4.6 real solution: activity of solute and solvent or by henry’s law judged by raoult’s law γ...
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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.