5장-3. 수업목표

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 A}^A

 

^A_{A A}



^A

A 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

₃

COCH

₃

. HNO

₃

/H

₂

O, HCl/H

₂

O

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

_mix

G - D

_mix

G

^ideal

Water/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

₂

T

_uc

= 573K

H

₂

O-(C

₂

H

₅

)

₃

N 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

₁

is 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

_p

X

_q

log  _   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

J

RT

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

A

RT 

A

    

 1



A

a

_A

 x

_A

*

For (real) so : ln

_A

l n

*^A A

A

RT p

    ^  p ^ 

   

^*A

 RT ^ln   a

A

For ideal soln,

*^A _A A

p x

p 

5E.2 Solute Activity

ideal dilute behavior as x _B → 0 not x _B → 1

B B

B

B

x

B

a p

K  



*

0

*

* *

ln ln

ln

^B _B ^B _B

ln

_B

B

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

_B

ln

B B

o o

B

RT x

B

RT a

 

    

(b) Real solute:

: A new standard state

B B B

p K a

 

B B

B

B

x

B

a 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

^





^H

H

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} _m^ideal





 

^{1 2}



_

  

_{ }

• For M

_p

X

_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  

¹

^ln  ^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 2

Ionic 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

log

1 2

1

A z z I BI CI



_

 

^{ }





where B and C are

dimensionless constants