2.7: Thermodynamic Potentials

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\)
\( \newcommand\VTheta{\vec\Theta}\)
\( \newcommand\VLambda{\vec\Lambda}\)
\( \newcommand\VXi{\vec\Xi}\)
\( \newcommand\VPi{\vec\Pi}\)
\( \newcommand\VSigma{\vec\Sigma}\)
\( \newcommand\VUps{\vec\Upsilon}\)
\( \newcommand\VPhi{\vec\Phi}\)
\( \newcommand\VPsi{\vec\Psi}\)
\( \newcommand\VOmega{\vec\Omega}\)
\( \newcommand\BA{\mib A}\)
\( \newcommand\BB{\mib B}\)
\( \newcommand\BC{\mib C}\)
\( \newcommand\BD{\mib D}\)
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\( \newcommand\BBA{\boldsymbol\RA}\)
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\( \def\texpect#1#2#3{\langle #1 |   #2  |  #3 \rangle}\)
\( \def\sket#1{|  \, #1 \,  \rangle}\)
\( \def\sbra#1{\langle \,  #1 \, |}\)
\( \def\sbraket#1#2{\langle \, #1  \, |  \, #2 \,  \rangle}\)
\( \def\sexpect#1#2#3{\langle \, #1 \, | \,  #2  \, | \, #3 \, \rangle}\)
\(\def\ket#1{\big| \, #1\, \big\rangle}\)
\( \def\bra#1{\big\langle \, #1 \, \big|}\)
\( \def\braket#1#2{\big\langle \, #1\, \big| \,#2 \,\big\rangle}\)
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\( \newcommand\pz{\partial}\)
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\( \newcommand\svph{\vphantom{\int}}\)
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\( \newcommand\bvph{\vphantom{\sum_N^N}}\)
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\(\newcommand \gtwid{\approx}\)

\( \newcommand\index{\textsf{ind}}\)
\( \newcommand\csch{\,{ csch\,}}\)
\( \newcommand\ctnh{\,{ ctnh\,}}\)
\( \newcommand\ctn{\,{ ctn\,}}\)
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\( \def\tmapright#1{\xrightarrow \limits^{#1}}\)
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\( \newcommand\Rep{\textsf{Re}\,}\)
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\( \newcommand\ncdot{\!\cdot\!}\)
\( \def\tmapright#1{ \smash{\mathop{\hbox to 35pt{\rightarrowfill}}\limits^{#1}}\ }\)
\( \def\bmapright#1{ \smash{\mathop{\hbox to 35pt{\rightarrowfill}}\limits_{#1}}\ }\)
\( \newcommand\bsqcap{\mbox{\boldmath{$\sqcap$}}}\)

\( \def\pabc#1#2#3{\left({\pz #1\over\pz #2}\right)\ns_{\!\!#3}}\)
\( \def\spabc#1#2#3{\big({\pz #1\over\pz #2}\big)\ns_{\!#3}}\)
\( \def\qabc#1#2#3{\pz^2\! #1\over\pz #2\,\pz #3}\)
\( \def\rabc#1#2#3#4{(\pz #1,\pz #2)\over (\pz #3,\pz #4)}\)
\( \newcommand\subA{\ns_\ssr{A}}\)
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\( \newcommand\subDA{\ns_\ssr{DA}}\)
\( \def\lmapright#1{\ \ \smash{\mathop{\hbox to 55pt{\rightarrowfill}}\limits^{#1}}\ \ }\)
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\( \newcommand\longrightleftharpoons{ \mathop{\vcenter{\hbox{\ooalign{\raise1pt\hbox{$\longrightharpoonup\joinrel$}\crcr  \lower1pt\hbox{$\longleftharpoondown\joinrel$}}}}}}\)
\( \newcommand\longrightharpoonup{\relbar\joinrel\rightharpoonup}\)
\( \newcommand\longleftharpoondown{\leftharpoondown\joinrel\relbar}\)
\( \newcommand\cds{\,\bullet\,}\)
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\( \newcommand\nsub{_{\vphantom{\dagger}}}\)
\( \newcommand\rhohat{\hat\rho}\)
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\( \newcommand\impi{\int\limits_{-\infty}^\infty\!\!\!}\)
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\( \newcommand\zbar{\bar z}\)
\( \newcommand\ftil{\tilde f}\)
\( \newcommand\XBE{\RXi\ns_\ssr{BE}}\)
\( \newcommand\XFD{\RXi\ns_\ssr{FD}}\)
\( \newcommand\OBE{\Omega\ns_\ssr{BE}}\)
\( \newcommand\OFD{\Omega\ns_\ssr{FD}}\)
\( \newcommand\veF{\ve\ns_\RF}\)
\( \newcommand\kF{k\ns_\RF}\)
\( \newcommand\kFu{k\ns_{\RF\uar}}\)
\( \newcommand\SZ{\textsf Z}}\) \( \newcommand\kFd{k\ns_{\RF\dar}\)
\( \newcommand\muB{\mu\ns_\ssr{B}}\)
\( \newcommand\mutB{\tilde\mu}\ns_\ssr{B}\)
\( \newcommand\xoN{\Bx\ns_1\,,\,\ldots\,,\,\Bx\ns_N}\)
\( \newcommand\rok{\Br\ns_1\,,\,\ldots\,,\,\Br\ns_k}\)
\( \newcommand\xhiOZ{\xhi^\ssr{OZ}}\)
\( \newcommand\xhihOZ

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\)
\( \newcommand\jhz{\HJ(0)}\)
\( \newcommand\nda{\nd_\alpha}\)
\( \newcommand\ndap{\nd_{\alpha'}}\)
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\)
\( \newcommand\msa{m\ns_\ssr{A}}\)
\( \newcommand\msb{m\ns_\ssr{B}}\)
\( \newcommand\mss{m\ns_\Rs}\)
\( \newcommand\HBx{\hat\Bx}\)
\( \newcommand\HBy{\hat\By}\)
\( \newcommand\HBz{\hat\Bz}\)
\( \newcommand\thm{\theta\ns_m}\)
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Thermodynamic systems may do work on their environments. Under certain constraints, the work done may be bounded from above by the change in an appropriately defined thermodynamic potential.

Energy \(E\)

Suppose we wish to create a thermodynamic system from scratch. Let’s imagine that we create it from scratch in a thermally insulated box of volume \(V\). The work we must to to assemble the system is then \(\CW=E\) . After we bring all the constituent particles together, pulling them in from infinity (say), the system will have total energy \(E\). After we finish, the system may not be in thermal equilibrium. Spontaneous processes will then occur so as to maximize the system’s entropy, but the internal energy remains at \(E\).

We have, from the First Law, \(dE=\dbar Q-\dbar W\)  and combining this with the Second Law in the form \(\dbar Q \le T\,dS\) yields \[dE\le T\,dS - \dbar W\quad. \label{dEeqn}\] Rearranging terms, we have \(\dbar W\le T\,dS-dE\) . Hence, the work done by a thermodynamic system under conditions of constant entropy is bounded above by \(-dE\), and the maximum \(\dbar W\) is achieved for a reversible process. It is sometimes useful to define the quantity \[\dbar W\ns_{free}=\dbar W- p\,dV\ ,\] which is the differential work done by the system other than that required to change its volume. Then we have \[\dbar W\ns_{free}\le T\,dS - p\,dV - dE \quad,\] and we conclude for systems at fixed \((S,V)\) that \(\dbar W_{free}\le - dE\).

In equilibrium, the equality in Equation [dEeqn] holds, and for single component systems where \(\dbar W = p\,dV - \mu\,dN\) we have \(E=E(S,V,N)\) with \[T=\pabc{E}{S}{V,N} \qquad,\qquad -p=\pabc{E}{V}{S,N} \qquad,\qquad \mu=\pabc{E}{N}{S,V}\quad .\] These expressions are easily generalized to multicomponent systems, magnetic systems,

Now consider a single component system at fixed \((S,V,N)\). We conclude that \(dE\le 0\) , which says that spontaneous processes in a system with \(dS=dV=dN=0\) always lead to a reduction in the internal energy \(E\). Therefore, spontaneous processes drive the internal energy \(E\) to a minimum in systems at fixed \((S,V,N)\).

Helmholtz free energy \(F\)

Suppose that when we spontaneously create our system while it is in constant contact with a thermal reservoir at temperature \(T\). Then as we create our system, it will absorb heat from the reservoir. Therefore, we don’t have to supply the full internal energy \(E\), but rather only \(E-Q\), since the system receives heat energy \(Q\) from the reservoir. In other words, we must perform work \(\CW=E-TS\) to create our system, if it is constantly in equilibrium at temperature \(T\). The quantity \(E-TS\) is known as the Helmholtz free energy, \(F\), which is related to the energy \(E\) by a Legendre transformation, \[F=E-TS\ .\] The general properties of Legendre transformations are discussed in Appendix II, §16.

Again invoking the Second Law, we have \[dF\le -S\,dT-\dbar W\quad. \label{dFeqn}\] Rearranging terms, we have \(\dbar W\le -S\,dT - dF\) , which says that the work done by a thermodynamic system under conditions of constant temperature is bounded above by \(-dF\), and the maximum \(\dbar W\) is achieved for a reversible process. We also have the general result \[\dbar W\ns_{free}\le -S\,dT - p\,dV - dF \ ,\] and we conclude, for systems at fixed \((T,V)\), that \(\dbar W_{free}\le -dF\).

Under equilibrium conditions, the equality in Equation [dFeqn] holds, and for single component systems where \(\dbar W = p\,dV-\mu\,dN\) we have \(dF=-S\,dT - p\,dV + \mu\,dN\) . This says that \(F=F(T,V,N)\) with \[-S=\pabc{F}{T}{V,N} \qquad,\qquad -p=\pabc{F}{V}{T,N} \qquad,\qquad \mu=\pabc{F}{N}{T,V}\quad .\] For spontaneous processes, \(dF \le -S\,dT - p\,dV + \mu\,dN\) says that spontaneous processes drive the Helmholtz free energy \(F\) to a minimum in systems at fixed \((T,V,N)\).

Enthalpy \(\CH\)

Suppose that when we spontaneously create our system while it is thermally insulated, but in constant mechanical contact with a ‘volume bath’ at pressure \(p\). For example, we could create our system inside a thermally insulated chamber with one movable wall where the external pressure is fixed at \(p\). Thus, when creating the system, in addition to the system’s internal energy \(E\), we must also perform work \(pV\) in order to make room for it. In other words, we must perform work \(\CW=E+pV\). The quantity \(E+pV\) is known as the enthalpy, \(\CH\). (We use the calligraphic font for \(\CH\) for enthalpy to avoid confusing it with magnetic field, \(H\).) The enthalpy is obtained from the energy via a different Legendre transformation than that used to obtain the Helmholtz free energy \(F\) , \[\CH=E+pV \quad.\]

Again invoking the Second Law, we have \[d\CH\le T\,dS -\dbar W + p\,dV + V dp\quad,\] hence with \(\dbar W\ns_{free}=\dbar W - p\,dV\), we have in general \[\dbar W\ns_{free} \le T\,dS + V dp -d\CH\ ,\] and we conclude, for systems at fixed \((S,p)\), that \(\dbar W_{free}\le -d\CH\).

In equilibrium, for single component systems, \[d\CH=T\,dS + V dp + \mu\,dN\ ,\] which says \(\CH=\CH(S,p,N)\), with \[T=\pabc{\CH}{S}{p,N} \qquad,\qquad V=\pabc{\CH}{p}{S,N} \qquad,\qquad \mu=\pabc{\CH}{N}{S,p}\quad.\] For spontaneous processes, \(d\CH\le T\,dS + V dp +\mu\,dN\), which says that spontaneous processes drive the enthalpy \(\CH\) to a minimum in systems at fixed \((S,p,N)\).

Gibbs free energy \(G\)

If we create a thermodynamic system at conditions of constant temperature \(T\) and constant pressure \(p\), then it absorbs heat energy \(Q=TS\) from the reservoir and we must expend work energy \(pV\) in order to make room for it. Thus, the total amount of work we must do in assembling our system is \(\CW=E-TS+pV\). This is the Gibbs free energy, \(G\). The Gibbs free energy is obtained from \(E\) after two Legendre transformations, \[G=E-TS+pV\] Note that \(G=F+pV=\CH-TS\). The Second Law says that \[dG\le -S\,dT + V dp + p\,dV -\dbar W\quad,\] which we may rearrange as \(\dbar W\ns_{free}\le -S\,dT + V dp - dG\) . Accordingly, we conclude, for systems at fixed \((T,p)\), that \(\dbar W_{free}\le - dG\).

For equilibrium one-component systems, the differential of \(G\) is \[dG=-S\,dT + V dp + \mu\,dN\ ,\] therefore \(G=G(T,p,N)\), with \[-S=\pabc{G}{T}{p,N} \qquad,\qquad V=\pabc{G}{p}{T,N} \qquad,\qquad \mu=\pabc{G}{N}{T,p}\quad.\] Recall that Euler’s theorem for single component systems requires \(E=TS-pV+\mu N\)  which says \(G=\mu N\), Thus, the chemical potential \(\mu\) is the Gibbs free energy per particle. For spontaneous processes, \(dG\le -S\,dT + Vdp + \mu\,dN\) , hence spontaneous processes drive the Gibbs free energy \(G\) to a minimum in systems at fixed \((T,p,N)\).

Grand potential \(\Omega\)

The grand potential, sometimes called the Landau free energy, is defined by \[\Omega=E-TS-\mu N\ .\] Under equilibrium conditions, its differential is \[d\Omega=-S\,dT -p\,dV - N\,d\mu\ ,\] hence \[-S=\pabc{\Omega}{T}{V,\mu} \qquad,\qquad -p=\pabc{\Omega}{V}{T,\mu} \qquad,\qquad -N=\pabc{\Omega}{\mu}{T,V}\ .\] Again invoking Equation [ETS], we find \(\Omega = -pV\), which says that the pressure is the negative of the grand potential per unit volume.

The Second Law tells us \[d\Omega \le - \dbar W - S\,dT - \mu\,dN - N\,d\mu\ ,\] hence \[\dbar {\widetilde W}\ns_{free}\equiv \dbar W\ns_{free}+\mu\,dN\le -S\,dT -p\,dV -N\,d\mu -d\Omega\ .\] We conclude, for systems at fixed \((T,V,\mu)\), that \(\dbar {\widetilde W}_{free}\le - d\Omega\).