1
Phase Transitions
Introduction
Phase – substance in a form of matter which is homogeneous throughout both in
chemical composition and physical state.
Transition – a change from one state to another at a characteristic temperature and a
given pressure
According to the 2nd law, free energy decreases for a spontaneous process.
Thus for an arbitrary phase transition   ,     or        0
Consider the chemical potentials of the different forms of water.
Below 0 C, the chemical potential of the solid must be lower than the chemical
potential of the liquid s  l .
At 0 C, the chemical potential of the solid is equal to the chemical potential of
the liquid s  l .
Above 0 C, the chemical potential of the solid must be higher than the chemical
potential of the liquid s  l .
Temperature Dependence of Free Energy
Recall the Gibbsian relationship for Gibbs free energy (which is true molar quantities also)
dG  SdT  V dp  dG  d  SdT  V dp
Considering only the temperature change yields
d
 S
dT
That is, the slope of a  vs. T is the negative of the molar entropy.

solid
liquid
Tm
T
Note the transition temperature occurs when s  l .
2
Pressure Dependence of Free Energy
A change in the pressure will cause a change in the free energy of a material.
dG  d  SdT  V dp 
d
V
dp
Since the molar volume (inverse of the molar density) is always positive, as pressure
increases, the free energy increases.
The chemical potential change with pressure is greater with liquids than solids since
the molar volume of a solid is usually smaller than the molar volume of a liquid. The
greatest change in chemical potential with pressure would be for a gas.
Since the changes in chemical potential for each phase are different, the temperatures
where the chemical potential for different phases are equal will be different when the
pressure changes. Thus melting point and boiling points of substances have a
dependence on the external pressure.

solid(P2 )
solid(P1 )
liquid (P2)
liquid (P 1)
Tm (P1)
Tm (P2)
T
3
Clapeyron Equation
Phase diagrams are give the state of a substance at a given temperature and pressure.
Since the phase diagram is a plot of pressure and temperature, the boundaries between
the phases are specific pressure-temperature relationships for a particular substance.
The boundaries between phases are described with the Clapeyron equation.
     S dT  V dp  S dT  V dp
S
S dT  S dT  V dp  V dp 





 S dT  V  V dp  S dT  V dp
dp S

dT V
Above is the most fundamental version of the Clapeyron equation.
We can the Clapeyron equation into another form by remembering that during the
phase transition, the free energy is unchanged; therefore, G    0
From the Gibbs-Helmholtz equation we have, H  TS  S 
H
T
Thus the boundaries between the phases (i.e. the pressure dependence of the phase
transition temperatures) depend on the molar enthalpy of the phase transition.
dp
H

dT TV
We can follow-through by integrating the equation (assuming that enthalpy of the
phase transition and the volume are temperature independent). [A reasonable
assumption for pressure change of a few bars.
dp 
H 1
dT 
V T
p2
 dp 
p1
H 2 1
dT 
V T1 T
T
 p2  p1  
H  T2 
ln  
V  T1 
Now we will use a clever algebra trick and the Maclaurin series expansion for
1
ln 1  x   x  x 2   x .
2
H  T2  T1 
p 2  p1 
V
T1
This equation is a useful relationship for the description of a solid – solid or the
solid – liquid boundary.
4
Clausius-Clapeyron Equation
The boundary between the liquid and gaseous phases deserves special attention
primarily because of the large difference between the molar volume of a gas and the
molar volume of a liquid.
We will start with the second version of the Clapeyron equation that we derived.
Hg  Hl
H vap
dp H



dT TV T Vg  Vg
T Vg  Vg




The molar volume of a liquid is very small compared to the molar volume of a gas;
therefore, we will assume that the change in the molar volume is the molar volume of
the gas.
H vap
H vap
dp


dT T Vg  Vl
T Vg


Substituting the ideal gas equation yields,
H vap
p H vap
dp H vap



dT T Vg T  RT p 
RT 2
Now the equation will be integrated, assuming that the enthalpy of vaporization is
temperature independent.
dp H vap
p R
p1
p2
 p 2  H vap  1  1  
1
dT

ln
  
 
 T2
R  T2  T1  
 p1 
T1
H vap  1 1 
p 
ln  2   
  
p
R
 1
 T2 T1 
T2
The differential form of the Clausius – Clapeyron can be put into an alternative form
with small amount of clever manipulation. First consider the following equalities:
1
d  ln p   dp
p
1
1
d     2 dT
T
T
Now we can use these manipulations to put the Clausius – Clapeyron equation in an
alternative (and sometimes more useful) equation.
dp p H vap

dT
RT 2

H vap  1 
dp H vap 1

dT

d
ln
p


d 


p
R T2
R
T
H vap
d  ln p 

R
1
d 
T
Note that the result allows the plot of logarithm pressure versus inverse temperature
to yield a straight line whose slope yields the enthalpy of vaporization.
5
Classification of Phase Transitions
1st Order Transitions
1. Heat capacity at transition temperature is infinite.
-  2 T 2 is discontinuous.
2. Derivative of chemical potential w.r.t. temperature at transition temperature is
discontinuous.
3. Molar volume, entropy and enthalpy are discontinuous at transition temperature.
4. Phase transition has a specific transition enthalpy.
5. Melting, evaporation and sublimations are first-order transitions.

V
Ttrans
T
H
Ttrans
T
Cp
Ttrans
T
Ttrans
T
2nd Order Transitions
1. Heat capacity at transition temperature is discontinuous.
2. Derivative of chemical potential w.r.t. temperature at transition temperature is
continuous.
3. Molar volume , entropy and enthalpy are continuous at transition temperature.
4. Temperature derivatives of volume and enthalpy are discontinuous at transition
temperature.
5. Second-order transitions include glass transitions in polymers, onset of
ferroelectricity is perovskite crystals, onset of superconductivity.

V
Ttrans
Ttrans
T
Cp
H
Ttrans
T
Ttrans
T
T
Lambda Transitions
1. Heat capacity at transition temperature is discontinuous.
2. Theoretically, the transition has a specific transition enthalpy, though in practice
the transition enthalpy exists only as a limit in infinite time.
3. Molar volume , entropy and enthalpy are continuous at transition temperature.
4. Lambda transitions include transitions between phases of liquid crystals or the
onset of ferromagnetism.
5. A lambda transition is a like a second-order transition except that it has an infinite
heat capacity at the transition temperature (theoretically!).

V
Ttrans
T
H
Ttrans
T
Cp
T trans
T
Ttrans
T
6
Thermodynamic versus Kinetic Control
Graphite and diamond
Some substances exist in a thermodynamically unstable state. To illustrate, let us
compare two allotropes of carbon, graphite and diamond. Gf 0 (C(graphite)) = 0
kJ/mol, because graphite is the most thermodynamically stable allotrope of carbon.
The Gibbs free energy of formation Gf 0 (C(diamond)) = 2.900 kJ/mol. Diamond
should spontaneous lose free energy to change into graphite. However, because the
carbon-carbon bonding in diamond is so strong, the activation energy to allow the
transformation is extremely high.
Thus diamond can exist for eons even though graphite is thermodynamically favored
because the carbons atoms don’t have enough energy to rearrange themselves to make
graphite.
As a side note, the third allotrope of carbon, “buckyballs”, has a Gibbs free energy of
formation of approximately 13.9 kJ/mol. Thus finding buckyballs in nature are
extremely rare.
White tin and grey tin
At 25 C, the allotrope of tin known as white tin (-tin) is thermodynamically favored
over the allotrope, grey tin (-tin) with Gf 0 (C(-tin)) = 0.1 kJ/mol. The solid-solid
phase transition temperature between the two phases is 13.2 C, though impurities
can cause the transition temperature to lowered considerably (> 0 C).
Extremely cold weather in 18th century Europe caused many organ pipes to turn to
dust over several years. The dust has been named tin blight, tin disease, tin pest or tin
leprosy. The dust is grey tin which lacks the malleability of its brother white tin. The
cold temperature conversion of tin from its white form to its grey form reflects a
change in the thermodynamic favorable alloptrope with temperature. Kinetically, the
reaction is autocatalytic. Formation of the tin blight lowers the activation for the
formation of more tin blight. Thus a piece of tin may survive a long time in cold
weather; however, it may fail catastrophically in a single season without warning.
7
Summary
Clapeyron equation
dp S

dT V
or
dp
H

dT TV
Clausius-Clapeyron Equation
H vap  1 1 
p 
ln  2   
  
R  T2 T1 
 p1 
H vap
d  ln p 

R
1
d 
T