The Thermodynamic Potentials
Four Fundamental Thermodynamic Potentials
Equilibrium
dU = TdS - pdV
U  U ( S ,V );
dU  0
fixed V,S
dH = TdS + Vdp
H  H ( S , P);
dH  0
fixed S,P
dG = Vdp - SdT
G  G( P, T );
dG  0
fixed P,T
dA = -pdV - SdT
A  A(V , T );
dA  0
fixed T,V
The appropriate thermodynamic potential
to use is determined by the constraints
imposed on the system. For example,
since entropy is hard to control (adiabatic
conditions are difficult to impose) G and A
are more useful. Also in the case of solids
p is a lot easier to control than V so G is
the most useful of all potentials for solids.
Our discussion of these thermodynamic potentials has considered only
“closed” (fixed size and composition) systems to this point. In this case
two independent variables uniquely defines the state of the system.
For example for a system at constant P and T the condition, dG = 0 defines
equilibrium, i.e., equilibrium is attained when the Gibbs potential or Gibbs Free
Energy reaches a minimum value.
If the composition of the system is variable in that the number of moles of the
various species present changes (e.g., as a consequence of a chemical reaction)
then minimization of G at fixed P and T occurs when the system has a unique
composition.
For example, for a system containing CO, CO2, H2 and H2O at fixed P and T,
minimization of G occurs when the following reaction reaches equilibrium.
CO  H 2O  CO2  H 2
Since G is an EXTENSIVE property, for multi-component or open system
it is necessary that the number of moles of each component be specified.
i.e.,
G  G T , P, n1 , n2 , n3 ,...ni 
 G 
dni



i 1  ni T , P , n
j ...
k
Then
 G 
 G 
 G 
 G 
dG  
dT  
dP  
dn1  
dn2  ...




 T  P , n1 , n2 ...
 P T , n1 , n2 ...
 n1  P ,T , n2 , n3 ...
 n2  P ,T , n1 , n3 ...
If the number of moles of each of the individual species remain fixed we
know that
dG  SdT  VdP
 G 
 S ;


 T  P , n1 , n2 ...
 G 
V


 P T , n1 , n2 ...
 G 
dG   SdT  VdP   
dni

i 1  ni T , P , n
j ...
k
 G 
 i  Gi
Chemical potential: The quantity 

 ni  T ,P ,n j ...
is called the chemical potential of component i. It correspond to the rate of
change of G with ni when the component i is added to the system at fixed P,T
and number of moles of all other species.
k
dG  SdT  VdP   i dni
i 1
One can add the same open system term, i dni for the other thermodynamic
potentials, i.e., U, H and A.
The chemical potential is the partial molal Gibbs Free Energy (or U,H, A) of
component i. Similar equations can be written for other extensive variables, e.g.,
 V 
 V 
dV  
dn

dn2  ...



1
 n1  P ,T , n2 , n3 ...
 n2  P ,T , n1 , n3 ...
dV  V1dn1  V2 dn2  ...
Physically this corresponds to how the volume in the system changes upon
addition of 1 mole of component ni at fixed P,T and mole numbers of other
components.
Maxwell relations: These mathematical relations are used to connect
experimentally measurable quantities to those that are not easily
accessible
Consider the relation for the Gibbs Free Energy:
dG  SdT  VdP
at fixed T
 G 

 V
 P T
at fixed P
 G 

  S
 T  P
Now take the derivative of these quantities at fixed P and T respectively,
   G  
 V 



 T  P  
T  P  T  P
 
   G  
 S 


 
 P  T  
 P T
 P T
 
   G  
 V 



 T  P  

T



P
T P

   G  
 S 


 
 P  T  


 P T
P T

If we compare the LHS of these equations, they must be equal since G is a state
function and an exact differential and the order of differentiation is inconsequential,
   G  
   G  
 T  P     P  T  
T  P  
 P T
 
So, the RHS of each of these equations must be equal,
 S 
 V 
   

 P T  T  P
Similarly we can develop a Maxwell relation from each of the other three
potentials:
A
 S 
 V 
   


P

T
 T 
P
B
 S 
 P 

 

 V T  T V
C
 T 
 V 





 P  S  S  P
D
 T 
 P 




 
 V  S
 S V
Let’s see how these Maxwell relations ca be useful. Consider the following
for the entropy.
S  S T , V 
 S 
 S 
dS  
dT



 dV
 T V
 V T
Using the definition of the constant volume heat capacity and the definition of
entropy for a reversible process
q 
 dU 
CV  




dT
dT

V 
V
TdS   qrev  dU  ncV dT
dS 
 qrev
T
Dividing by dT, the entropy change with temperature at fixed P is
ncV
 S 



T
 T V
Then,
 S 
 S 
dS  
dT



 dV
 T V
 V T
dS 
ncV
 S 
dT  
 dV
T
 V T
For the entropy change with volume at fixed T we can use the Maxwell relation B
 S 
 P 






V

T

T 
V
dS 
ncV
 P 
dT  
 dV
T
 T V
Now from the ideal gas law, PV = nRT
nR
 P 

 
 T V V
dS 
ncV
nR
dT 
dV
T
V
Integrating between states 1 and 2,
 T2
S2  S1  ncV ln 
 T1

 V2 
  nR ln  

 V1 
This equation can be used to evaluate the entropy change at fixed T,
problem 4.1.
Some important bits of information
For a mechanically isolated system kept at constant temperature and volume
the A = A(V, T) never increases. Equilibrium is determined by the state of
minimum A and defined by the condition, dA = 0.
For a mechanically isolated system kept at constant temperature and pressure
the G = G(p, T) never increases. Equilibrium is determined by the state of
minimum G and defined by the condition, dG = 0.
Consider a system maintained at constant p. Then
G   SdT
T2
T2
T1
T1
S   S T dT   C p T d ln T
T2
S T2   S T1    C p T d ln T
T1
T2

 T2


G  G T2   G T1    T2  T1 S T1    dT  C p T d ln T 
T

T1

 1

Temperature dependence of H, S, and G
Consider a phase undergoing a change in temp @ const P
 q   dH 
Cp  
 

 dT  p  dT  p
dS 
q
T
and

C p dT
T
 C p d ln T
dG  Vdp  SdT
@ Const. P
dG   SdT
H   C p ( t )dT
S   C p (T )d lnT
Temperature dependence of H, S, and G
T2
G    S(T )dT
T1
S   C p (T ) d ln T
T2
S (T2 )  S (T1 )   C p (T )d ln T
T1
T2

 T2

G  G(T2 )  G(T1 )  T2  T1 S (T1 )   dT  C p (T )d ln T 
T


T1
1

T2


G    T  S( T1 )   SdT 


T1
Temperature Dependence of the Heat Capacity
Cp
Dulong and Petit value
1
CV
3R
T (K)
Contributions to Specific Heat
1.
2.
3.
4.
5.
6.
Translational motion of free electrons ~ T1
Lattice vibrations ~ T3
Internal vib. within a molecule
Rotation of molecules
Excitation of upper energy levels
Anomalous effects
Temp. dep. of H, S, and G
H
S
S0 ≡ 0 pure elemental
solids, Third Law
T
H0
298K
T
ref. state for H is arbitrarily
set@ H(298) = 0 and P = 1 atm
for elemental substances
H
G  H  TS
H
G
G0
slope = Cp
T
T
TS
G
G
slope = -S
Thermodynamic Description of Phase Transitions
Gs  H s  TSs
1. Component Solidification
G  
Gl  Hl  TSl
LT
Tm
G
G(T )  Gs (T )  Gl (T )  H  TS
@ T= Tm
Gsolid
Gl =Gs
dGl = dGs ; G = 0
Gliquid
T*
Tm
T
ΔS =
H
Tm

L
Tm
Where L is the enthalpy change
of the transition or the heat of
fusion (latent heat).
For a small undercooling to say T*
ΔH and ΔS are constant (zeroth order approx.)
ΔG  H  TS  H  T (
H
)
Tm
T*
L
ΔG  H(1 
)
T
Tm
Tm
Where ΔT = T m – T*
* Note that L will be negative
The location of the transition temp Tm will change with pressure
dGl  Vl dp  Sl dT
dGs  Vs dp  Ss dT
dGl  dGs
dp Sl  S s S


dT Vl  Vs V
dp
L

dT Tm V
L
S 
Tm
Clapeyron equation
For a small change in melting point ΔT, we
can assume that ΔS & ΔH are constant so
 TV 
T  
P
 H 
The Clapeyron eq. governs the vapor pressure in any first order transition.
Melting or vaporization transitions are called first-order transitions
(Ehrenfest scheme) because there is a discontinuity in entropy, volume etc
which are the 1st partial derivatives of G with respect to Xi i.e.,
 G 
 G 

  S ; 
 V
 T  p
 p T
S  0
V  0
There are phase transition for which ΔS = 0 and ΔV = 0 i.e., the first
derivatives of G are continuous. Such a transition is not of first-order.
According to Ehrenfest an nth order transition if at the transition point.
 nG1  nG2

n
T
T n
Whereas all lower derivatives are equal.
There are only two transitions known to fit this scheme
gas – liquid
2nd order trans. in superconductivity
Notable exceptions are;
Curie pt. trans in ferromag.
Order-disorder trans. in binary alloys
λ – transition in liquid helium