1. Solid-vapor equilibrium (SVE) and Solid-liquid equilibrium (SLE)
Chapter 14-Part VIII
Solid-vapor equilibrium (SVE)
and
Solid-liquid equilibrium (SLE)

2. A solid can vaporize at T < T triple point; pressures along the sublimation curve are called
saturation pressures of the solid

3. Lets consider SVE of a pure solid (1) and a vapor mixture
Species 2 does not dissolve in the solid phase;
In the vapor phase usually 2 is the solvent; 1 is the solute
Vapor mixture
of 1 and 2
Solid 1
We want to calculate the solubility of 1 in the vapor phase as a
function of T and P

4. SVE for component 1:
We model the solid phase with the same equation of the liquid
(is a condensed phase)

5. Solubility of 1 in the vapor phase
How this expression may reduce at low pressures?

6. Simplifications to the equation
Solubilities of solids in fluids at high pressures important for separation processes
Examples: extraction of caffeine from coffee, separation of asphaltenes from heavy
petroleum fractions
Usually P1sat is very small;
Saturated vapor can be considered ideal gas
Also if y1 is very small,

7. Simplified equation
The fugacity coefficient at infinite dilution can be calculated from an EOS
Where aij is calculated as
aij =(1-lij)(aiaj)1/2
lij is a cross-coefficient for the mixture

9. Solubility of a solid in a gas
Estimate the solubility of naphthalene in carbon dioxide at 1 bar and
temperatures of 35 and 60.4 oC assuming that the solid is incompressible,
and the solid and fluid phases may be considered ideal.

10. Solving at each temperature
T = 35 oC; Pnsat = x10-4 bar; yN =
T = 60.4 oC; Pnsat = x10-3 bar; yN =0.0024

11. Lets consider the effect of pressure
Estimate the solubility of naphthalene in carbon dioxide at 1 bar and
temperatures of 35 oC and pressures from 1 bar to 60 bar using the virial
equation of state with the following values for the second virial coefficient
B(CO2-naphthalene) = -345 cc/mol
B (CO2-CO2) = cc/mol
B(naphthalene-naphthalene) = cc/mol
Assume that CO2 is insoluble in solid naphthalene, and therefore only equate the
fugacities of naphthalene in the solid and vapor phases.
Since the fugacity coefficient
is a function of y1
iterate

12. SLE: solid-liquid equilibria

13. Solid-liquid equilibria (SLE)

14. Solid-Liquid Equilibria (SLE)

15. SLE-cont.
T-effect on fugacity

16. SLE-cont. The enthalpies are functions of T
(through the Cp dependence on T)

17. SLE (cont.)

18. Simplifications Triple integral I is usually neglected
Heat capacity change of melting usually not available
Therefore
For all components

19. SLE-Typical cases A) Assume ideal-solution behavior for both phases
for all T and compositions
Gives a normal
T-x phase diagram

20. SLE-typical cases B) Assume ideal behavior for the liquid phase
and complete immiscibility for all species in the solid state=>

21. For case B: Liquid + solid 1 Liquid + solid 2 Both equations apply
at the eutectic point

22. Example: SLE
Estimate the solubility of solid naphthalene in liquid n-hexane at 20 oC.
Data
Naphthalene MW =
Melting point: 80.2 oC
Heat of fusion: kJ/mol
Density of the solid: g/cc at 20oC
Density of the liquid: g/cc at 100 oC
Vapor pressure of the solid: log P (bar) = /T (T in K)
The heat capacities of liquid and solid naphthalene may be assumed to be equal
If DCp =0,
The result is x1 = 0.269, the experimental result is x1 = 0.09
How can we correct the answer?

23. Same example using UNIFAC to estimate the naphthalene activity coefficient
Naphthalene has 8 aromatic CH (subgroup 10) and 2 aromatic C (subgroup 11)
N-hexane has 2 CH3 (subgroup 1) and 4 CH2 (subgroup 2) groups.

24. Since g1 is a function of x1, we need to iterate in x1
Solving for x1 yields x1 =0.124,
38% larger than the experimental value

25. Calculation of activity coefficient from solubility data
The following data has been reported for benzo-pyrene and
its solubility in water at 25 oC:
Melting point: oC
Heat of fusion: 15.1 kJ/mol
Solubility in water: xBP =3.37x10-10
Estimate the activity coefficient of benzo-pyrene in water at 25 oC

26. Calculation is similar to previous example
Note that in this case the correction given by the activity coefficient is HUGE !!!!
Important in applications when we are interested in the distribution of a chemical species
between air, water, soil. Since the concentration of benzopyrene is small, the
calculated activity coefficient is the infinite diluted activity coefficient.