Physical Chemistry
Lecture 18
Maxwell Relations
Spontaneity under arbitrary
conditions
Constant U and V
(∆S )U ,V
≥ 0
Constant T and V
(∆A)T ,V
≤ 0
Constant T and P
(∆G )T , P
≤ 0
Must calculate changes to make
predictions
Free-energy changes
Helmholtz free energy, A
dA = − SdT
− PdV
Gibbs free energy, G
dG = − SdT
+ VdP
Must know how entropy depends on
temperature to evaluate free-energy
 ∂S 
 ∂S 
change
dS = 
 dT + 
 dV
dS
 ∂T V
 ∂S 
= 
 dT
 ∂T  P
 ∂V T
 ∂S 
+   dP
 ∂P T
Evaluation by integration
Evaluate changes of a function such as
A or G in a process requires evaluating
an integral of a derivative
T2
∆A = − ∫ S (T )dT
−
T1
T2
∆G = − ∫ S (T )dT
T1
V2
∫ P(V )dV
V1
+
P2
∫ V ( P)dP
P1
Maxwell relation from
Helmholtz energy
dA = − SdT − PdV
By definition
Equality of crossed second partial
derivatives of the state function, A
 ∂

 ∂T
 ∂

 ∂T
 ∂A  
 

 ∂V T V
(− P )
V
 ∂S 


 ∂V T
 ∂  ∂A  
= 
 

 ∂V  ∂T V T
 ∂
(− S )
= 
T
 ∂V
 ∂P 
=  
 ∂T V
Maxwell relation from Gibbs
energy
dG = − SdT + VdP
By definition
Equality of crossed second partial
derivatives of the state function, G
 ∂
 ∂T

 ∂

 ∂T
 ∂G  

 
 ∂P T  P
(V )
P
 ∂S 
 
 ∂P T
 ∂  ∂G  
=  
 
 ∂P  ∂T  P T
 ∂

=  (− S )
 ∂P
T
 ∂V 
= −

 ∂T  P
Evaluating entropy at a
temperature
By definition
 ∂S 
S (T , P) = S (Tref , P) + ∫ 
 dT
∂T  P
Tref 
T
T
= S (Tref , P) +
∫
Tref
C P (T )
dT
T
Determined by the way the heat
capacity changes
Must specify entropy at the reference
temperature (Third Law)
Evaluation by numerical
integration
∆Si
 C P (Ti +1 ) C P (Ti ) 
+


Ti +1
Ti 
(Ti +1 − Ti )
= 


2




Summation gives
change in entropy with
temperature
Can also use trapezoid
to approximate area
Heat Capacity of Platinum
0.1
0.09
0.08
0.07
CP (joule gm -1 K-1)
Treat integral as a sum
over small intervals
Area of rectangle
0.06
0.05
0.04
0.03
0.02
0.01
0
0
10
20
30
40
50
T (K)
60
70
80
90
100
Evaluating entropy at a pressure
By definition of the change in a state
P
function
 ∂S 
S (T , P) = S (T , Pref ) +
∫  ∂P 
Pref
dP
T
 ∂V 
∫P  ∂T  P dT
ref
P
= S (T , Pref ) −
Evaluation by change of volume
 ∂S 
S (T , V ) = S (T , Vref ) + ∫ 
 dV
∂V T
Vref 
V
 ∂P 
= S (T , Vref ) + ∫   dT
∂T V
Vref 
V
Standard molar entropies
at 298.15 K
Evaluate entropies by numerical
integration of heat capacities
Solid
Ag
Sθ (J/K)
42.55
Liquid
Br2
Sθ (J/K)
Gas
Sθ (J/K)
152.2
H2
130.7
AgCl
96.2
H2O
69.9
N2
191.6
Graphite
5.74
Hg
76.0
O2
205.1
Diamond
2.377
CH3OH
126.8
CO2
213.7
I2
116.1
C2H5OH
160.7
Ar
154.9
173.
Xe
169.7
Hg2Cl2
196.
C6H6
Internal energy derivatives
For a reversible process
dU = TdS − PdV
Replace entropy differential by definition
dU
 ∂S 
= T 
 dT
 ∂T V

 ∂S 
+ 
 dV  − PdV
 ∂V T 
 ∂S 
= T
 dT
 ∂T V
  ∂S 

+ T 
 − P dV
  ∂V T

 ∂S 
= T
 dT
 ∂T V
  ∂P 

+ T 
 − P dV
  ∂T V

The last substitution uses a Maxwell relation
Indentifying partial derivatives
Compare equation to standard form
 ∂U 


 ∂T V
 ∂S 
= T 
 ∂T V
 ∂U 


 ∂V T
⇒
 ∂P 
= T 
 ∂T V
 ∂S 
 
 ∂T V
=
CV
T
− P
Gives evaluable forms for derivatives
Other forms of work
Stretching an elastic network
dwelastic
=
fdl
Moving charge against an electric potential
dwelecrtic
= VdQ
Magnetizing a material
dwmagnetic
= BdM
Creating a surface
dwsurface
= γdA
Summary
Free-energy changes specify
spontaneity


(dA)T,V ≤ 0
(dG)T,P ≤ 0
Must calculate the changes, ∆A and/or
∆G, to determine spontaneity of a
process
Maxwell relations provide connection
between derivatives to allow proper
integration