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diff --git a/content/know/concept/thermodynamic-potential/index.pdc b/content/know/concept/thermodynamic-potential/index.pdc new file mode 100644 index 0000000..5d154d5 --- /dev/null +++ b/content/know/concept/thermodynamic-potential/index.pdc @@ -0,0 +1,279 @@ +--- +title: "Thermodynamic potential" +firstLetter: "T" +publishDate: 2021-07-07 +categories: +- Physics +- Thermodynamics + +date: 2021-07-03T14:40:22+02:00 +draft: false +markup: pandoc +--- + +# Thermodynamic potential + +**Thermodynamic potentials** are state functions +whose minima or maxima represent equilibrium states of a system. +Such functions are either energies (hence *potential*) or entropies. + +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. + +The link from natural variables to potentials +is established by thermodynamic ensembles. + +Once enough natural variables have been found, +the appropriate potential can be selected from the list below. +All non-natural variables can then be calculated +by taking partial derivatives of the potential +with respect to the natural variables. + +Mathematically, the potentials are related to each other +by [Legendre transformation](/know/concept/legendre-transform/). + + +## Internal energy + +The **internal energy** $U$ represents +the capacity to do both mechanical and non-mechanical work, +and to release heat. +It is simply the integral +of the [fundamental thermodynamic relation](/know/concept/fundamental-thermodynamic-relation/): + +$$\begin{aligned} + \boxed{ + U(S, V, N) \equiv T S - P V + \mu N + } +\end{aligned}$$ + +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: + +$$\begin{aligned} + \boxed{ + \dd{U} = T \dd{S} - P \dd{V} + \mu \dd{N} + } +\end{aligned}$$ + +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 $N$: + +$$\begin{aligned} + \boxed{ + T = \Big( \pdv{U}{S} \Big)_{V,N} + \qquad + P = - \Big( \pdv{U}{V} \Big)_{S,N} + \qquad + \mu = \Big( \pdv{U}{N} \Big)_{S,V} + } +\end{aligned}$$ + +It is convention to write those subscripts, +to help keep track of which function depends on which variables. +They are meaningless; these are normal partial derivatives. + + +## Enthalpy + +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: + +$$\begin{aligned} + \boxed{ + H(S, P, N) \equiv U + P V + } +\end{aligned}$$ + +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: + +$$\begin{aligned} + \boxed{ + \dd{H} = T \dd{S} + V \dd{P} + \mu \dd{N} + } +\end{aligned}$$ + +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 $N$: + +$$\begin{aligned} + \boxed{ + T = \Big( \pdv{H}{S} \Big)_{P,N} + \qquad + V = \Big( \pdv{H}{P} \Big)_{S,N} + \qquad + \mu = \Big( \pdv{H}{N} \Big)_{S,P} + } +\end{aligned}$$ + + +## Helmholtz free energy + +The **Helmholtz free energy** $F$ represents +the capacity of a system to +do both mechanical and non-mechanical work, +and is given by: + +$$\begin{aligned} + \boxed{ + F(T, V, N) \equiv U - T S + } +\end{aligned}$$ + +It depends on the temperature $T$, volume $V$, and particle count $N$: +these are natural variables. +An infinitesimal change $\dd{H}$ is as follows: + +$$\begin{aligned} + \boxed{ + \dd{F} = - P \dd{V} - S \dd{T} + \mu \dd{N} + } +\end{aligned}$$ + +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 $N$: + +$$\begin{aligned} + \boxed{ + S = - \Big( \pdv{F}{T} \Big)_{V,N} + \qquad + P = - \Big( \pdv{F}{V} \Big)_{T,N} + \qquad + \mu = \Big( \pdv{F}{N} \Big)_{T,V} + } +\end{aligned}$$ + + +## Gibbs free energy + +The **Gibbs free energy** $G$ represents +the capacity of a system to do non-mechanical work: + +$$\begin{aligned} + \boxed{ + G(T, P, N) + \equiv U + P V - T S + } +\end{aligned}$$ + +It depends on the temperature $T$, pressure $P$, and particle count $N$: +they are natural variables. +An infinitesimal change $\dd{G}$ is as follows: + +$$\begin{aligned} + \boxed{ + \dd{G} = V \dd{P} - S \dd{T} + \mu \dd{N} + } +\end{aligned}$$ + +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 $N$: + +$$\begin{aligned} + \boxed{ + S = - \Big( \pdv{G}{T} \Big)_{P,N} + \qquad + V = \Big( \pdv{G}{P} \Big)_{T,N} + \qquad + \mu = \Big( \pdv{G}{N} \Big)_{T,P} + } +\end{aligned}$$ + + +## Landau potential + +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: + +$$\begin{aligned} + \boxed{ + \Omega(T, V, \mu) \equiv U - T S - \mu N + } +\end{aligned}$$ + +It depends on temperature $T$, volume $V$, and chemical potential $\mu$: +these are natural variables. +An infinitesimal change $\dd{\Omega}$ is as follows: + +$$\begin{aligned} + \boxed{ + \dd{\Omega} = - P \dd{V} - S \dd{T} - N \dd{\mu} + } +\end{aligned}$$ + +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 $\mu$: + +$$\begin{aligned} + \boxed{ + S = \Big( \pdv{\Omega}{T} \Big)_{V,\mu} + \qquad + P = - \Big( \pdv{\Omega}{V} \Big)_{T,\mu} + \qquad + N = - \Big( \pdv{\Omega}{\mu} \Big)_{T,V} + } +\end{aligned}$$ + + +## Entropy + +The **entropy** $S$, in units of energy over temperature, +is an odd duck, but nevertheless used as a thermodynamic potential. +It is given by: + +$$\begin{aligned} + \boxed{ + S(U, V, N) \equiv \frac{1}{T} U + \frac{P}{T} V - \frac{\mu}{T} N + } +\end{aligned}$$ + +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: + +$$\begin{aligned} + \boxed{ + \dd{S} = \frac{1}{T} \dd{U} + \frac{P}{T} \dd{V} - \frac{\mu}{T} \dd{N} + } +\end{aligned}$$ + +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 $N$: + +$$\begin{aligned} + \boxed{ + \frac{1}{T} = \Big( \pdv{S}{U} \Big)_{V,N} + \qquad + \frac{P}{T} = \Big( \pdv{S}{V} \Big)_{U,N} + \qquad + \frac{\mu}{T} = - \Big( \pdv{S}{N} \Big)_{U,V} + } +\end{aligned}$$ + + + +## References +1. H. Gould, J. Tobochnik, + *Statistical and thermal physics*, 2nd edition, + Princeton. |