, pressure $P$, and chemical potential $\mu$. -They can be recovered by differentiating $U$ -with respect to the natural variables $S$, $V$, and $N$: +temperature $$T$$, pressure $$P$$, and chemical potential $$\mu$$. +They can be recovered by differentiating $$U$$ +with respect to the natural variables $$S$$, $$V$$, and $$N$$: $$\begin{aligned} \boxed{ @@ -78,7 +78,7 @@ They are meaningless; these are normal partial derivatives. ## Enthalpy -The **enthalpy** $H$ of a system, in units of energy, +The **enthalpy** $$H$$ of a system, in units of energy, represents its capacity to do non-mechanical work, plus its capacity to release heat. It is given by: @@ -89,9 +89,9 @@ $$\begin{aligned} } \end{aligned}$$ -It is a function of the entropy $S$, pressure $P$, and particle count $N$: +It is a function of the entropy $$S$$, pressure $$P$$, and particle count $$N$$: these are its natural variables. -An infinitesimal change $\dd{H}$ is as follows: +An infinitesimal change $$\dd{H}$$ is as follows: $$\begin{aligned} \boxed{ @@ -100,9 +100,9 @@ $$\begin{aligned} \end{aligned}$$ The non-natural variables are -temperature $T$, volume $V$, and chemical potential $\mu$. -They can be recovered by differentiating $H$ -with respect to the natural variables $S$, $P$, and $N$: +temperature $$T$$, volume $$V$$, and chemical potential $$\mu$$. +They can be recovered by differentiating $$H$$ +with respect to the natural variables $$S$$, $$P$$, and $$N$$: $$\begin{aligned} \boxed{ @@ -117,7 +117,7 @@ $$\begin{aligned} ## Helmholtz free energy -The **Helmholtz free energy** $F$ represents +The **Helmholtz free energy** $$F$$ represents the capacity of a system to do both mechanical and non-mechanical work, and is given by: @@ -128,9 +128,9 @@ $$\begin{aligned} } \end{aligned}$$ -It depends on the temperature $T$, volume $V$, and particle count $N$: +It depends on the temperature $$T$$, volume $$V$$, and particle count $$N$$: these are natural variables. -An infinitesimal change $\dd{H}$ is as follows: +An infinitesimal change $$\dd{H}$$ is as follows: $$\begin{aligned} \boxed{ @@ -139,9 +139,9 @@ $$\begin{aligned} \end{aligned}$$ The non-natural variables are -entropy $S$, pressure $P$, and chemical potential $\mu$. -They can be recovered by differentiating $F$ -with respect to the natural variables $T$, $V$, and $N$: +entropy $$S$$, pressure $$P$$, and chemical potential $$\mu$$. +They can be recovered by differentiating $$F$$ +with respect to the natural variables $$T$$, $$V$$, and $$N$$: $$\begin{aligned} \boxed{ @@ -156,7 +156,7 @@ $$\begin{aligned} ## Gibbs free energy -The **Gibbs free energy** $G$ represents +The **Gibbs free energy** $$G$$ represents the capacity of a system to do non-mechanical work: $$\begin{aligned} @@ -166,9 +166,9 @@ $$\begin{aligned} } \end{aligned}$$ -It depends on the temperature $T$, pressure $P$, and particle count $N$: +It depends on the temperature $$T$$, pressure $$P$$, and particle count $$N$$: they are natural variables. -An infinitesimal change $\dd{G}$ is as follows: +An infinitesimal change $$\dd{G}$$ is as follows: $$\begin{aligned} \boxed{ @@ -177,9 +177,9 @@ $$\begin{aligned} \end{aligned}$$ The non-natural variables are -entropy $S$, volume $V$, and chemical potential $\mu$. -These can be recovered by differentiating $G$ -with respect to the natural variables $T$, $P$, and $N$: +entropy $$S$$, volume $$V$$, and chemical potential $$\mu$$. +These can be recovered by differentiating $$G$$ +with respect to the natural variables $$T$$, $$P$$, and $$N$$: $$\begin{aligned} \boxed{ @@ -194,7 +194,7 @@ $$\begin{aligned} ## Landau potential -The **Landau potential** or **grand potential** $\Omega$, in units of energy, +The **Landau potential** or **grand potential** $$\Omega$$, in units of energy, represents the capacity of a system to do mechanical work, and is given by: @@ -204,9 +204,9 @@ $$\begin{aligned} } \end{aligned}$$ -It depends on temperature $T$, volume $V$, and chemical potential $\mu$: +It depends on temperature $$T$$, volume $$V$$, and chemical potential $$\mu$$: these are natural variables. -An infinitesimal change $\dd{\Omega}$ is as follows: +An infinitesimal change $$\dd{\Omega}$$ is as follows: $$\begin{aligned} \boxed{ @@ -215,9 +215,9 @@ $$\begin{aligned} \end{aligned}$$ The non-natural variables are -entropy $S$, pressure $P$, and particle count $N$. -These can be recovered by differentiating $\Omega$ -with respect to the natural variables $T$, $V$, and $\mu$: +entropy $$S$$, pressure $$P$$, and particle count $$N$$. +These can be recovered by differentiating $$\Omega$$ +with respect to the natural variables $$T$$, $$V$$, and $$\mu$$: $$\begin{aligned} \boxed{ @@ -232,7 +232,7 @@ $$\begin{aligned} ## Entropy -The **entropy** $S$, in units of energy over temperature, +The **entropy** $$S$$, in units of energy over temperature, is an odd duck, but nevertheless used as a thermodynamic potential. It is given by: @@ -242,9 +242,9 @@ $$\begin{aligned} } \end{aligned}$$ -It depends on the internal energy $U$, volume $V$, and particle count $N$: +It depends on the internal energy $$U$$, volume $$V$$, and particle count $$N$$: they are natural variables. -An infinitesimal change $\dd{S}$ is as follows: +An infinitesimal change $$\dd{S}$$ is as follows: $$\begin{aligned} \boxed{ @@ -252,9 +252,9 @@ $$\begin{aligned} } \end{aligned}$$ -The non-natural variables are $1/T$, $P/T$, and $\mu/T$. -These can be recovered by differentiating $S$ -with respect to the natural variables $U$, $V$, and $N$: +The non-natural variables are $$1/T$$, $$P/T$$, and $$\mu/T$$. +These can be recovered by differentiating $$S$$ +with respect to the natural variables $$U$$, $$V$$, and $$N$$: $$\begin{aligned} \boxed{ |