Physical Chemistry
Lecture 20
Phase Equilibrium
Multi-phase system
A system containing
more than one phase
has the possibility of
exchange of material
between phases
Exchange at equilibrium
must not affect state
variables
Example: liquid-vapor
equilibrium
General effects of change of
material in a biphasic system
A transfer of material into a phase may
change its internal energy
dU α
= TdS α
− PdV α
+ µ α dnα
µα is the chemical potential in the α phase
nα is the number of moles of material in the α
phase
A similar equation could be written for the β
phase
The chemical potential
A measure of the change of energy of a
phase by transfer of material into (or
out of) the phase
µα
 ∂U 
=  α
 ∂n  S ,V
 ∂H 
=  α
 ∂n  S , P
 ∂A 
=  α
 ∂n T ,V
 ∂G 
=  α
 ∂n T , P
A function of state variables
Example process at constant T and P:
(∆G )αT , P
=
nα
2
α
α
µ
dn
∫
n1α
Equilibrium exchange of
material in a biphasic system
In a biphasic system, change has two components
dG = dGα + dG β
Under constant-P, constant-T conditions
dG = µ α dnα + µ β dn β
If the system is closed, then exchange is only between
the two phases
dG = µ α dnα − µ β dnα = (µ α − µ β )dnα
At equilibrium, the Gibbs energy does not change
dG = 0 = (µ α − µ β )dnα
This gives the equilibrium requirement: the chemical
potentials in the two phases must be equal
Criteria for phase equilibrium
under various conditions
At constant V and T
(dA)T ,V
= 0
⇒
µα
= µβ
At constant P and T
(dG )T ,P
= 0
⇒
µα
= µβ
At constant S and V
(dU )S ,V
= 0
⇒
µα
= µβ
The general requirement is the equality of the
chemical potentials in the two phases
Variation of the chemical
potential
The chemical potential depends on other state
variables, e.g. T and P
µ = µ (T , P )
Differential behavior like other state variables
dµ
 ∂µ 
= 
 dT
 ∂T  P
 ∂µ 
+   dP
 ∂P T
Must evaluate derivatives to find out how
chemical potential changes as other variables
change
Derivatives of the chemical
potential
Temperature derivative
 ∂µ 


∂
T

P
 ∂  ∂G  

= 

  =
 ∂T  ∂n T , P  P
 ∂S 
= −  
= −
 ∂n T , P
 ∂  ∂G  
 
 
 ∂n  ∂T  P T , P
Sm
Pressure derivative
 ∂µ 
 
 ∂P T
 ∂  ∂G  
 ∂  ∂G  


=  
  =  
 
 ∂n  ∂P T T , P
 ∂P  ∂n T , P T
 ∂V 
= 
= Vm

 ∂n T , P
Differential of the chemical
potential in a phase
The change in the potential as a
function of pressure and temperature
dµ α
= Vmα dP − S mα dT
Integration of this equation gives the
change in the chemical potential in a
process that changes T and P
∆µ α
T2
= − ∫ S mα dT
T1
+
P2
α
V
∫ m dP
P1
Variation of phase equilibrium
A requirement of phase equilibrium: chemical
potentials of two phases must change in
tandem when, e.g., temperature and pressure
are changed
dµ
α
= dµ
β
This is the equation of all points where
equilibrium exists -- a coexistence curve
Clapeyron’s equation
Consider, when T is changed, how P is
changed so that phase equilibrium is
maintained
α
β
dµ
= dµ
− S mα dT
+ Vmα dP = − S mβ dT
+ Vmβ dP
The differential relationship between T
and P is known as Clapeyron’s equation
dP
dT
=
( S mβ − S mα )
(Vmβ − Vmα )
=
∆S m , transition
∆Vm.transition
Clapeyron’s equation applied
The equilibrium coexistence curve of a material gives
thermodynamic information
∆S m , transition
dP
=
dT
∆Vm.transition
Clausius-Clapeyron equation
Applicable to equilibrium with the gas
phase (Vm, gas >> Vm, liquid or Vm. solid)
dP
dT
∆H m , transition
≅
T Vm , gas
If the gas phase is ideal,
dP
dT
=
∆H m , transition P
RT
2
⇒
Integrated form
 P2 
ln  =
 P1 
1 dP
P dT
=
∆H m , transition
1 2 ∆H m , transition
dT
2
∫
R T1
T
T
RT 2
Clausius-Clapeyron plot
Plot of ln(P) versus 1/T
Slope at any point gives ∆Hm for the phase transition at
that temperature
Carbon dioxide phase
behavior
ln(P/torr)
12
8
4
0
0.003
0.004
0.005
0.006
0.007
0.008
1/T (1/K)
Liquid and solid behavior plotted according to the
Clausius-Clapeyron equation [ln(P/torr) vs. 1/T]
Determine ∆vHm and ∆sHm for CO2 from the slopes of
the lines
Approximate ClausiusClapeyron equation
Sometimes over a restricted range, the
Clausius-Clapeyron equation is
integrated, with the assumption that
∆vHm is independent of temperature
 P2 
∆v H m  1 1 
 − 
ln  = −
R  T2 T1 
 P1 
Only approximate
Use cautiously
∆ vHm (joule/mole)
Enthalpies of vaporization of
alkanes
25000
20000
15000
10000
260
280
300
320
340
Temperature (K)
n-Butane
Isobutane
Propane
The enthalpies of vaporization are not
independent of temperature
Antoine equation
Empirical equation for vapor pressure
B
 P 
ln θ  = −
C +T
P 
Implies a particular form for ∆vHθ(T)
BR
θ
∆v H
=
2
C 
 + 1
T

Standard enthalpies of
transition
∆fHm (kJ/mole)
Tv (K)
∆vHm kJ/mole)
Substance
Tf (K)
H2
13.95
0.12
20.38
0.90
O2
54.39
0.44
90.18
6.82
N2
63.14
0.72
77.33
5.58
Sublimes at 194.5 K with ∆sHm ≅ 25 kJ/mole
CO2
NH3
195.39
5.653
239.72
23.33
CH4
90.67
0.941
111.66
8.18
C2H6
89.88
2.86
184.52
14.72
CH3OH
175.4
3.17
337.9
35.27
C2H5OH
156
5.02
351.7
38.58
H2O
273.15
6.01
373.15
40.66
Al
933.52
2.56
Summary
A biphasic system has the possibility of exchange of
material between the phases
If the exchange occurs at equilibrium, a criterion is
 µα = µβ
Chemical potential depends on system variables, like
T and P
Clapeyron’s equation describes the relation of
temperature and pressure at phase equilibrium
If one phase has a much larger molar volume than
the other, the Clausius-Clapeyron equation gives the
criterion for phase equilibrium