1. Liquid-Liquid Equilibrium

2. The use of the stability criteria
If we test some liquid mixtures (we think they are a single liquid phase), and we discover that they do not satisfy the stability criteria
Such mixtures will split into two liquid phases of different compositions
This is important for many separation processes such as solvent extraction

3. Liquid-Liquid Extraction
As a lab process
As an industrial process

4. LLE equilibrium criteria
and b are two liquid phases
how do we model fugacity in a liquid phase?

5. Notice: Activity coefficients are functions of compositions in the respective phases, T, and P

6. LLE at constant P or at low temperature (weak P-dependence)—binary system
How many equations and how many unknowns?

7. T-x1a-x1b diagram (LLE solubility diagram)

8. Computing a T-x diagram
Fix T and solve for compositions of component 1 in the two liquid phases

9. Prediction of LLE is strongly dependent on the chosen GE model
Since we obtain the activity coefficients from GE, the selection of the GE model is crucial to appropriately describe LLE
Some GE models cannot describe LLE (for example the Wilson equation, see example 14.7)
The most advanced UNIQUAC, UNIFAC are able to describe LLE

10. Example: Very low miscibility
ΔGmix = x1lnx1 + x2lnx2 + GE
A=4.0
ΔGmix
X1α X1
X1β

11. Example: very low miscibility
System: oil-water. The amount of water dissolved in oil is extremely small, so the “oil” phase (a) is very dilute in component 1 (water), and the “aqueous” phase (b) is very dilute in component 2 (hydrocarbon)

12. LLE equations for two almost immiscible liquids:
these equations give us estimates of the compositions in both liquid phases based on a model for GE
Usually the activity coefficients at infinite dilution are related in a straightforward way to the model parameters
Alternatively, if we have measured compositions in the liquid phases, we can determine the activity coefficients at infinite dilution

13. The simplest GE model that predicts LLE
For two phases and LLE:

14. With Margules 1-parameter model:
Notice: Solubility curves for this model are symmetric with respect to x1 =0.5

15. Symmetry of 1-parameter Margules model
spreadsheet
ΔGmix = x1lnx1 + x2lnx2 + GE
ΔGmix X1
X1α
X1β
Phase α
X1α
X2α
Phase β
X2β
X1β

16. Only because the curves are symmetric:
Notice:
A is a function of temperature !!!
A > 2  3 roots: x1a, x1b, ½
A=2  3 roots =1/2
A < 2  only one root =1/2

18. A=3.0
A=2.5
ΔGmix
ΔGmix X1 X1
A=2.0
A=1.5
ΔGmix
ΔGmix X1 X1

19. 1. How equilibrium compositions vary with temperature?
2. Does A increase or decrease with temperature?

20. Temperature dependence of A
For example:
The excess enthalpy and the T-dependence of A
are directly related

21. When A =2 we have a consolute point (could be upper or lower or both)
dA/dT < 0, is a UCST
HE > 0  endothermic
HE <0  exothermic
dA/dT > 0, is a LCST
When A =2 we have a consolute point
(could be upper or lower or both)

23. Depending on the values of a, b, and c this equation may have 0, 1, 2, or 3
Temperature roots
For example
TL = K and TU =391.2 K

24. HE changes sign in the temperature interval of LLE
dA/dT<0
dA/dT>0
A=2
From the graph A vs T we know if there is an UCST or a LCST

25. Only one root: T = 346 K; it is a UCST; HE is >0
A>2, dA/dT <0
A<2, dA/dT <0
Only one root: T = 346 K; it is a UCST; HE is >0

26. Only one root, T = 339.7 K, there is a LCST
A>2, dA/dT >0
A<2, dA/dT >0
Only one root, T = K, there is a LCST

28. Conclusion
The model GE/RT =A x1 x2 cannot predict LLE for values of A <2
From the stability criteria we said that

29. When x1 = x2 =1/2 ,minimum value rhs

30. Note that if the model is written as GE = A x1 x2, then we can obtain an expression
for the critical solution temperature:

31. Lever rule VLE

32. Lever rule LLE