User:EverettYou/Jacobian in Thermodynamics

Introduction to Jacobian

The Jacobian of a transform (x,y) → (u,v) is defined as a determinant of derivatives

$J(u,v)={\frac {\partial (u,v)}{\partial (x,y)}}={\begin{vmatrix}{\big (}{\frac {\partial u}{\partial x}}{\big )}_{y}&{\big (}{\frac {\partial v}{\partial x}}{\big )}_{y}\\{\big (}{\frac {\partial u}{\partial y}}{\big )}_{x}&{\big (}{\frac {\partial v}{\partial y}}{\big )}_{x}\end{vmatrix}}$.

When v = y, the Jacobian is reduced to a partial derivative

${\frac {\partial (u,y)}{\partial (x,y)}}=\left({\frac {\partial u}{\partial x}}\right)_{y}.$

This makes Jacobian useful in thermodynamics. All the partial derivatives in thermodynamics can be converted to Jacobians, and then treated systematically.

Basic Properties

The Jacobian has the following properties:

1. Permutation sign: $\partial (x,y)=-\partial (y,x)$, which implies $\partial (x,x)=0$.

2. Linearity: $\partial (x_{1}+x_{2},y)=\partial (x_{1},y)+\partial (x_{2},y)$.

3. Product rule: $\partial (xy,z)=x\,\partial (y,z)+y\,\partial (x,z)$.

4. Chain rule: ${\frac {\partial (x,y)}{\partial (u,v)}}{\frac {\partial (u,v)}{\partial (s,t)}}={\frac {\partial (x,y)}{\partial (s,t)}}$ or in short as ${\frac {\partial (u,v)}{\partial (u,v)}}=1$. This means one can really cancel out identical Jacobians in the fraction, like division of numbers.

Regroup Variables

When two Jacobians are multiplied together, variables can be regrouped by

$\partial (a,b)\partial (c,d)=\partial (a,c)\partial (b,d)-\partial (a,d)\partial (b,c)$.

To remember the signs, one can draw under-brackets between regrouped variables. If the brackets intersect, the regrouped term is positive; if the brackets do not intersect, the regrouped term has a minus sign in front.

Application to Thermodynamics

Maxwell Relation

The Maxwell relations are reduced to a single formula

$\partial (S,T)=\partial (V,p)$.

To remember, just match the intensive quantity with the intensive quantity and the extensive quantity with the extensive quantity. Maxwell relation is often used to reduce the entropy S to measurable quantities p, V, T.

Thermodynamic Potentials

The fundamental equations of the thermodynamic potentials can be reformulated in terms of Jacobians as

$\partial (U,x)=T\partial (S,x)-p\partial (V,x)$,

$\partial (H,x)=T\partial (S,x)+V\partial (p,x)$,

$\partial (F,x)=-S\partial (T,x)-p\partial (V,x)$,

$\partial (G,x)=-S\partial (T,x)+V\partial (p,x)$,

where x is an arbitrary thermodynamic quantity. These equations are used to express thermodynamic potentials in terms of first order quantities.

Material Properties

Relation between Jacobians by thermodynamic quantities.

Compressibility at constant temperature (isothermal) or constant entropy (adiabatic)

$\beta _{T}=-{\frac {1}{V}}{\frac {\partial (V,T)}{\partial (p,T)}}$, $\beta _{S}=-{\frac {1}{V}}{\frac {\partial (V,S)}{\partial (p,S)}}$.

Heat capacity at constant pressure or constant volume

$C_{p}=T{\frac {\partial (S,p)}{\partial (T,p)}}$, $C_{V}=T{\frac {\partial (S,V)}{\partial (T,V)}}$.

Coefficient of thermal expansion

$\alpha ={\frac {1}{V}}{\frac {\partial (V,p)}{\partial (T,p)}}$.

Their (multiplicative) relations are concluded in the diagram on the right. The arrow $A{\xrightarrow {c}}B$ denotes the relation $cA=B$. Following the arrows, all Jacobians can be expressed as a multiple of $\partial (T,p)$ (in terms of thermodynamic coefficients), such that their ratios can be evaluated straight forwardly.

The ratio of compressibilities and the ratio of heat capacities must be equal, such that the diagram commute. This defines the adiabatic index

${\frac {C_{p}}{C_{V}}}={\frac {\beta _{T}}{\beta _{S}}}=\gamma$.

The difference between compressibilities and the difference between heat capacities are given by

$\beta _{T}-\beta _{S}=VT{\frac {\alpha ^{2}}{C_{p}}}$,

$C_{p}-C_{V}=VT{\frac {\alpha ^{2}}{\beta _{T}}}$,

which can be proved by taking the common denominator and regrouping the variables on the numerator.

Derived Properties

Joule–Thomson coefficient

$\mu _{\text{JT}}={\frac {\partial (H,T)}{\partial (H,p)}}={\frac {V}{C_{p}}}(\alpha T-1).$