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# 2.16: Appendix III- Useful Mathematical Relations


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Consider a set of $$n$$ independent variables $$\{x\ns_1,\ldots,x\ns_n\}$$, which can be thought of as a point in $$n$$-dimensional space. Let $$\{y\ns_1,\ldots,y\ns_n\}$$ and $$\{z\ns_1,\ldots,z\ns_n\}$$ be other choices of coordinates. Then ${\pz x_i\over \pz z\ns_k}={\pz x_i\over\pz y\ns_j}\,{\pz y_j\over\pz z\ns_k}\ .$ Note that this entails a matrix multiplication: $$A\ns_{ik}=B\ns_{ij}\,C\ns_{jk}$$, where $$A\ns_{ik}=\pz x\ns_i/\pz z\ns_k$$, $$B\ns_{ij}=\pz x\ns_i/\pz y\ns_j$$, and $$C\ns_{jk}=\pz y\ns_j/\pz z\ns_k$$. We define the determinant ${det}\bigg({\pz x\ns_i\over\pz z\ns_k}\bigg)\equiv {\pz (x\ns_1,\ldots,x\ns_n)\over \pz (z\ns_1,\ldots,z\ns_n)}\ .$ Such a determinant is called a Jacobian. Now if $$A=BC$$, then $${det}(A)={det}(B)\cdot{det}(C)$$. Thus, ${\pz (x\ns_1,\ldots, x\ns_n)\over \pz( z\ns_1,\ldots, z\ns_n)}= {\pz( x\ns_1,\ldots, x\ns_n)\over \pz( y\ns_1,\ldots, y\ns_n)}\cdot {\pz( y\ns_1,\ldots, y\ns_n)\over \pz( z\ns_1,\ldots, z\ns_n)}\ . \label{chain}$ Recall also that ${\pz x\ns_i\over \pz x\ns_k}=\delta\ns_{ik}\ .$

Consider the case $$n=2$$. We have ${\pz( x, y)\over \pz( u, v)}={det}\begin{pmatrix} \pabc{x}{u}{v} & \pabc {x}{v}{u} \\ & \\ \pabc{y}{u}{v} & \pabc{y}{v}{u} \end{pmatrix} = \pabc{x}{u}{v} \pabc{y}{v}{u} - \pabc {x}{v}{u} \pabc{y}{u}{v} \ .$ We also have ${\pz( x, y)\over \pz( u, v)}\cdot {\pz( u, v)\over \pz( r, s)} = {\pz( x, y)\over \pz( r, s)}\ .$ From this simple mathematics follows several very useful results.

1) First, write ${\pz(x, y)\over \pz(u, v)}=\Bigg[{\pz(u, v)\over \pz(x, y)}\Bigg]^{-1}\ .$ Now let $$v=y$$ : ${\pz(x, y)\over \pz(u, y)}=\pabc{x}{u}{y}={1\over\pabc{u}{x}{y}}\ .$ Thus, $\pabc{x}{u}{y}=1\Big/\pabc{u}{x}{y} \label{boxone}$

2) Second, we have ${\pz(x, y)\over \pz(u, y)}=\pabc{x}{u}{y} \nonumber\\ ={\pz(x, y)\over \pz(x, u)}\cdot {\pz(x, u)\over \pz(u, y)}\\ =-\pabc{y}{u}{x}\pabc{x}{y}{u}\ ,$ which is to say $\pabc{x}{y}{u} \pabc{y}{u}{x} = - \pabc{x}{u}{y}\ . \label{boxtwob}$ Invoking Equation [boxone], we conclude that $\pabc{x}{y}{u} \pabc{y}{u}{x} \pabc{u}{x}{y} = -1\ . \label{boxtwo}$

3) Third, we have ${\pz(x, v)\over \pz(u, v)}={\pz(x, v)\over \pz(y, v)}\cdot {\pz(y, v)\over \pz(u, v)}\ ,$ which says $\pabc{x}{u}{v}=\pabc{x}{y}{v}\pabc{y}{u}{v} \label{boxthree}$ This is simply the chain rule of partial differentiation.

4) Fourth, we have $\begin{split} {\pz(x,y)\over\pz(u,y)}&={\pz(x,y)\over\pz(u,v)}\cdot{\pz(u,v)\over\pz(u,y)}\\ &=\pabc{x}{u}{v}\pabc{y}{v}{u}\pabc{v}{y}{u}-\pabc{x}{v}{u}\pabc{y}{u}{v}\pabc{v}{y}{u}\ , \end{split}$ which says $\pabc{x}{u}{y}=\pabc{x}{u}{v}-\pabc{x}{y}{u}\pabc{y}{u}{v} \label{boxfour}$

5) Fifth, whenever we differentiate one extensive quantity with respect to another, holding only intensive quantities constant, the result is simply the ratio of those extensive quantities. For example, $\pabc{S}{V}{p,T}={S\over V}\ .$ The reason should be obvious. In the above example, $$S(p,V,T)=V\phi(p,T)$$, where $$\phi$$ is a function of the two intensive quantities $$p$$ and $$T$$. Hence differentiating $$S$$ with respect to $$V$$ holding $$p$$ and $$T$$ constant is the same as dividing $$S$$ by $$V$$. Note that this implies $\pabc{S}{V}{p,T}=\pabc{S}{V}{p,\mu}=\pabc{S}{V}{n,T}={S\over V}\ ,$ where $$n=N/V$$ is the particle density.

6) Sixth, suppose we have a function $$\Phi(y,v)$$ and we write $d\Phi=x\,dy + u\,dv\ .$ That is, $x=\pabc{\Phi}{y}{v}\equiv \Phi\ns_y \qquad,\qquad u=\pabc{\Phi}{v}{y}\equiv \Phi\ns_v\ .$ Now we may write \begin{aligned} dx&=\Phi\ns_{yy}\,dy + \Phi\ns_{yv}\,dv \label{dxe} \\ du&=\Phi\ns_{vy}\,dy + \Phi\ns_{vv}\,dv\ .\label{due}\end{aligned} If we demand $$du=0$$, this yields $\pabc{x}{u}{v}={\Phi\ns_{yy}\over \Phi\ns_{vy}}\ . \label{pxuv}$ Note that $$\Phi\ns_{vy}=\Phi\ns_{yv}$$ . From the equation $$du=0$$ we also derive $\pabc{y}{v}{u}=-{\Phi\ns_{vv}\over \Phi\ns_{vy}}\ . \label{pyvu}$ Next, we use Equation [due] with $$du=0$$ to eliminate $$dy$$ in favor of $$dv$$, and then substitute into Equation [dxe]. This yields $\pabc{x}{v}{u}=\Phi\ns_{yv}-{\Phi\ns_{yy}\,\Phi\ns_{vv}\over \Phi\ns_{vy}}\ . \label{pxvu}$ Finally, Equation [due] with $$dv=0$$ yields $\pabc{y}{u}{v}={1\over \Phi\ns_{vy}}\ . \label{pyuv}$

Combining the results of eqns. [pxuv], [pyvu], [pxvu], and [pyuv], we have $\begin{split} {\pz(x,y)\over\pz(u,v)}&=\pabc{x}{u}{v}\pabc{y}{v}{u} - \pabc{x}{v}{u}\pabc{y}{u}{v}\\ &=\bigg({\Phi\ns_{yy}\over \Phi\ns_{vy}}\bigg)\bigg(-{\Phi\ns_{vv}\over \Phi\ns_{vy}}\bigg)- \bigg(\Phi\ns_{yv}-{\Phi\ns_{yy}\,\Phi\ns_{vv}\over \Phi\ns_{vy}}\bigg)\bigg({1\over \Phi\ns_{vy}}\bigg)\bvph=-1\ . \label{jacob} \end{split}$ Thus, if $$\Phi=E(S,V)$$, then $$(x,y)=(T,S)$$ and $$(u,v)=(-p,V)$$, we have ${\pz(T,S)\over\pz(-p,V)}=-1\ . \label{detTSpV}$

Nota bene: It is important to understand what other quantities are kept constant, otherwise we can run into trouble. For example, it would seem that Equation [jacob] would also yield ${\pz(\mu,N)\over\pz(p,V)}=1\ . \label{detmuNpV}$ But then we should have ${\pz(T,S)\over\pz(\mu,N)}={\pz(T,S)\over\pz(-p,V)}\cdot {\pz(-p,V)\over\pz(\mu,N)} = +1 \qquad\hbox{(WRONG!)}$ when according to Equation [jacob] it should be $$-1$$. What has gone wrong?

The problem is that we have not properly specified what else is being held constant. In Equation [detTSpV] it is $$N$$ (or $$\mu$$) which is being held constant, while in Equation [detmuNpV] it is $$S$$ (or $$T$$) which is being held constant. Therefore a naive application of the chain rule for determinants yields the wrong result, as we have seen.

Let’s be more careful. Applying the same derivation to $$dE=x\,dy + u\,dv + r\,ds$$ and holding $$s$$ constant, we conclude ${\pz(x,y,s)\over\pz(u,v,s)}=\pabc{x}{u}{v,s}\pabc{y}{v}{u,s}-\ \pabc{x}{v}{u,s}\pabc{y}{u}{v,s} = -1\ .$ Thus, if $dE= T\,dS + y\,dX + \mu\,dN\quad,$ where $$(y,X)=(-p,V)$$ or $$(H^\alpha,M^\alpha)$$ or $$(E^\alpha,P^\alpha)$$, the appropriate thermodynamic relations are \begin{aligned} {\pz(T,S,N)\over\pz(y,X,N)}&=-1 & {\pz(T,S,\mu)\over\pz(y,X,\mu)}&=-1 \nonumber \\ {\pz(\mu,N,X)\over\pz(T,S,X)}&=-1 & {\pz(\mu,N,y)\over\pz(T,S,y)}&=-1 \bvph \label{TSyXN} \\ {\pz(y,X,S)\over\pz(\mu,N,S)}&=-1 & {\pz(y,X,T)\over\pz(\mu,N,T)}&=-1 \nonumber\end{aligned} For example, ${\pz(T,S,N)\over\pz(-p,V,N)}={\pz(-p,V,S)\over\pz(\mu,N,S)}={\pz(\mu,N,V)\over\pz(T,S,V)}=-1$ and ${\pz(T,S,\mu)\over\pz(-p,V,\mu)}={\pz(-p,V,T)\over\pz(\mu,N,T)}={\pz(\mu,N,-p)\over\pz(T,S,-p)}=-1\ .$

If we are careful, then the results in eq. [TSyXN] can be quite handy, especially when used in conjunction with Equation [chain]. For example, we have $\pabc{S}{V}{T,N}={\pz(T,S,N)\over\pz(T,V,N)}=\stackrel{=\,1}{\overbrace ParseError: invalid DekiScript (click for details) Callstack: at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.16:_Appendix_III-_Useful_Mathematical_Relations), /content/body/p[14]/span[1], line 1, column 1  } \cdot{\pz(p,V,N)\over\pz(T,V,N)}=\pabc{p}{T}{V,N}\ ,$ which is one of the Maxwell relations derived from the exactness of $$dF(T,V,N)$$. Some other examples include \begin{aligned} \pabc{V}{S}{p,N}&={\pz(V,p,N)\over\pz(S,p,N)}=\stackrel{=\,1}{\overbrace ParseError: invalid DekiScript (click for details) Callstack: at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.16:_Appendix_III-_Useful_Mathematical_Relations), /content/body/p[14]/span[2], line 1, column 1  }\cdot{\pz(S,T,N)\over\pz(S,p,N)}=\pabc{T}{p}{S,N}\\ \pabc{S}{N}{T,p}&={\pz(S,T,p)\over\pz(N,T,p)}=\stackrel{=\,1}{\overbrace ParseError: invalid DekiScript (click for details) Callstack: at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.16:_Appendix_III-_Useful_Mathematical_Relations), /content/body/p[14]/span[3], line 1, column 1  } \cdot{\pz(\mu,N,p)\over\pz(N,T,p)}=-\pabc{\mu}{T}{p,N}\ ,\bvph\end{aligned} which are Maxwell relations deriving from $$d\CH(S,p,N)$$ and $$dG(T,p,N)$$, respectively. Note that due to the alternating nature of the determinant – it is antisymmetric under interchange of any two rows or columns – we have ${\pz(x,y,z)\over\pz(u,v,w)}=-{\pz (y,x,z)\over\pz (u,v,w)} = {\pz (y,x,z)\over\pz (w,v,u)} = \ldots \ .$

In general, it is usually advisable to eliminate $$S$$ from a Jacobian. If we have a Jacobian involving $$T$$, $$S$$, and $$N$$, we can write ${\pz(T,S,N)\over\pz(\cds,\cds,N)}={\pz(T,S,N)\over\pz(p,V,N)}\,{\pz(p,V,N)\over\pz(\cds,\cds,N)} ={\pz(p,V,N)\over\pz(\cds,\cds,N)}\ ,$ where each $$\cds$$ is a distinct arbitrary state variable other than $$N$$.

If our Jacobian involves the $$S$$, $$V$$, and $$N$$, we write ${\pz(S,V,N)\over\pz(\cds,\cds,N)}={\pz(S,V,N)\over\pz(T,V,N)}\cdot{\pz(T,V,N)\over\pz(\cds,\cds,N)}={C\ns_V\over T}\cdot {\pz(T,V,N)\over\pz(\cds,\cds,N)}\ .$

If our Jacobian involves the $$S$$, $$p$$, and $$N$$, we write ${\pz(S,p,N)\over\pz(\cds,\cds,N)}={\pz(S,p,N)\over\pz(T,p,N)}\cdot{\pz(T,p,N)\over\pz(\cds,\cds,N)}={C\ns_p\over T}\cdot {\pz(T,p,N)\over\pz(\cds,\cds,N)}\ .$

For example, \begin{aligned} \pabc{T}{p}{S,N}&={\pz(T,S,N)\over\pz(p,S,N)}=\stackrel{=\,1}{\overbrace ParseError: invalid DekiScript (click for details) Callstack: at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.16:_Appendix_III-_Useful_Mathematical_Relations), /content/body/p[18]/span, line 1, column 1  } \cdot{\pz(p,V,N)\over\pz(p,T,N)}\cdot {\pz(p,T,N)\over\pz(p,S,N)}={T\over C\ns_p}\pabc{V}{T}{p,N}\bvph\\ \pabc{V}{p}{S,N}&={\pz(V,S,N)\over\pz(p,S,N)}={\pz(V,S,N)\over\pz(V,T,N)}\cdot{\pz(V,T,N)\over\pz(p,T,N)}\cdot{\pz(p,T,N)\over\pz(p,S,N)} ={C\ns_V\over C\ns_p}\,\pabc{V}{p}{T,N}\ .\bvph\end{aligned} With $$\kappa\equiv -{1\over V}\,{\pz V\over\pz p}$$ the compressibility, we see that the second of these equations says $$\kappa\ns_T\,c\ns_V=\kappa\ns_S\,c\ns_p$$ , relating the isothermal and adiabatic compressibilities and the molar heat capacities at constant volume and constant pressure. This relation was previously established in Equation [cpcvktks]

2.16: Appendix III- Useful Mathematical Relations is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Daniel Arovas.