Onecomponent Phase Equilibrium

1
Chapter 7

• One-component Phase Equilibrium

2
The Phase Rule
Physical Chemistry
Chapter 7
Phase a state of matter that is uniform
throughout in chemical composition and physical
state. (Gibbs)
Number of phase (p) Gas or gaseous mixture
single phase Liquid one, two and three
phases two tally miscible liquids single
phase a slurry of ice and water two
phases Solid a crystal is a single phase an
alloy of two metals two phases (immiscible) -
one phase (miscible)
3
The Phase Rule
Physical Chemistry
Chapter 7
(a)
(b)
The difference between (a) a single-phase
solution, in which the composition is uniform on
a microscopic scale, and (b) a dispersion, in
which regions of one component are embedded in a
matrix of a second component.
4
The Phase Rule
Physical Chemistry
Chapter 7
The difference between (a) constituent and (b)
component
constituent a chemical species (an ion or a
molecule) that is present in a system
component a chemically independent constituent
of a system
number of components (c) the minimum number of
independent species necessary to define the
composition of all the phase present in the system
5
The Phase Rule
Physical Chemistry
Chapter 7
When no reaction takes place,
The number of constituent the number of
component
Pure water
a one-component system
water
Mixture of ethanol and water
a two-component system
ethanol
water
6
The Phase Rule
Physical Chemistry
Chapter 7
When a reaction takes place,
CaCO3(s) ? CaO(s) CO2(g)
Phase 1
Phase 2
Phase 3
a two-component system
CaO
CO2
CaO CO2 ? CaCO3
The number of constituent ? the number of
component
7
The Phase Rule
Physical Chemistry
Chapter 7
Counting components
How many components are present in a system in
which ammonium chloride undergoes thermal
decomposition?
NH4Cl(s) ? NH3(g) HCl(g)
Phase 1
Phase 2
three constituents
two-component
a one-component system
additional NH3 or HCl
NH4Cl
NH4Cl ? NH3 HCl
8
The Phase Rule
Physical Chemistry
Chapter 7
Degree of freedom or Variance (f) the number of
intensive variables that can be changed
independently without disturbing the number of
phases in equilibrium.
The phase rule a general relation among the
variance f, the number of components c and the
number of phases p at equilibrium for a system of
any composition.
no reactions
9
The Phase Rule
Physical Chemistry
Chapter 7
Two assumptions
(1) no chemical reactions occur
(2) every chemical species is present in every
phase
Counting the total number of intensive variables
(properties that do not depend on the size of the
system). The pressure P and temperature T count
as 2.
Specify the composition of a phase by giving the
mole fractions of c-1 components (because
x1x2xc1, and all mole fractions are known if
all except one are specified.)
There are p phases, the total number of
composition variables is p(c-1). At this stage,
the total number of intensive variables is
p(c-1)2.
10
The Phase Rule
Physical Chemistry
Chapter 7
At equilibrium, the chemical potential of a
component j must be the same in every phase
?j,? ?j,? for p phase
That is, there are p-1 equations to be satisfied
for each component j. as there are c components,
the total number of equations is c(p-1).
Each equation reduces the freedom to vary one of
the p(c-1)2 intensive variables. It follows
that the total variance is
f p(c-1) 2 - c(p-1) c p 2
11
Physical Chemistry
Chapter 7
One-component phase equilibrium
For a one-component system (pure water)
f1-p23-p,(C1)
f 0, p 1, 3p1
p1,f2
p2,f1
p3,f0
12
The Phase Rule
Physical Chemistry
Chapter 7
Phase diagram shows the regions of pressure and
temperature at which its various phases are
thermodynamically stable.
Phase boundary a boundary between regions, shows
the values of P and T at which two phases coexist
in equilibrium.
13
The Phase Rule
Physical Chemistry
Chapter 7
Tf
Tb
P
?
solid
Critical point
liquid
Triple point
vapor
Gas stable
Liquid stable
Solid stable
T3
Tc
T
T
Solid-liquid phase boundary a plot of the
freezing point at various P.
Liquid-vapor phase boundary a plot of the vapor
P of liquid against T.
Solid-vapor phase boundary a plot of the
sublimation vapor P against T.
14
The Phase Rule
Physical Chemistry
Chapter 7
P
solid
Critical point
liquid
Triple point
vapor
T3
Tc
T
Triple point at which three different phases (s,
l, g) all simultaneously coexist in equilibrium.
It occurs at a single definite pressure and
temperature characteristic of the substance
(outside our control).
Critical point at which (critical P and critical
T) the surface disappears.
15
Physical Chemistry
Chapter 7
H2O phase diagram P T
D
C
218 atm
Y
I
solid
liquid
Line Point Region
S
P / 10 5 Pa
1 atm
R
gas
0.00611
A
O
0.01
374.2
99.974
0.0024
Tb
T3
Tf
t/?
16
Physical Chemistry
Chapter 7
H2O phase diagram P T
Region (s, l, g) f2, one phase
D
C
218 atm
Y
I
Line (OA, AD, AC) f1, two phases in equilibrium
solid
liquid
S
P / 10 5 Pa
1 atm
R
gas
0.00611
A
Point (A) f0, three phases in equilibrium
O
0.01
374.2
99.974
0.0024
Tc
Tb
T3
Tf
t/?
17
Physical Chemistry
Chapter 7
Difference between triple point and freezing point
18
Physical Chemistry
Chapter 7
Difference between triple point and freezing point
Why the freezing point is lower than the triple
point?
The higher pressure lowers the freezing point
compared with that of pure water
The dissolved air (i.e. N2 and O2) lowers the
freezing point compared with that of pure water
19
The Clapeyron Equation
Physical Chemistry
Chapter 7
P
? ?
Phase equilibrium
Phase ?
dP
2
For a pure substance
1
Phase ?
At point 1,
T
dT
At point 2,
Fig. 7.5 two neighboring points on a two-phase
line of a one-component system.
20
The Clapeyron Equation
Physical Chemistry
Chapter 7
For a single phase
(7.14)
pure phase
one-phase, one-component
21
The Clapeyron Equation
Physical Chemistry
Chapter 7
For any point on the ?-? equilibrium line
22
The Clapeyron Equation
Physical Chemistry
Chapter 7
For a reversible (equilibrium) phase change
one component two-phase equilibrium
Clapeyron Equation (Clausius-Clapeyron equation)
23
The Clapeyron Equation
Physical Chemistry
Chapter 7
P
? ?
Phase ?
dP
2
1
Phase ?
T
The slope of the phase boundaries
dT
Fig. 7.5 two neighboring points on a two-phase
line of a one-component system.
Any phase equilibrium of any pure substance
24
The liquid-vapor boundary The solid-vapor
boundary
Physical Chemistry
Chapter 7
Solid-gas or liquid-gas equilibrium, not near Tc
25
The liquid-vapor boundary The solid-vapor
boundary
Physical Chemistry
Chapter 7
Solid-gas or liquid-gas equilibrium, not near Tc
liquid-gas equilibrium, not near Tc
26
The solid-liquid boundary
Physical Chemistry
Chapter 7
Solid-liquid equilibrium, small temperature range
27
Constructing a solid-liquid phase boundary
Physical Chemistry
Chapter 7
Example construct the ice-liquid phase boundary
for water at temperature between 1oC and 0oC.
What is the melting temperature of ice under a
pressure of 1.5 kbar? ?fusH 6.008 kJ/mol,
?fusV -1.7 cm3/mol.
Answer
28
The Phase Rule
Physical Chemistry
Chapter 7
The formula gives the following values
T/oC -1.0 -0.8 -0.6 -0.4 -0.2 0.0 P/bar 130 105
79 53 27 1.0
What is the melting temperature of ice under a
pressure of 1.5 kbar?
Rearrange the formula into
Then, with P1.5 kbar, T262 K or 11oC.
29
The Phase Rule
Physical Chemistry
Chapter 7
P11.0 bar, T1273 K P21.5 kbar, T2262 K
Comment notice the decrease in melting
temperature with increasing pressure water is
denser than ice, so ice responds to pressure by
tending to melt.
30
Solid-solid Phase Transitions
Physical Chemistry
Chapter 7
Polymorphism
Many substances have more than one solid form
which has a different crystal structure and is
thermodynamically stable over certain ranges of T
and P.
Allotropy
Polymorphism in elements.
Metastable
The rate of conversion of ? to ? is slow enough
to allow ? to exist for a significant period of
time.
31
Solid-solid Phase Transitions
Physical Chemistry
Chapter 7
Phase diagram of S (part)
Three triple points
B 95 oC
C 119 oC
E 151 oC
Fig. 7.9 (a)
32
Solid-solid Phase Transitions
Physical Chemistry
Chapter 7
There are many different types of Phase
Transition.
Fusion, vaporization
Ehrenfest Classification
Changes of enthalpy and volume
33
Solid-solid Phase Transitions
Physical Chemistry
Chapter 7
First-order phase transition
Because ?trsV and ?trsS are non-zero for melting
and vaporization for such transitions, the slopes
of the chemical potential plotted against either
pressure or temperature are different on either
side of the transition. The first derivatives of
the chemical potentials with respect to pressure
and temperature are discontinuous at the
transition.
34
Solid-solid Phase Transitions
Physical Chemistry
Chapter 7
First-order phase transition
CP is the slope of H-T. at Tt, the slope of H and
Cp are infinite.
A first-order phase transition is also
characterized by an infinite heat capacity at the
transition temperature.
35
Solid-solid Phase Transitions
Physical Chemistry
Chapter 7
Second-order phase transition
The first derivative of the chemical potential
with respect to temperature is continuous but its
second derivative with respect to temperature is
discontinuous at the transition.
A continuous slope of ? (a graph with the same
slope on either side of the transition) implies
that the volume and entropy (and hence the
enthalpy) do not change at the transition.
The heat capacity is discontinuous at the
transition but does not become infinite.
36
Solid-solid Phase Transitions
Physical Chemistry
Chapter 7
First-order phase transition
Second-order phase transition
37
Solid-solid Phase Transitions
Physical Chemistry
Chapter 7
First-order
Second-order
Lambda
not first-order