School of Chemical & Biological Engineering, Konkuk University

School of Chemical & Biological Engineering, Konkuk University

Lecture 21

• Vapor pressure diagrams (p-x)

• Lever Rule

• Temperature-composition diagrams (T-x)

Ch. 6 Phase Diagrams

Prof. Yo-Sep Min Physical Chemistry I, Spring 2009 Ch. 6-3

• According to Raoult’s law, the partial

vapor pressures of the components of an ideal solution of two volatile liquids (A and B) are related to the composition of the

mixture. _*

A A

A x p

p  and p_B  x_Bp_B^*

• The total vapor pressure (p) of the mixture is expressed as:

 

*

*

* _{B B A A} 1 _{A B}

A A B

A p x p x p x p x p

p

p       





B p p x

p

p  ^*  ^*  ^*



y-intercept Slope

• The total vapor pressure (at a fixed T) increases linearly from to as x_A changes from 0 to 1.

*

pB

*

pA

• The compositions of the liquid (x_J) and vapor (y_J) in mutual equilibrium are not necessarily the same.

• In the vapor phase, the more volatile component should be richer than the less volatile.

• From Dalton’s law for a mixture of gas A and B, the mole fraction in the gas phase (y_A and y_B) are given by:

p

y_A  p^A and

p y_B  p^B

• If the mixture of A and B is an ideal solution, y_J may be expressed in terms of x_J. ^*

A A Ap x p

y  and y_B p  x_B p^*_B

 









A A

A p p p x

p

y _* x_{* *}

*



  y_B 1 y_A

Prof. Yo-Sep Min Physical Chemistry I, Spring 2009 Ch. 6-5





B

A A

A p p p x

p

y _* x_{* *}

*



 

*

*

B A

p

 p

• y-x plot for various values of (A is more volatile than B).

• In all cases, y_A > x_A when .

• y_A = x_A when .

• If B is non-volatile ( ), B does not contribute to the vapor (y_B = 0 and y_A = 1).

* 1

* 

B A

p p

*  0 pB

* 1

* 

B A

p p

* 1

* 

B A

p p





A A

A p p p x

p

y _* x_{* *}

*



 

A A A

y p p x

 *





*

A A A

A B A

B

Ap p p x y x p

y   





A B

Ap x p p p x y

y ^*  ^*  ^*  ^*

 





A B

Ap x p p p y

y ^*  ^*  ^*  ^*





A

B A

A p p p y

p

x _* y_{* *}

*



 

 





B A

A

A A A B

A

B A

y p p

p

p p

y

y p p p

p

p y p

*

*

*

*

*

*

*

*

*

*



 



 







B A

y p p

p

p

p _* p_{* *}

*

*



 

Prof. Yo-Sep Min Physical Chemistry I, Spring 2009 Ch. 6-7





A

B A

y p p

p

p

p _* p_{* *}

*

*



 

• p-y plot for various values of (A is more volatile than B).

• When (equal volatility), the total vapor pressure is independent of y_A.

• When , the total vapor pressure increases with y_A.

* 1

* 

B A

p p

*

*

B A

p

 p

* 1

* 

B A

p p

* 1

* 

B A

p p

• The pressure-composition diagram of a two-component mixture can be drawn by the combination as below:





B A

y p p

p

p p _* p_{* *}

*

*



 

* 

*

B A

p p





B p p x

p

p  ^*  ^*  ^*

Prof. Yo-Sep Min Physical Chemistry I, Spring 2009 Ch. 6-9

• For a mixture of two volatile liquids (C = 2), when two phases (P = 2) are in equilibrium,

2 2

2 2

2    





 C P F

• Therefore if the composition (x_A or y_A) is specified, the

pressure at which the two phases are in equilibrium is fixed.

• However, at a given T, the remaining variance is one (F’ = 1).

• And also, if the pressure of the coexisting two phases is specified, the composition (x_A and y_A) is fixed.

Because the applied pressure is higher than the vapor pressure of the system, only liquid phase exists.

Because the applied pressure is lower than the vapor pressure of the system, only vapor phase exists.

L+V

• Note that there is a similar story for p-y_A diagram.

Prof. Yo-Sep Min Physical Chemistry I, Spring 2009 Ch. 6-11

• In the region between the upper and the lower lines, two phases (L + V) are present.

• Therefore, in this region if p is also given, at constant T, the

remained variance is zero (F = C – P + 2 = 2; F’ = 2 – 1 – 1 = 0).

 The liquid and vapor phases coexisting in this region have fixed compositions.

• The horizontal axis shows the overall mole fraction (z_A) of A in the entire system.

• Above the upper line, only liquid is present.

 z_A = x_A

• Below the lower line, only vapor is present.

 z_A = y_A

L + V

• Consider lowering the applied pressure on a liquid mixture of overall composition z_A = a.

• The vertical line through “a” is called an isopleth (from Greek words for ‘equal

abundance’) in which the overall composition is constant.

• p > p₁: A single liquid phase.

• p = p₁ (point a₁): A tiny amount of vapor coexists with liquid.

The line from a₁ to a₁’ is called a tie line.

y_A is given by point a₁’ x_A = z_A

Prof. Yo-Sep Min Physical Chemistry I, Spring 2009 Ch. 6-13

• p₃ < p < p₁: for example, consider p₂.

• Because the remaining variance is zero for a given p₂ at the point a₂’’, x_A and y_A are fixed.

x_A is given by a₂ y_A is given by a₂’.

• For a mixture of two volatile liquids (C = 2),

P P

C

F   2  4

• T must be specified for the pressure-composition diagram.

One phase: F = 3, F’ = 2 Two phase: F = 2, F’= 1

• If p is also specified, F’ should be further decreased by 1.

F’=2

F’=2

• p = p₃ (point a₃’):

A tiny amount of liquid coexists with vapor.

x_A is given by point a₃. y_A = z_A

• p < p₃ (point a₄): only vapor phase present.

Prof. Yo-Sep Min Physical Chemistry I, Spring 2009 Ch. 6-15

• A point in the two-phase region indicates not only qualitatively that both liquid and vapor are present, but represents

quantitatively the relative amounts of each phase.

• To find the relative amounts of two phases  and  in

equilibrium, we measure the distance l_ and l_ along the tie line.

m_ m_

l_ l_







l m l

m 

Lever Rule: n_l_  n_l_

n_: total amount of A and B in phase .

n_: total amount of A and B in phase .

• The lever rule is readily proved as below:



 n

n n  

n_: total amount of A and B in phase .

n_: total amount of A and B in phase .

The overall amount of A in both  and  phases (nz_A) is the sum of its amounts in the two phases.

A A

A n x n y

nz  _  _

Multiplying by z^{n  n}_ ^{ n}_{ A}, ^nz_A ^{ n}_^z_A ^n_ ^z_A

A A

A A

A n x n y n z n z

nz  _  _  _  _











n_   _  n_l_  n_l_

Prof. Yo-Sep Min Physical Chemistry I, Spring 2009 Ch. 6-17

• Another choice for phase diagram of a two-component mixture is the temperature-composition diagram (at a fixed pressure).

• This diagram is used to discuss distillation of the mixture.

• Consider an ideal mixture of A and B (A is more volatile than B).

• Note that the liquid phase now lies in the lower part of the diagram.

• The region between the lines is a two-

phase (V + L) region where F’ = 1 (fixed p).

V

L

V+L

• Therefore, at a given T (F’=0), the compositions of the phases in equilibrium are fixed.

V

L

V+L

• Consider heating the ideal mixture of A and B (A is more volatile than B).

• By heating a liquid (a₁), when T reaches T₂, the liquid mixture boils.

x_A = a₁ = a₂ (Most of A and B are liquid.) y_A = a₂’ (A trace of A and B is vapor.)

If p = 1 atm, this T is T_b,A. T_b,B

Bubble point line Dew point line

Prof. Yo-Sep Min Physical Chemistry I, Spring 2009 Ch. 6-19

• Reading: page 183 ~ 191

• The 2^nd exam: May 23 (Fri), 19:00, B566, Ch. 3~5