5장-3. 수업목표
• The phase rule: F = C − P + 2
A A B
A
B A
y p p
p
p p p
)
(
* **
*
*
A B A
B
p p x
p
p
* (
*
*)
y A , y B
x A , x B
• The lever rule: n l n l
• Vapor pressure diagram
5C. Phase diagrams of binary systems
F = 4 − P ( T, p, x )
For an ideal solution of two volatile liquids Raoult’s law:
x A : mole fraction in the liquid
5C.1 Vapor pressure diagrams
• At constant T, F΄ = 3 − P : (p, x)
• At constant p, F΄ = 3 − P : (T, x)
total vapor p:
* * *
* *
* *
1
( )
A
B A
A B B
A A A
B A B
p x p x p
x p x p
p p p x
*
*
1
A A A
B B B
A B
p x p p x p
x x
* * *
( )
B
A B
A B A
p p p
p p p
p x
The mole fractions in the gas, y A , y B
* ** * *
( )
A B
A B A A
p p p
p p p y
y A , y B x A , x B
*
* A B
p p
* * * *
*
*
*
* * *
*
*
* * *
* *
* * *
* *
*
*
* * *
* * *
* *
,
1
( )
( )
( )
(
( )
( )
( )
( )
B A B A
B A B A
A B
A B
A A B B A A A B
A A A
A
A A A
A B A
A B A A
A B
B A B
A B A A
B A
B A B A
B A
B
B A
p p p x
p p p x
p p
y y
p p
p x p x p x p x p
p x p
y p
y x p
x y p
p p
p p p x
p
p y p
p p p y p
p p p y
p p
p p x
p
*
* * ** * *
) ( )
( )
A A A B A B
A B A A
p y p p y p
p p p y
5C.1 Vapor pressure diagrams
* *
* * *
( )
A B
A B A A
p p p
p p p y
A B A
B
p p x
p
p
* (
*
*)
5C.1(b) The interpretation of the diagram
• Isopleth: Greek “equal abundance”
• Tie line
(a)
(b)
(c)
5C.1(c) The lever rule
Overall amount of A: nz A x A : mole fraction in phase y A : mole fraction in phase
A AA
AA A
AA A
A
y z
n z
x n
y n x
n z
n n
y n x
n nz
n n
n
l n l
n
5장-4. 수업목표
• The lever rule:
• Fractional distillation: the boiling and
condensation cycle is repeated successively
• Steam distillation: distillation of heat sensitive, water insoluble organic compound
• Azeotrope: boiling without changing
• Partially miscible liquids
• Eutectic, peritectic
• Congruent, incongruent melting
l n l
n
5C.2 Temperature-composition diagrams
p = f(x A ) and g(y A )→ T = h 1 (x A ) and h 2 (y A ) Ideal solution
Fractional distillation: the boiling and condensation cycle is repeated successively.
more similar partial p
→ more theoretical
plates needed
Fractional distillation-1
Vigreux column
Fractional distillation-2
5C.2(b) Azeotropes (“boiling without changing”) (nonideal solution: G E 0, distillation cannot separate the two liquids )
G E < 0
Favorable interaction between A and B stabilize the liquid.
Ex, CHCl
3/CH
3COCH
3. HNO
3/H
2O, HCl/H
2O
Ex, dioxaxane/water, ethanol/water
G E > 0
vapor:
pure Azeotrope in the pot
liquid: pure in the pot
vapor:
azeotrope vapor:
pure
Recall that G
E= D
mixG - D
mixG
idealWater/ethanol azeotrope
Ethylene hydration or brewing
produces an ethanol–water mixture.
For most industrial and fuel uses, the ethanol must be purified. Fractional distillation can concentrate ethanol to 95.6% by volume (89.5 mole%). This mixture is an azeotrope with a boiling point of 78.1 °C, and cannot be further purified by distillation. Addition of an entraining agent, such as benzene, cyclohexane, or heptane, allows a new ternary azeotrope comprising the
ethanol, water, and the entraining agent to be formed. This lower-boiling
ternary azeotrope is removed
preferentially, leading to water-free
ethanol.
5C.2(c) Immiscible liquids
(p = p A + p B p A * + p B * )
Mixture boils at a lower temperature than either would alone
Steam distillation: heat sensitive, water insoluble organic compound
Oils of low volatility distil in low abundance
Each component should be kept saturated in the other component for this to work. Without contact, (b) would not boil at the same
temperature
5C.3 Liquid-liquid phase diagrams
(partially miscible liquids)
T uc : upper critical (solution) T T lc : lower critical T
Above T
uc(or below T
lc) the two components are fully miscible
As T↑, each phase in equilibrium richer in its minor component
B-rich phase A-rich
phase
a’
a’’
T →
Partially miscible liquids: l-l phase diagrams
nitrobenzene-hexane T
uc= 294K
Pd-PdH
2T
uc= 573K
H
2O-(C
2H
5)
3N T
lc= 292K
nicotine-water T
uc= 483K,
T
lc= 334K
Example 5C.2 (p209)
A mixture of 50 g hexane (0.58 mol) and 50 g nitrobenzene (0.41 mol) at 290 K. x(C 6 H 5 NO 2 ) = 0.41
What are the compositions of the phases?
To what T the sample be heated to obtain a single phase?
7 times more hexane-rich phase( ) than nitrobenzene-rich phase( )
0.83 0.35
06 7 . 0
42 . 0 35
. 0 41 . 0
41 . 0 83 .
0
l l n
n
0.41 hexane-rich
phase()
nitrobenzene -rich phase()
β
α
5C.3(c) The distillation of partially miscible liquids
(in case of partially miscible and form a low-boiling azeotrope)
No T uc
Isopleth e: Two phases persist up to the T
b.
The mixture vaporizes like a single substance Three phases: T is fixed
Two kinds of molecule avoid each other Eutectic (Greek, “easily melted”)
Example 5C.3 (p212)
State the changes that occur when a
1is boiled and the vapor is condensed
5C.4 Liquid-solid phase diagrams
Eutectic mixture (C=2, P=3, F΄= 0):
solidifies at a single definite T without gradually unloading one or other of the components from the liquid
A pair of solids that are almost completely immiscible right up to their melting point
Eutectic (Greek, “easily melted”)
Ex, 67% Sn + 33% Pb = solder (T
m=183℃)
23% NaCl + 77% water (T
m= −21.1℃)
5C.4(b) reacting systems
Congruent melting: the composition of the liquid it forms is the same as that of the solid compound
A + B → C
Ga GaAs As
Ga + As → GaAs
5C.4(c) Incongruent melting
compound melts into its components and does not itself form a liquid phase
Peritectic line
(3 phases)
Peritectic line
(3 phases)
Mole fraction of Si, x Si
5D. Phase diagrams of ternary systems
3 components system: x A x B x C 1 → equilateral triangle
Bl 5D.1 (p217)
5D.2 Ternary systems
Ternary phase diagrams are widly used in metallurgy and materials science
(A,W) (A, C) fully miscible (W,C) partially miscible
As A is added, single phase forms
5D.2(a) Partially miscible liquids
Plait point:
The compositions of the
two phases in equilibrium
become identical
: , , 0.88, 0.12, 0 : , , 0.05, 0.95, 0
W C A
W C A
W x x x
C x x x
"
2 ' 2
0.18
: , , 0.07, 0.82, 0.11
: , , 0.57, 0.20, 0.23
A
W C A
W C A
When x
a x x x a x x x
0.12, 0.61, 0.2 0.3
7 0.28, 0.37, 0.3 4
, ,
, , 5
A
W C A
W C A
When x x x x x x x
5D.2(c) Ternary solids
a. 74% Fe, 18% Cr, 8% Ni b. Two phases (γ-FeNi, Cr)
c. Three phases (γ-FeNi, Fe, Ni)
Stainless steel phase diagram
(using % instead of mole fraction)
• Activity:
• Mean activity coefficient:
• Debye-Huckel limiting law:
5장-5 수업목표. Activities
= 1: ideal
o ln
R T a
ln ln ln
o ideal ideal
RT a RT RT
o : standard state
ideal
a a
p q 1 s
ideal ln
i i RT
for M
pX
qlog z z AI 1 2
Ionic strength: I 1 z
i2 m
i/ m
o 1 z m
2 z m
2
5E. Activities
to take into account deviations from ideal behavior
= 1: ideal
A * : standard state
* A
A A
A A
a p p
x
5E.1 The solvent activity
• For real gas: fugacity (effective p)
0 0
ln ln ,
o o
J J
RT
JRT
p p
p f p
f
*
* l n A A ln
A A RT x RT a A
a A : effective mole fraction
A : activity coefficient
,
1 x
A*
ln ln
A A
RT x
ART
A
1
Aa
A x
A*
For (real) so : ln
Al n
*A AA
RT p
p
*A RT ln a
AFor ideal soln,
*A A Ap x
p
5E.2 Solute Activity
ideal dilute behavior as x B → 0 not x B → 1
B B
B
B
x
Ba p
K
*
0
*
* *
ln ln
ln
B B B Bln
BB
B B
B B
RT K
p RT x
T p x
R p RT
,
0
x
B
B 1 a
B x
B*
ln
*o B
B B
B
RT K
p
(a) Ideal dilute solutions: p B K B x B
ln
Bln
B B
o o
B
RT x
BRT a
(b) Real solute:
: A new standard state
B B B
p K a
B B
B
B
x
Ba p
K
B
o B B
o B
B
RT ln x RT ln a
(b) Real solute:
(c) Activity in terms of molality, m B :
B o
B B
o B
B RT ln m RT ln a
o
B B B
B m
a m
(d) The biological standard state (pH=7),
Conventional: pH = 0
ln ln10
7 ln10
o o
H H H H
o H H
RT a RT pH
RT
10 ln log
ln
HH
a a pH
At 298 K, 7RT ln10 = 39.96 kJ/mol
H
Activities, Standard states
= 1: ideal
o ln
RT a
ln ln ln
o ideal ideal
RT a RT RT
o : standard state
ideal
a a
5F. The activities of ions
ideal ideal
ideal
G
m
5F.1 Mean activity coefficients ,
There is no experimental way of separating the product into contributions from the cations and anions →
Interactions between ions are so strong that activity must be used except in very dilute solutions
• For MX (M
+and X
-)
ideal
ln
RT
RT ln RT ln G RT ln
G
m ideal ideal mideal
1 2
• For M
pX
q(p cations and q anions)
ln ln ln
ideal ideal ideal p q
m m
G p
q
p
q
pRT
qRT
G RT
p q
1 s
where s = p + q
ideal ln
i i RT
Recall that for ions in solutions
ln
1ln ln
ideal ideal ideal
m m i
p q p q
p q s
G RT
G RT
s RT
For each ion,
5F.1(b) The Debye-Huckel limiting law
• At very low concentration:
Due to ionic atmosphere, of any given ion is lowered
log
z z AI
1 2Ionic strength:
A = 0.509 for aqueous solution at 25 ℃
2 2 2
2
1 /
2
o
1
i i
i
z m m
I z m
z m
The long range and strength of the Coulombic
interaction between ions is primarily responsible for
the departures from ideality in ionic solutions
5F.1(c) The extensions of the limiting law
when I of the solution is too high for the limiting law to be valid
1 2