Introduction to Jacobian [ edit ] The Jacobian of a transform (x ,y ) → (u ,v ) is defined as a determinant of derivatives
J
(
u
,
v
)
=
∂
(
u
,
v
)
∂
(
x
,
y
)
=
|
(
∂
u
∂
x
)
y
(
∂
v
∂
x
)
y
(
∂
u
∂
y
)
x
(
∂
v
∂
y
)
x
|
{\displaystyle 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
∂
(
u
,
y
)
∂
(
x
,
y
)
=
(
∂
u
∂
x
)
y
.
{\displaystyle {\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 [ edit ] The Jacobian has the following properties:
1. Permutation sign:
∂
(
x
,
y
)
=
−
∂
(
y
,
x
)
{\displaystyle \partial (x,y)=-\partial (y,x)}
∂
(
x
,
x
)
=
0
{\displaystyle \partial (x,x)=0}
2. Linearity:
∂
(
x
1
+
x
2
,
y
)
=
∂
(
x
1
,
y
)
+
∂
(
x
2
,
y
)
{\displaystyle \partial (x_{1}+x_{2},y)=\partial (x_{1},y)+\partial (x_{2},y)}
3. Product rule:
∂
(
x
y
,
z
)
=
x
∂
(
y
,
z
)
+
y
∂
(
x
,
z
)
{\displaystyle \partial (xy,z)=x\,\partial (y,z)+y\,\partial (x,z)}
4. Chain rule:
∂
(
x
,
y
)
∂
(
u
,
v
)
∂
(
u
,
v
)
∂
(
s
,
t
)
=
∂
(
x
,
y
)
∂
(
s
,
t
)
{\displaystyle {\frac {\partial (x,y)}{\partial (u,v)}}{\frac {\partial (u,v)}{\partial (s,t)}}={\frac {\partial (x,y)}{\partial (s,t)}}}
∂
(
u
,
v
)
∂
(
u
,
v
)
=
1
{\displaystyle {\frac {\partial (u,v)}{\partial (u,v)}}=1}
Regroup Variables [ edit ] When two Jacobians are multiplied together, variables can be regrouped by
∂
(
a
,
b
)
∂
(
c
,
d
)
=
∂
(
a
,
c
)
∂
(
b
,
d
)
−
∂
(
a
,
d
)
∂
(
b
,
c
)
{\displaystyle \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 [ edit ] Maxwell Relation [ edit ] The Maxwell relations are reduced to a single formula
∂
(
S
,
T
)
=
∂
(
V
,
p
)
{\displaystyle \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 [ edit ] The fundamental equations of the thermodynamic potentials can be reformulated in terms of Jacobians as
∂
(
U
,
x
)
=
T
∂
(
S
,
x
)
−
p
∂
(
V
,
x
)
{\displaystyle \partial (U,x)=T\partial (S,x)-p\partial (V,x)}
∂
(
H
,
x
)
=
T
∂
(
S
,
x
)
+
V
∂
(
p
,
x
)
{\displaystyle \partial (H,x)=T\partial (S,x)+V\partial (p,x)}
∂
(
F
,
x
)
=
−
S
∂
(
T
,
x
)
−
p
∂
(
V
,
x
)
{\displaystyle \partial (F,x)=-S\partial (T,x)-p\partial (V,x)}
∂
(
G
,
x
)
=
−
S
∂
(
T
,
x
)
+
V
∂
(
p
,
x
)
{\displaystyle \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 [ edit ] Relation between Jacobians by thermodynamic quantities. Compressibility at constant temperature (isothermal ) or constant entropy (adiabatic )
β
T
=
−
1
V
∂
(
V
,
T
)
∂
(
p
,
T
)
{\displaystyle \beta _{T}=-{\frac {1}{V}}{\frac {\partial (V,T)}{\partial (p,T)}}}
β
S
=
−
1
V
∂
(
V
,
S
)
∂
(
p
,
S
)
{\displaystyle \beta _{S}=-{\frac {1}{V}}{\frac {\partial (V,S)}{\partial (p,S)}}}
Heat capacity at constant pressure or constant volume
C
p
=
T
∂
(
S
,
p
)
∂
(
T
,
p
)
{\displaystyle C_{p}=T{\frac {\partial (S,p)}{\partial (T,p)}}}
C
V
=
T
∂
(
S
,
V
)
∂
(
T
,
V
)
{\displaystyle C_{V}=T{\frac {\partial (S,V)}{\partial (T,V)}}}
Coefficient of thermal expansion
α
=
1
V
∂
(
V
,
p
)
∂
(
T
,
p
)
{\displaystyle \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
→
c
B
{\displaystyle A{\xrightarrow {c}}B}
c
A
=
B
{\displaystyle cA=B}
∂
(
T
,
p
)
{\displaystyle \partial (T,p)}
The ratio of compressibilities and the ratio of heat capacities must be equal, such that the diagram commute. This defines the adiabatic index
C
p
C
V
=
β
T
β
S
=
γ
{\displaystyle {\frac {C_{p}}{C_{V}}}={\frac {\beta _{T}}{\beta _{S}}}=\gamma }
The difference between compressibilities and the difference between heat capacities are given by
β
T
−
β
S
=
V
T
α
2
C
p
{\displaystyle \beta _{T}-\beta _{S}=VT{\frac {\alpha ^{2}}{C_{p}}}}
C
p
−
C
V
=
V
T
α
2
β
T
{\displaystyle 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 [ edit ] Joule–Thomson coefficient
μ
JT
=
∂
(
H
,
T
)
∂
(
H
,
p
)
=
V
C
p
(
α
T
−
1
)
.
{\displaystyle \mu _{\text{JT}}={\frac {\partial (H,T)}{\partial (H,p)}}={\frac {V}{C_{p}}}(\alpha T-1).}
