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5.60 Thermodynamics & Kinetics
Spring 2008
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5.60 Spring 2008
Lecture #22
Two-Component Phase Equilibria III
Ideal and Non-Ideal Solutions
Free energy change ∆Gmix in ideal solutions
∆G
A (liq)
B (liq)
G1 (l) = nAxAµA* (l) + nBxBµB* (l)
A + B (liq)
G2 (l) = nAxAµAmix (l) + nBxBµBmix (l
) ∆Gmix = G2 (l) − G1 (l)
= nxA ⎡⎣µA∗ (l) + RTln xA ⎤⎦ + nxB ⎡⎣µB∗ (l) + RTln xB ⎤⎦ − nxAµA∗ (l) − nxBµB∗ (l)
∆Gmix = nRT ( xA ln xA + xB ln xB )
Purely entropic, as in gas mixture
G = Vdp – SdT
⎛ ∂∆Gmix ⎞
∆Smix = − ⎜⎜
⎟⎟ = −nR ( xA ln xA + xB ln xB )
∂T
⎝
⎠p
∆Hmix = ∆Gmix + T∆Smix = 0
No enthalpy change, all of ∆G is due to entropy of mixing
Change in volume ∆Vmix
⎛ ∂∆Gmix ⎞
⎟⎟ = 0
⎝ ∂p ⎠T
∆Vmix = ⎜⎜
No volume change, just like ideal gas.
page 1
5.60 Spring 2008
Lecture #22
page 2
Non-Ideal Solutions:
In reality, molecules interact:
A
B
A
A
A
B
B
B
uAA < 0
PURE
uBB < 0
uAB
MIXED
uAB
∆u = 2uAB- (uAA + uBB)
This difference determines how solutions depart from ideality.
I.
Positive Deviations: ∆u > 0 (most common)
Mixing is energetically not favorable in liquid phase.
∆H = ∆U + ∆(PV) ≈ ∆U
∆Gmix =
n
∆u + nRT ( xA ln xA + xB ln xB ) > ∆Gmix (ideal)
4
e.g. acetone & carbon disulphide
CH3
CH3
C=O
A
+
S=C=S
B
5.60 Spring 2008
Lecture #22
page 3
p = pA + pB
p
Real
*
pCS
= pB*
2
*
pacetone
= pA*
Raoult
pB
xCS2 = xB
0
1
pA
*
pCS > xCS pCS
2
2
2
*
pacetone > xacetonepacetone
⇒ ptot (real) > ptot (Raoult )
⇒ vapor pressure is higher than expected by Raoult’s Law
II.
Negative Deviations: ∆u < 0
e.g. acetone & chloroform
CH3
CH3
Cl
C=O
+
H - C - Cl
Cl
A
B
H-bonding
attraction
Mixing is energetically favorable in liquid phase.
5.60 Spring 2008
Lecture #22
page 4
p = pA + pB
p
Real
*
pCHCl
= pB*
3
*
pacetone
= pA
*
Raoult
pB
xCHCl3 = xB
0
pA
1
*
pCS < xCS pCS
2
2
2
*
pCHCl < xCHCl pCHCl
3
3
3
⇒ ptot (real) < ptot (Raoult )
Ideal Dilute Solutions and Henry’s Law:
Non-ideal solutions are difficult to describe completely
⇒ Describe limits xB → 1 and xB → 0 ⇒ “Ideal Dilute Solution”
I. xCS2 = xB → 1
(B is the “solvent”)
Then Raoult’s Law applies for CS2
CS2 molecules see mostly other CS2 molecules
*
pCS = xCS pCS
2
2
2
5.60 Spring 2008
Lecture #22
page 5
II. xB → 0 (B is the “solute”) Then Henry’s Law applies:
pCS = xCS KCS
2
2
2
pB = xBKB
Henry’s Law constant
KB ≡ Henry’s Law constant, depends on the solvent/solute mixture and the temperature.
Labeled just KB, even though it depends on A also. KCS2 = KB
Real
p
Positive deviation ⇒ KB > pB*
Henry
*
*
pCS
=
p
B
2
(Negative deviation would
have KB < pB* )
Raoult
0
xCS2 = xB
Ideal dilute solution:
Solvent, e.g. A:
Solute, e.g. B:
xA ~ 1
xB ~ 0
⇒
⇒
Raoult’s Law
Henry’s Law
pA = xA pA*
p B = x B KB
1
5.60 Spring 2008
Lecture #22
page 6
Total phase diagram:
Ideal
p
Real
p
Liquid phase
pA*
pB*
pA*
gas
phase
xB,yB 0
Liquid
phase
pB*
1
gas
phase
xB,yB
0
1
Stronger non-ideality: Azeotropes
Liquid
phase
p
pB*
*
pA
At this point
xB = yB
gas phase
xB,yB
0
1
Cannot distill
past azeotrope
xB = yB
Similar for constant p phase diagram
Can have positive or negative azeotropes
T
T
gas
phase
TA*
TA*
TB*
Liquid
phase
0
gas
phase
xB,yB
1
TB*
Liquid
phase
0
xB,yB
1