# Thermodynamic potentials

Thermodynamic potentials are extremely useful tools, whose name derives from an analogy with mechanical potential energy: as we will later see, in certain circumstances the work obtainable from a macroscopic system is related to the change of an appropriately defined function, the thermodynamic potential. They are useful because they allow one to define quantities which are experimentally more easy to control and to rewrite the fundamental thermodynamic relations in terms of them.

Mathematically all the thermodynamic potentials are the result of a Legendre transformation of the internal energy, namely they are a rewriting of the internal energy so that a variable has been substituted with another. For a ${\displaystyle PVT}$ system ${\displaystyle U=U(S,V,N)}$; more in general we can write:

${\displaystyle U=U(S,\lbrace x_{i}\rbrace ,\lbrace N_{j}\rbrace )}$
where ${\displaystyle x_{i}}$ are called generalized displacements and ${\displaystyle N_{j}}$ is the number of the ${\displaystyle j}$-th type of particle, and so we have:
${\displaystyle dU=TdS+\sum _{i}X_{i}dx_{i}+\sum _{j}\mu _{j}dN_{j}}$
where ${\displaystyle X_{i}}$ are called generalized forces. The thermodynamic potentials that are generally used are:

• Helmholtz free energy. It is defined by:
${\displaystyle F=U-TS}$

Let's see that it is indeed a function of ${\displaystyle T}$ instead of ${\displaystyle S}$:

${\displaystyle dF=dU-d(TS)=dU-SdT-TdS=TdS-PdV+\mu dN-SdT-TdS\quad \Rightarrow }$
${\displaystyle \Rightarrow \quad dF=-SdT-PdV+\mu dN\quad \Rightarrow \quad F=F(T,V,N)}$
Let us note that from this we have:
${\displaystyle S=-{\frac {\partial F}{\partial T}}_{|V,N}\quad \qquad P=-{\frac {\partial F}{\partial V}}_{|T,N}\quad \qquad \mu ={\frac {\partial F}{\partial N}}_{|T,V}}$

We now show that ${\displaystyle F}$ is in effect a thermodynamic potential. For a general infinitesimal process we have:

${\displaystyle dF=dU-d(TS)=\delta Q-\delta W-TdS-SdT}$
and thus:
${\displaystyle \delta W=\delta Q-TdS-SdT-dF}$
For a reversible transformation ${\displaystyle \delta Q-TdS=0}$, and if it is also isothermal then ${\displaystyle dT=0}$; thus, for a reversible isothermal process ${\displaystyle \delta W=-dF}$: the quantity ${\displaystyle -\Delta F}$ is the amount of work that can be obtained from a reversible isothermal process, and hence the name "thermodynamic potential". However, if the process is isothermal but irreversible then ${\displaystyle \delta Q-TdS\leq 0}$, so:
${\displaystyle (\delta W)_{\text{irr.,isot.}}=\delta Q-TdS-dF\leq -dF}$
therefore in general ${\displaystyle -\Delta F}$ is the maximum work that can be extracted from a system at constant temperature. We also see that if the process is spontaneous ${\displaystyle \delta W=0}$ and so ${\displaystyle dF\leq 0}$: a spontaneous transformation can only decrease the Helmholtz free energy of a system, and thus we can conclude that equilibrium states of a system at fixed ${\displaystyle T}$, ${\displaystyle \lbrace x_{i}\rbrace }$, ${\displaystyle \lbrace N_{j}\rbrace }$ are the global minima of ${\displaystyle F}$.

• Gibbs free energy. It is defined by:

${\displaystyle G=U-TS+PV=F+PV\quad \Rightarrow \quad G=G(T,P,N)}$
and as we have done for ${\displaystyle F}$ it can be shown that ${\displaystyle G}$ is the thermodynamic potential for transformations at fixed temperature and pressure. Note that in general, if a system can be characterised by a set ${\displaystyle \lbrace x_{i}\rbrace }$ of generalized displacements (and ${\displaystyle \lbrace X_{i}\rbrace }$ are the corresponding generalized forces) and is composed of different types of particles, then:
${\displaystyle G(T,\lbrace X_{i}\rbrace ,\lbrace N_{j}\rbrace )=\sum _{j}\mu _{j}N_{j}}$
In order to show that, we must consider the Euler identity associated tu the internal energy ${\displaystyle U}$. As a reminder, ${\displaystyle f(x_{1},\dots ,x_{N})}$ is a homogeneous function of degree ${\displaystyle \ell }$ if and only if we have:
${\displaystyle \sum _{k=1}^{N}{\frac {\partial f}{\partial x_{k}}}x_{k}=\ell f(x)}$
Taking ${\displaystyle f}$ as the internal energy ${\displaystyle U(S,\lbrace x_{i}\rbrace ,\lbrace N_{j}\rbrace )}$, since it is homogeneous of degree one we have:
${\displaystyle {\frac {\partial U}{\partial S}}_{|x_{i},N}S+\sum _{i}{\frac {\partial U}{\partial x_{i}}}_{|S,N}x_{i}+\sum _{j}{\frac {\partial U}{\partial N_{j}}}_{|S,x_{i}}N_{j}=U}$
Namely:
${\displaystyle U=TS+\sum _{i}X_{i}x_{i}+\sum _{j}\mu _{j}N_{j}}$
Considering the general definition of the Gibbs free energy, we get precisely ${\textstyle G(T,\lbrace X_{i}\rbrace ,\lbrace N_{j}\rbrace )=\sum _{j}\mu _{j}N_{j}}$ .

• Entalpy. It is defined as:

${\displaystyle H=U+PV\quad \Rightarrow \quad H=H(S,P,N)}$
and it is the thermodynamic potential for isobaric transformations.

• Grand potential. It is defined as:

${\displaystyle \Phi =U-TS-\mu N=F-\mu N\quad \Rightarrow \quad \Phi =\Phi (T,V,\mu )}$
and is useful for the description of open systems, namely systems that can exchange particles with their surroundings.