MIT 3.00 Fall 2002
154
c W.C Carter
°
Lecture 22
Mathematical Relations and Changing Variables
Last Time
Reaction Equilibria
Exact Differentials
Legendre Transformations
Lechatelier’s Principle
MIT 3.00 Fall 2002
155
c W.C Carter
°
Maxwell’s Relations
µ
df =
∂f
∂x
¶
µ
dx +
y
∂f
∂y
¶
dy
(22-1)
x
A property of a perfect differential is:
∂ 2f
∂ 2f
=
∂x∂y
∂y∂x
(22-2)
If Equation 22-2 is applied to dU = T dS − P dV :
µ
¶
µ
¶
∂ 2U
∂P
∂ 2U
∂T
=−
=
∂V ∂S
∂S V
∂S∂V
∂V S
(22-3)
This can be summarized in the following tables (you should be able to derive these tables
on your own):
Second Law
Formulation
dU =
T dS
−P dV
P
+ C
i=1 µi dNi
Internal Energy U
Independent
Conjugate
Variables
Variables
¡ ¢
S
T = ∂U
∂S
¡ ∂U ¢V,Ni
V
−P = ∂V S,Ni
³ ´
∂U
Ni
µi = ∂N
i
S,V,Nj 6=Ni
Maxwell
¡ ∂T ¢ Relations
¡ ∂P ¢
=
−
∂V
∂S
¡ ∂µi ¢ S,Ni ³ ∂P ´ V,Ni
= − ∂Ni
∂V S,Ni
³ ´S,V,Nj 6=Ni
¡ ∂µi ¢
∂P
= − ∂N
∂V S,Ni
i
S,V,Nj 6=Ni
MIT 3.00 Fall 2002
156
c W.C Carter
°
Enthalpy H
H = U + PV
H = G + TS
H = F + PV − TS
Second Law
Independent
Conjugate
Maxwell
Formulation
Variables
Variables
Relations
¡ ¢
¡ ∂T ¢
¡ ¢
dH =
T dS
S
T = ∂H
= ∂V
∂S V,Ni
∂P S,Ni ³
∂S
´ P,Ni
¡ ∂H ¢
¡ ∂µi ¢
∂V
+V dP
P
V = ∂P S,Ni
= ∂Ni
∂P S,Ni
³
´
³ ´S,P,Nj 6=Ni
¡ ∂µi ¢
PC
∂H
∂T
+ i=1 µi dNi
Ni
µi = ∂Ni
= ∂N
∂S P,Ni
i
S,P,Nj 6=Ni
S,P,Nj 6=Ni
Helmholtz Free Energy F
F = U − TS
F = H − PV − TS
F = G + PV
Second Law
Independent
Conjugate
Maxwell
Formulation
Variables
Variables
¡ ∂F ¢
¡ ∂S ¢ Relations
¡ ∂P ¢
dF =
−SdT
T
−S = ∂T V,Ni
=
∂V T,Ni ³ ∂T´ V,Ni
¡ ∂F ¢
¡ ∂µi ¢
∂P
−P dV
V
−P = ∂V T,Ni
= − ∂N
∂V T,Ni
³
´
³ i ´T,V,Nj 6=Ni
¡
¢
PC
∂µ
∂F
∂S
i
+ i=1 µi dNi
Ni
µi = ∂N
= − ∂N
∂T V,Ni
i
i
T,V,Nj 6=Ni
T,V,Nj 6=Ni
Gibbs Free Energy G
G = U − TS + PV
G = F + PV
G = H − TS
Second Law
Independent
Conjugate
Maxwell
Formulation
Variables
Variables
Relations
¡ ¢
¡ ∂S ¢
¡ ¢
dG =
−SdT
T
−S = ∂G
= − ∂V
∂T P,Ni
∂P T,Ni
³ ∂T
´ P,Ni
¡ ∂G ¢
¡ ∂µi ¢
∂S
+V dP
P
V = ∂V T,Ni
= − ∂Ni
∂T P,Ni
³
´
³ ´ T,P,Nj 6=Ni
¡ ∂µi ¢
PC
∂G
∂V
+ i=1 µi dNi
Ni
µi = ∂Ni
= ∂N
∂P T,Ni
i
T,P,Nj 6=Ni
T,P,Nj 6=Ni
Change of Variable
Sometimes it is more useful to be able to measure some quantity, such as
µ ¶
∂S
CP = T
= f1 (T, P )
∂T P
or
µ ¶
∂S
CV = T
= f2 (T, V )
∂T V
(22-4)
(22-5)
under different conditions than those indicated by their natural variables.
It would be easier to measure CV at constant P , T , so a change of variable would be useful.
MIT 3.00 Fall 2002
157
c W.C Carter
°
To change variables, a useful scheme using Jacobians can be employed:23
¯ ∂u ∂u ¯
¯
¯
∂(u, v)
∂y ¯
≡ det ¯¯ ∂x
∂v
∂v ¯
∂(x, y)
∂x
∂y
∂u ∂v ∂u ∂v
=
−
∂x ∂y ∂y ∂x
µ ¶ µ ¶
µ ¶ µ ¶
∂u
∂v
∂u
∂v
=
−
∂x y ∂y x
∂y x ∂x y
=
(22-6)
∂u(x, y) ∂v(x, y) ∂u(x, y) ∂v(x, y)
−
∂x
∂y
∂y
∂x
∂(u, v)
∂(v, u)
∂(v, u)
=−
=
∂(x, y)
∂(x, y)
∂(y, x)
µ ¶
∂(u, v)
∂u
=
∂(x, v)
∂x v
∂(u, v)
∂(u, v) ∂(r, s)
=
∂(x, y)
∂(r, s) ∂(x, y)
(22-7)
To see where the last rule comes from:
For example,
µ
CV = T
23
∂S
∂T
¶
=T
V
∂(S, V )
∂(T, V )
An alternative scheme is presented in Denbigh, Sec. 2.10(c)
(22-8)
MIT 3.00 Fall 2002
158
c W.C Carter
°
Using the Maxwell relation:
¡ ∂S ¢
∂P T
=−
¡ ∂V ¢
∂T
P
:
¡ ¢ 2
[ ∂V
]
P
¢
CP − CV = −T ¡∂T
∂V
∂P
T
(22-9)