### 2 + 2 = 2

```Chapter 7 Equilibrium rela
1
1. Phase Rule
C–Φ+2
F = number of degrees of freedom, or variance
C= number of components
Φ= number of phases
2 —— only temperature and pressure may affect th
e.g.: In systems of two components, C=2；Φ=2；
therefore, F = 2 - 2 + 2 = 2
•F =
2
•For binary distillation, F=2, there are
four variables of interests: pressure,
temperature, and the mole fractions of
component A in liquid and vapor phases.
If the pressure is fixed,only one variable,
e.g., liquid phase mole fraction, can be
changed independently, and temperature
and vapor-phase mole fraction follow.
3
2. Equilibrium of Gas and Liquid
and equilibrium stage
Vapor
Heater

Liquid
4
P=pA+pB
yA+yB=1
Vapor
xA+xB=1

Heater
Liquid
5
Equilibrium State:
No further changes in composition, temperature, o
The chemical potential of the vapor and liquid pha
The apparatus is performing as if it were an “equil
6
3. Thermodynamic relationships
1)Equilibrium ratio ( or equilibrium constant or K
yA
KA 
xA
Where KA=EquilibriumKratio
A

yA=mole fractionof
A in vapor
ABcomponent
KB
xA= mole fraction of component
A in liquid
7
•
•
The more volatile
components in a
mixture will have the higher values of KA,
whereas less volatile components will have
lower values of KA.
2)Relative volatility（相对挥发度）----key
separation factor in distillation.
（易挥发）
 AB
KA

KB
(b)
Where  AB  relative volatility ( A relative to B)
8
•
3)Ideal system
•
Ideal systems of vapor and liquid mixtures obey
Dalton’s and Raoult’s law.
Dalton’s law relates the concentration of a
component present in an ideal gas or vapor
mixture to its partial pressure.
•
p A  PyA
(a)
2
Where P=total pressure,force/length
pA
pA=partial pressure P
of component A,force/length2
yA=mol fraction of component A,dimensionless(无因次)
yA
9
•
Raoult’s law
p A  p A0 x A
p B  p xB  p (1  x A )
0
B
(b)
0
B
Where pA=partial pressure of component A,
force/length2
xA=molar fraction of component A,
dimensionless(无因次)
10
Combining Equation(a) and (b) yields
Therefore,
Py

P
x
A
PyAA  PP xx
Py
PPP
KK

A
KAA  P
PP
PPPA 
AB  AA
AB P 
AB
B
PP

A  A
AA AA

A 
AA
BB
BBB
i
ln
P

A

i
ln PP  AAii C  iTi
ln
Ci TT
C

i 
ii
11
P

Vapor pressure of pure
PA component i as a
 
function 
ofAB
temperature
is correlated byAthe
i
P
B
Antoine equation:
Bi

Ai
PyA  PA x A 
Bi
Ci
Ai ln Pi Ai  Ai 
Bi

C

T
T
B
BP
i
Ci
A
A
i
i
i
KA 

Where ACiiPBi C i are Antoine constants
R
T


T
isCabsolute
temperature,
or
K
BTi P
R
i
A


K Ain Ktemperature
CRless
T sensitive
•  AB is
to
changes
R
i


AB
P
 A or

B
than is
K
K
K
R  KB
TK
K

A
B
AB

(当温度变化不大时，可认为是常数，或取平均值)
K B  AB  relative vol
K
 R
i Where
i
i

12
Where
 AB  relative volatility
(
K
i
B
ln P  A 
C T
P  p A  pB
since
pA  P xA

A
pB  P x  P (1  x A )

B B

B
13
Bubble point equation:
P p
P  f 2 (t )
xA  0

pA  p
f1 (t )  f 2 (t )
0
B
0
B
Dew point equation:
p
p P p
f1 (t ) P  f 2 (t )
yA 
xA 
 0


P
P pA  p
P f1 (t )  f 2 (t )
0
A
0
A
0
B
0
B
14
Combining Dalton’s law and
Raoult’s
law

Py

P
A x A
yields
A
Py  P x
p A  PyAA  PAA xAA
P
A

P
K

Therefore,
A
P
A
KKA  PA
A
PP
P
A

P


A
P
AB
AB  PA
AB
PPBB
B
Though not a strong function
of temperature,
B

i
B
ln
P

A


i
B
generally decreaseslnwith
increasing
temperature
i
i

i
P

A

i
i
ln
P

A

C

T
i
i
i
and increases with decreasing temperature
CCi TT
i
15
For a binary system, the relative volatility of the
two components (A and B) is given by:
y A / xA

y B / xB
y B  1  y A , xB  1  x A
The above equations can be rearranged
give
(重新整理)
16
to
 AB x A
yA 
1  ( AB  1) x A
Let xA=x；yA=y:
Phase equilibrium
equation:
x
y
1  (  1) x
The above equation is used to express the
concentration of component A in the vapor as
a function of its concentration in the liquid
and relative volatility.表示在总压一定时,气液平衡时

17
4. Phase equilibrium diagrams for ideal systems
•1)
t-x-y diagram (boiling-point diagram)
(Total pressure is constant)
t
•2 lines ; 3 regions
•2 lines: Bubble point line (t-x curve, plotted from Bubble po
Bubble point line

x
18
Dew point line
t point equation)
(t-y curve,plotted（图示） from Dew

Dew point line
y
19
t-x-y diagram
(Total pressure is constant)
The diagram gives the relation of composition

t

x(y)
20
Superheated
vapor region
Two phase
vapor/liquid
region
Dew

Point
t
Bubble

Point
Liquid
region
x(y)
21
Hint：
(1)At equilibrium,
temperature of liquid
and vapor phases is
the same.
Dew point
t
Bubble point
x
t superheated
tdew
Dew point
y
(2)When x=y:
tdew > tbubble
tbubble
bubble point
(3) In two phase region:
tcold
x(y)
y>x
22
2) x-y diagram
The most used equilibrium
curve in distillation is x-y
diagram. It can be plotted
from the t-x-y diagram.
Question: What are the
temperature relationships
of points 1, 2 and 3？
23
Hint: (1)The higher the x
or y is, the lower the
temperature is;
(2) When α>1, y>x,
equilibrium line is above
diagonal;
（3）When α=1, y=x
equilibrium line is just
diagonal;
y
α=4
α=2
α=1
α<1
x
（4）α<1, equilibrium
line is below diagonal.
24
5. Henry’s law
•
For low concentrations, the relationship
between the concentration of a component in
a liquid mixture and its partial pressure in
the vapor (or gas) phase can be expressed as
a linear relationship. The modified form of
Raoult’s law, is known as Henry’s law.
p A  ExA
(c)
E law constant for component
Where E=Henry’s
A, force/lengthy2,A xAmolar
m xA fraction of liquid.
25
E
p A  ExA
•
If both sides of eq.(c) are divided by the total
pressure, thenE
y A  m xA
E
E
m

m P
P
Where m=modified Henry’s law constant(中文又

molar fraction of liquid
26
6. Non-ideal systems
Most liquid mixture are nonideal. Eq.(b) must
be modified to include a correction factor
called the liquid-phase activity coefficient.
pA  p  xA   A
0
A
(d)
Where γA=liquid phase activity coefficient,
dimensionless
27
•If γ>1, positive deviations from ideality in the liquid
occur;
•if γ<1, there are negative deviations from ideality.
 P
KA 
P


 A PA

P
 AB K A A A P
 B PB

 A PA
 dependent
KA and  AB are
on


P
B
B
composition.

A A
28
Azeotropes （恒沸物）
•When azeotropes are encountered, vapor and liquid
compositions are equal, and components cannot be
separated by conventional distillation.
Minimum Boiling Azeotrope
（最低恒沸点的溶液）
Maximum Boiling Azeotrope
（最高恒沸点的溶液） 29
7. Effect of total pressure on
vapor/liquid equilibrium
(1) Total Pressure↑→Bubble point↑ → α↓ → Separation
difficulty↑
(2) Total Pressure changed→Azeotropic compositions
changed
•Total Pressure Effect on Water-Ethanol（乙醇） system
P（kPa）
101.3 52.6
25.3
12.7
Azeotrope(mol fraction) 0.894 0.915 0.994 0.997
Assignments：21.1(a), (b)[p.706]
30
```