1. Activity Coefficient Estimation Methods
Bharat Chandramouli
February 5, 2002

2. Activity Coefficient
The activity coefficient is a measure of the non-ideality of mixing
Two components, Enthalpic and Entropic

3. Estimation/Measurement
Activity coefficients in single component/simple mixtures easy to measure
Activity coefficients in water or octanol can be calculated from solubility given sufficiently sensitive methods

4. Need for estimation What about complex mixtures?
What about dynamic systems with changing compositions?
It becomes more practical to use estimation methods to approximate g in these cases

5. Estimation Methods
Group contribution methods are most common because they have predictive ability
There are two group contribution methods commonly used for iom
calculation from solubility parameters
UNIFAC calculation

6. UNIFAC The activity coefficient is calculated from two components
Combinational
(V, SA)
Residual (interactions)
(Experiment Fit)

7. UNIFAC The group contribution components consist of
volume contributor -Rk
surface area contribution -Qk
interaction parameter between functional groups amk
To calculate interactions, similar sub-groups are assigned to groups and interactions are between these groups
Calculate activity coefficients by summing all contributions and interactions

8. UNIFAC-Simple example
Ethanol CH3-CH2-OH
Main Group.
Subgroup
Rk (vol)
Qk (SA)
Amk
CH3
“CH2”
CH3 (1)
0.9011
0.848
0, 0
CH2
CH2 (2)
0.6744
0.540 OH
“OH”
OH (2)
1.000
1.200
986.5, 156.4

9. UNIFAC Methods Interaction parameters are fit from experimental data
This work is still ongoing and many parameters still not available

10. Hansen Solubility Parameter
This method calculates activity coefficients from the solubility parameter
Theory of cohesive energy developed by Hildebrand for dispersive systems and extended by Hansen for polar and hydrogen bonding

11. Hansen Activity Coefficient
The activity coefficient is given by
Cohesive energy density
Molar Volume
Size effect term
Enthalpy
Entropy

12. The Size Effect Term
i,omd is a measure of the effect of differing sizes of i and om on their entropy of mixing
This was derived by Flory and Huggins using statistical thermodynamics
For an infinitely dilute solution

13. Cohesive Energy (Ecoh)
Closely linked to the heat of evaporation
It is a measure of a the ability of a liquid molecule to stay together
Theory of cohesive energy developed by Hildebrand for dispersive systems and extended by Hansen for polar and hydrogen bonding

14. Solubility parameter
Solubility parameters are measures of cohesive energy
cohesive energy
solubility parameter
coh. energy density

15. Calculating solubility parameters
Hansen and others compiled molar attraction constants for functional groups, which are additive contributions to the solubility parameter

16. Attraction Constants (F)
The product of V was found to vary linearly across homologous series
Additivity of structural sub-groups
F = V values compiled for dispersion and polar components of 
Hansen later compiled additive contributions to Eh

17. Multi-component Mixtures
How are om parameters calculated?
Parameters weighted using component mole fraction and molar volume to get “average om”

18. Cohesive Energy Density
i,omA can be derived as
ib is a weighting factor based on dispersive forces, has been tabulated for a variety of compounds
ib corrects for the fact that polar and H bonding forces are localized

19. Activity Coefficient
Putting the two components together
ln iom = +

20. Calculation
First, calculate group contributions for each component in the mixture
Calculate “om” parameters by weighting with mole fraction and molar volume
Calculate parameters for compound of interest
Calculate activity coefficient

21. Hansen or UNIFAC? UNIFAC more powerful interaction
UNIFAC not universal–missing parameters
Hansen has certain inconsistencies as certain parameters have to be culled from different sources. Very sensitive to parameter choice
ib not widely available for many compounds, so estimation may be difficult

22. Where do you use this? Water solubility estimation
Solvent-Water partitioning (Kow)

23. Gas/Particle Partitioning
What happens when a semivolatile organic (SOC) encounters a particle??
particle
gas
Thermodynamic Equilibrium?
Temperature
Humidity
Particle type
Compound
This picture is a simplistic overview of gas/particle partitioning. What we study

24. Partitioning Modes
Mode of SOC-particle interaction depends on the particle
Adsorption Solid particle, no organic liquid layer (dust, inorganic salts)
Absorption Particle either liquid, or has substantial liquid layer (combustion particles, secondary organic aerosol)
SOCs such as PAHs, and alkanes primarily partition to organic or carbonaceous aerosols rather than to mineral-based aerosols
To go beyond vapor pressure, we first have to consider the mode of partitioning, which depends on the type of particle we are dealing with. Adsorption, where the particles are solid and the SOC interacts with the surface. Or, with liquid particles with a substantial organic layer, the SOC interacts with the liquid layer and the partitioning is more of a bulk phenomenon. Many studies have shown that SOCs tend to partition to organic aerosols to a much greater extent, because Kp values tend to be higher.

25. Predictive Partitioning models
Pankow (1994) for absorptive partitioning
This is the model we use to predict partitioning on aerosols with an organic liquid layer.
fom- fraction extractable organic matter
igom - activity coefficient of SOC in om
MWom - molecular weight of om