Physical
Transformations of
Pure Substances
Chapter 4
Stabilities of Phase



A phase of a substance is a form of matter that is
uniform throughout in chemical composition and
physical state.
A phase transition is the spontaneous conversion of
one phase into another.
Phase transitions occur at a characteristic
temperature and pressure.
Stabilities of Phase



At 1 atm, < 0 °C, ice is the stable phase of H2O,
but > 0 °C, liquid water is the stable phase.
The transition temperature, Ttrs, is the temperature
at which two phases are in equilibrium.
So what happens to Gibbs energy?
Stabilities of Phase





At 1 atm, < 0 °C, ice is the stable phase of H2O,
but > 0 °C, liquid water is the stable phase.
The transition temperature, Ttrs, is the temperature
at which two phases are in equilibrium.
So what happens to Gibbs energy?
< 0 °C Gibbs energy decreases as liquid  solid.
> 0 °C Gibbs energy decreases as solid  liquid.
Stabilities of Phase



Thermodynamics does not provide information
regarding the rate of phase change.
Diamond  graphite
Thermodynamically unstable phases that persist due
to slow kinetics are called metastable phases.
Phase Diagrams

Phase boundaries show the
values of p and T at which
two phases coexist in
equilibrium.
Vapor Pressure


The pressure of a vapor in
equilibrium with a liquid is
called the vapor pressure.
The pressure of a vapor in
equilibrium with a solid is
called the sublimation vapor
pressure.
Boiling Point





Liquid can vaporize from a liquid surface below it’s
boiling point – as we learnt from the Drinking Bird.
In an open vessel, the temperature at which the vapor
pressure equals the external pressure, is called the
boiling temperature.
At 1 atm, it’s called the normal boiling temperature, Tb.
At 1 bar, it’s called the standard boiling point.
Normal point of H2O is 100.0 °C; it’s standard
boiling point is 99.6 °C.
Critical Point





In a closed rigid vessel, boiling does not occur.
As the temperature is raised the density of vapor
increases and the density of the liquid decreases.
When the density of the vapor and liquid phases are
equal the surface between the two phases disappears.
The temperature at which this occurs is called the
critical temperature, Tc.
The vapor pressure at the critical temperature is called
the critical pressure, pc.
Critical Point
Melting and Freezing






The temperature at which, under a specified pressure, the liquid
and solid phases of a substance coexist in equilibrium is called
them melting temperature.
The freezing temperature is the same as the melting point.
At 1 atm, the freezing temperature is called the normal freezing
point, Tf.
At 1 bar, it’s called the standard freezing point.
The difference is negligible in most cases.
The normal freezing point is also called the normal melting
point.
Triple Point




There is a set of conditions under which three different
phases of a substance (typically solid, liquid and vapor)
all simultaneously coexist in equilibrium.
This point is called the triple point.
For any pure substance the triple point occurs only at
single definite pressure and temperature.
The triple point of water lies at 273.16 K and 611 Pa.
Triple Point

The triple point marks the
lowest pressure at which a
liquid phase can exist.
Carbon Dioxide
Water
Helium
Thermodynamics of Phase
Transitions


The molar Gibbs energy, Gm, is also called chemical
potential, m. Phase transitions will be investigated
primarily considering the change in m.
Thermodynamic definition of equilibrium: At
equilibrium the chemical potential of a substance is the
same throughout the sample, regardless of how many
phases are present.
Thermodynamics of Phase
Transitions
Thermodynamics of Phase
Transitions


At low temperatures, and provided the pressure is not
too low, the solid phase of a substance has the lowest
chemical potential and is therefore the most stable.
Chemical potentials change with temperature: this
explains why different phases exist.
Temperature Dependence of
Phase Transitions
Gm 
m 

  Sm    Sm
 T p
T p


As temperature increases,
chemical potential
decreases.
Melting and Applied Pressure
Gm 
m 

  Vm    Vm
 p T
p T


Molar volume of solid is
smaller than that of the
liquid.
Melting and Applied Pressure
Gm 
m 

  Vm    Vm
 p T
p T


Molar volume of solid is
greater than that of the
liquid.
Melting and Applied Pressure
m 
   Vm
p T
m  Vm p

Melting and Applied Pressure

Calculate the effect on the chemical potentials of
ice and water of increasing pressure from 1.00 to
2.00 bar at 0 °C. The density of ice is 0.917 g
cm-3 and that of liquid water is 0.999 g cm-3.
Vm 
M

m 
Mp

M  18.02 g mol -1  0.01802 kg mol -1
p  1 bar = 10 5 Pa
ice  917 kg m -3 water  999 kg m -3
Melting and Applied Pressure

Calculate the effect on the chemical potentials of
ice and water of increasing pressure from 1.00 to
2.00 bar at 0 °C. The density of ice is 0.917 g
cm-3 and that of liquid water is 0.999 g cm-3.
(0.01802 kg mol -1 )  (1.00 Pa)
-1
u(ice) =

1.97
J
mol
917 kg m -3
(0.01802 kg mol -1 )  (1.00 Pa)
-1
u(liquid) =

1.80
J
mol
999 kg m -3
Vapor Pressure and Applied
Pressure


When pressure is applied to a condensed phase,
its vapor pressure rises.
This is interpreted as molecules get squeezed out
of the condensed phase and escape as a gas.
p  p*e Vm (l )P
RT
p  vapor pressure
p*  vapor pressure of condensed phase
in the absence of an additional pressure
P  pressure applied
Vapor Pressure and Applied
Pressure
Location of Phase Boundaries

Locations of phase boundaries – pressures and
temperatures - can be located precisely by
making use of the fact that at when two phases
are in equilibrium, their chemical potentials must
be equal
m ( p,T)  m ( p,T)
Location of Phase Boundaries
dG  Vdp  SdT
dm  Vm dp  Sm dT
V ,m dp  S ,m dT  V ,m dp  S ,m dT
(S ,m  S ,m )dT  (V ,m  V ,m )dp
 trsS dp

 trsV dT
 Clapeyron equation
Location of Phase Boundaries
Solid-liquid boundary
 trsS dp

 trsV dT
 trsH
 trsS 
T
dp  fusH

dT T trsV
 Clapeyron equation
Solid-liquid boundary
Solid-liquid boundary
dp  fusH

dT T fusV
 fusH dT
dp 
 fusV T
 fusH dT
 p* dp  T *  V T
fus
p
T
 fusH
dT
 p* dp   V T * T
trs
p
T
 fusH  T 
p p 
ln  * 
 fusV T 
*
Solid-liquid boundary
 fusH  T 
p p 
ln  * 
 fusV T 
*
When T and T * do not differ much
p p 
*
 fusH
T * fusV
(T  T * )
Liquid-vapor boundary
 trsS dp

 trsV dT
 trsH
 trsS 
T
dp  vap H

dT T vapV
 Clapeyron equation
Liquid-vapor boundary
dp
dT
dT
dp

 "small"
 "large"

Liquid-vapor boundary
dp  vap H

dT T vapV
 vapV  Vm (g)
 vap H
dp

dT T(RT p)
d ln p  vap H

dT
RT 2
- Clausius - Clapeyron equation
Liquid-vapor boundary
d ln p  vap H

dT
RT 2
 vap H
d ln p 
dT
2
RT
ln p
T  vap H
 ln p* d ln p  T * RT 2 dT
 vap H T dT
 vap H 1 1 
ln p
d
ln
p



  * 
 ln p*

*
2
T
R
T
R T T 
 vap H 1 1 
*
ln p p  
  * 
R T T 
 vap H 1 1 
p

e

  * 
*
p
R T T 
p  p*e 
Liquid-vapor boundary
d ln p  vap H

dT
RT 2
 vap H
d ln p 
dT
2
RT
ln p
T  vap H
 ln p* d ln p  T * RT 2 dT
 vap H T dT
 vap H 1 1 
ln p
d
ln
p



  * 
 ln p*

*
2
T
R
T
R T T 
 vap H 1 1 
*
ln p p  
  * 
R T T 
 vap H 1 1 
p

e

  * 
*
p
R T T 
p  p*e 
Solid-gas boundary
* 
p p e

 sub H 1 1 

  * 
R T T 