MSEG 803
Equilibria in Material Systems
5: Maxwell Relations & Stability
Prof. Juejun (JJ) Hu
hujuejun@udel.edu
Second order derivatives

Heat capacity
 2G
 Q 
 S 
CP  
 T 
  T
2
dT

T

T

P

P

Coefficient of thermal expansion
1  V 
1  2G
 
  
V  T  P V T P

2 F
 S 
CV  T 
  T
2

T

T

V
3  L 
Isotropy:   3 L  

L  T  P
Isothermal compressibility
1  V 
1  2G
T   
  
V  P T
V P 2
Maxwell relations
2

V

G


 S 
V  

 


T

T

P

P

P

T




LHS: volume change as a function of
temperature (thermal expansion)
RHS: entropy change as a function of pressure
Connects two seemingly unrelated quantities!
General form
 conj (Y ) 
 X 




 Y conj ( X )  conj ( X ) Y
Maxwell relations in single component
simple systems

Energy representation dU  TdS  PdV   dN
 2U
 T 
 P 

 




V

S

V

S

S , N

V , N
 2U
 T 
  


 


N

S

N

S

 S ,V

V , N
 2U
 P 
  


 


N

P

N

V

 S ,V

S , N
Note the sign difference
for terms involving P in
the U representation
Second order derivatives


Of all second order derivatives, only 3 can be
independent, and any given derivative can be
expressed in terms of an arbitrarily chosen set
of 3 basic derivatives
The conventional choice:
 2G
 S 
CP  T 
 T
T 2
 T  P
1  V 
1  2G
 
  
V  T  P V T P
1  V 
1  2G
T   
  
V  P T
V P 2
3 physical observables in the
Gibbs potential representation
Procedures for reducing derivatives





If the derivative contains any potentials, bring them to the
numerator and eliminate using the differential form of
fundamental equations
If the derivative contains the chemical potential, bring it to the
numerator and eliminate by means of the Gibbs-Duhem
relation: d = -sdT + udP
If the derivative contains the entropy, bring it to the numerator.
If one of the Maxwell relations now eliminates the entropy,
invoke it. If the Maxwell relations do not eliminate the entropy
take the deriative of S with respect to T. The numerator will
then be expressible as one of the specific heats (cv or cP)
Bring the volume to the numerator. The remaining derivative
will be expressible in terms of  and T
cv can be eliminated by the equation: cv = cP - Tv2/T
Some useful relations
 X 
 Y 

  1  X 

Z
 Y Z
 X 
 X 

   W 
Z
 Y Z 
 X 
 Z 

    Y 

X
 Y  Z
 Y 


 W  Z
 Z 


 X Y
Connecting CV and CP
 2G
 S 
CP  T 
  T
T 2
 T  P
2 F
 S 
CV  T 
  T
T 2
 T V
 S 
 S 
 S   P  
CV  T 
  T  
 
 
 

T

P

T
 T V





V 
P
T

 S 
 V   V 
 T  
 
   T 
P
 T  P  T  P 
 V  

 

P

T 
 S 
V 2 
TV  2
 T  
  CP 
 
T 
T
 T  P
Molar heat capacity: cV  cP 
Tv 2
T
Joule-Thomson (throttling) process
 T 
 H 

    P 

T
 P  H
 H 


 T  P
Isenthalpic curves
 T  S 
   S  
  
  V  T  
 

P

T



P 
T

 
  V 

 T  
  V  CP
  T  P

 T   1  V CP
Inversion temperature:
T  1 
V
0
CP
 T 
 0
 P  H
Inversion point: 
P > Pinv, J-T process leads to heating
P < Pinv, J-T process leads to cooling
Magnetic systems



Quasi-static magnetic work:
HdM
M is an unconstrainable
parameter!
Fundamental equation:
d   SdT   dN 
 V
i x, y , z
M 

1
V
1
V
Unmagnetized
(H = 0)
ii
Magnetized
(H > 0)
d ii  MdH
1  V 
 V 
 V   H  








 
 
 T  M V  T  H  H T  T  M 
 V 
 V   M 
 


 



H

T

T



H


T
H

thermal expansion
1
 M  




 
H
V
 H T 
 V   M 

 

 H T  T  H
1  M 
  xx 
 
Magnetostriction and piezomagnetic effect: 


V   xx  H
 H  M
Magnetic refrigeration
S
H=0
H = H1
0
T
Fundamental equation: d   SdT   dN 
 S 
 M 






H

T

T 
H
 V
i x, y , z
ii
d ii  MdH
Note that here H denotes the field not enthalpy
Optomechanical force in science fictions: solar sail
Count Dooku’s solar sailer:
Star Wars Episode II: Attack
of the Clones
Optomechanical force
x

Photons confined in a resonant cavity (bouncing back
and forth between two mirrors)
Fundamental equation: dN 
 dN  Fdx
Optomechanical force exerted by photons on the movable mirror:
 F 
 





 N  x  x  N
FN 
 
F  N 

 x  N
The sign of , CV and T

 can either be positive or negative



Positive : most materials
Negative : liquid water (< 4 °C), cubic zirconium tungstate,
quartz (over certain T range)
CV and T are positive in stable TD systems
Q
If CV < 0, thermal
fluctuation will be amplified
leading to instability
dV
If T < 0, volume
fluctuation will be amplified
leading to instability
Stability criteria
 2 S   1 T  


2
U

U

V , N
Molar
entropy s

Ext.
var. x
x0 - Dx
x0
1  T 
1


0

2 
2
T  U V , N
NT cV
2 F
1
 P 



0


2
V
 V T , N V  T
x0 + Dx
1
1
S
(
x

D
x
)

S ( x0  D x)  S ( x0 )
Stability:
0
2
2

cV  0
T  0
General
criterion:

2S
0
2
x
2
2S 2S  2S 


 0
U 2 V 2  U V 
Stable thermodynamic function
Molar
entropy s
E
G
A to B, E to F: stable
F
D
 2 S x 2  0
C to D: locally unstable
 2 S x 2  0
B
A
C
Ext.
var. x
B to C, D to E: locally stable,
globally unstable
 2 S x 2  0
 The stable thermodynamic function S is the envelope of
tangents everywhere above the underlying function S
 The line BGE corresponds to inhomogeneous mixtures of
the two phases B and E: 1st order phase transition
Le Chatelier-Braun principle


If a chemical system at equilibrium experiences
a change in concentration, temperature, volume,
or partial pressure, then the equilibrium shifts to
counteract the imposed change and a new
equilibrium is established.
Indirectly induced secondary processes also act
to attenuate the initial perturbation