4.6 Real solution: activity of solute and solvent

i i i ix a x

*ii i

P

P x

ii

i i

P

k x

Or by Henry’s LawJudged by Raoult’s law

γcs2

γacetone

By R.L.

B

,,

11lim( ) 1lim

i

x ix i

xxi

a

x

The activity factor

Example: Acetic acid(A)/benzene(B) solutionxA 0 0.0835 0.2973 0.6604 0.9931 1.000

pA /Pa 1535 3306 5360 7293 7333

pB /Pa 35197 33277 28158 18012 466.6

*

18012 / 315791.507

0.3396B B

BB B B

a P

x P x

18012 / 676230.7845

0.3396B A

BB B A

a P

x k x

P

xA0 1

Pa

PA

PB

By R.L

By H.L

*

5360 / 73330.731

0.6604A A

AA A A

a P

x P x

By R.L:

Pa

dx

dpK

B

B

xBxh

B

676230069.0

6.466

lim0

,,

mixGm=RT(xA ln aA + xB ln aB)= – 1167J.mol-1

Example: Chloroform in actone, 298K

BA

BA ad

x

xad lnln

According to R.L.:

According to H.L.:

Calculate aA, γA from aB, γB---Gibbs-Duhem equation

0lnln BBAA adxadx

Osmotic coefficient: represent the activity of solvent molecule

*A A AlnRT x

*A A A AlnRT x

A A

A

ln

ln

x

x

0.9328Ax

2(H O) 1.004

2(H O) 0.9364a

0.944

For example: 298K, KCl aqueous solution

4.7 Excess functions: represent the nonideality of the whole solution The difference between the observed thermodynamic

function of mixing and the function for an ideal solution

, ,E

mix real mix idealX X X excess

For example:

, ,E

mix re mix idS S S

The significance: Related to molecular interaction

A-A B-B A-B

Excess entropy

* *mix re 1 1 2 2 1 1 2 2

B B BBB B

B B B

=( ) ( )

= lnl lnn

G n n n n

n n RT x n RTRT a

Excess Gibbs energy

EB B

B

lnG n RT

)()( 22112211 nnnn

B

BB xRTnxRTnxRTn lnlnln 2211

Excess enthalpyEH

mix rE

mix ree mix id H HH H

mix id 0H

2

( )[ ]p

GHT

T T

Gibbs-Helmholtz equation

E

E 2 2 BB

B

( ) ln[ ] ( )p p

G

TH T RT nT T

Excess volume

Tetrachloroethene/cyclopentane

( )T

GV

p

EV

Emixmix re mix id idre mix( ) 0 V V VV V

EE

B BB

BB

mix re

( ) [ ( ln )]

ln = ( )

so 0

T T

BT

GV n RT

p p

RT np

V

Regular solutionsE

EB B

B

BB B B

B B

( ) [ ( ln ]

ln = ln ( )

p p

p

GS n RT

T T

R n RT nT

E E EG H TS E

E 2 2 BB

B

( ) ln[ ] ( )p p

G

TH T RT nT T

IfE EH TSE 0S

E EG Hor

The nonideality is totally coursed by HE ---the regular solutions

Activity of regular solutions

ΔmixG = nRT(xA ln xA + xB ln xB + βxAxB)

GE = nβRTxAxB

Margules equations2ln BA x 2ln AB x

EA BH n RTx x

Benzene/CyclohexaneFor different β

Understanding meaning of β

)1( 2

2

A

B

xA

xA

AAA

ex

ex

xa

Ideal solution

Follow raoult’s LawFor different β

Positive β

Negative β

)1(* 2Ax

AAA exPP

About regular solution

and E

BB

E

B( ) ln p

GRT

n

so2 E

B

( ) 0p

G

n T

E 0S E E

E

B

( ) 0 ( )p p

S GS

n T

Excess chemical potentialB

B

B

( ln )[ ] 0

ln tan

1ln

p

RT

TRT cons t

T

Athermal solutions

E EG TS

E E EG H TS

The nonideality of the solution coursed by excess entropy

E ETS H E 0H or

， soE 0H E

B

( ) 0p

H

n

lnso ( ) 0B

pT

E2

E

B B

( ) 1and [ ] ( ) 0p p

GGT

n T T T n

4.8 Thermodynamic discussion about extraction

B

B

cK

c

distribution coefficient of B in two different solvent αand β

B B

* *B B B B( ) ln ( ) lnRT a RT a

* *B B B

B

( ) ( )exp[ ] ( , )

aK T p

RTa

Example 291K, extraction of methylamine in chloroform by

water

49.8chloroform

water

c

c

V1, volume of chloroform solution

V2, volume of water

21

11

VKV

KV

m

m

nn

VKV

KV

m

m)(

21

1

n, the times of extraction

Review: About solution Solute B+ solvent A= solution

L(xA,xB)

Gas(yA,yB)

S(A)

B B B BB B

d 0 or d 0n z x z Gibbs-Duhem equation

A A B B

BA B

A

d d 0

d d

x V x Vx

V Vx

A B

A B

A A B B

A A B B

ln ln ( ) ( )

ln ln

or ( ) ( )

T T

T T

p p

x x

x p x p

p x p x

Duhem-Margulesequation

The chemical potential

Ideal gas i = i + RT ln ( pi /p

)

solution i (sln) = i(g) = i (g) + RT ln (pi / p

)

2 2 ((slnsln)= )= 22((slnsln)) + + RTRT ln( ln(aa22))

2 2 ((slnsln)= )= 22((slnsln)) + + RTRT ln( ln(xx22) )

Real gas i = i + RT ln ( fi /p

)

pi = pi*(pure liquid) = pi*xi (ideal solution) = Kh,x xi (diluted solution)

= Kh,mm

= Kh,c c

Nonideal solution：p2 = p2*a2 (R.L) = Kh,x a2 (H.L)

= Kh,m a2

= Kh,c a2

x2, m/m , c/c

Diluted solution

Real solution

jnpTii

def

i n

GG

,,

Homework

Discussion: Group I: Thermodynamics in Dialysis technique

Group II: Osmosis phenomenon in biological systems

Group III: If the solvent of an diluted solution follow Raoult’s Law, the solute would follow the Henry’s Law.