Activity Coefficient Estimation
Methods
Bharat Chandramouli
February 5, 2002
Activity Coefficient

The activity coefficient is a measure of
the non-ideality of mixing
Gs  RT ln x w  RT ln  w



Gmix  ideal




Gse nonideal
Two components, Enthalpic and
Entropic
Estimation/Measurement

Activity coefficients in single
component/simple mixtures easy to
measure
 sat 

1
x sat
Activity coefficients in water or octanol
can be calculated from solubility given
sufficiently sensitive methods
Need for estimation



What about complex mixtures?
What about dynamic systems with
changing compositions?
It becomes more practical to use
estimation methods to approximate  in
these cases
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
UNIFAC

The activity coefficient is calculated
from two components
ln   ln 
i
i
C
 ln 
i
R
Combinational
Residual (interactions)
(V, SA)
(Experiment Fit)
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
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”
CH2 (2)
0.6744
0.540
0, 0
OH
“OH”
OH (2)
1.000
1.200
986.5, 156.4
UNIFAC Methods


Interaction parameters are fit from
experimental data
This work is still ongoing and many
parameters still not available
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
Hansen Activity Coefficient

The activity coefficient is given by
i
i
ln  om ( V / RT )
i ,om
A
i ,om
d
Cohesive energy density
Molar Volume
Size effect term
Enthalpy
Entropy
The Size Effect Term
d is a measure of the effect of
 i,om


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
i ,om
i
i
d  ln( V /V )  1 V /V
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
i

i
i
i
Ecoh  Ed ( ispersive)  E p( olar )  Eh( ydrogen )
Theory of cohesive energy developed by
Hildebrand for dispersive systems and
extended by Hansen for polar and hydrogen
bonding
Solubility parameter

Solubility parameters are measures of
cohesive energy
1/ 2
  ( e coh )
solubility parameter
i
1/ 2
 ( Ecoh / V )
cohesive energy
coh. energy density
Calculating solubility
parameters

Hansen and others compiled molar
attraction constants for functional
groups, which are additive contributions
to the solubility parameter
i
 d   Fd ,k / i V
i
 p  (  F p2,k )1/ 2 / i V
i
 p  (  Eh ,k / V )
i
1/ 2
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
Multi-component Mixtures


How are om parameters calculated?
Parameters weighted using component
mole fraction and molar volume to get
“average om”
om
 d   ( i x  Fd ,k ) /Vom
om
 p  [( x 
om
 h  [  ( i x  Eh ,k ) /Vom ]1/ 2
i
2
1/ 2
F p ,k )] /Vom
Cohesive Energy Density
i,omA

om
i
can be derived as
2
i
om
i
2
i
om
i
(  d   d )  b(  p   p )  b(  h   h )
 i

b is a weighting factor based on
2
dispersive forces, has been tabulated
for a variety of compounds
ib corrects for the fact that polar and H
bonding forces are localized
Activity Coefficient

Putting the two components together
ln iom =
i
om
i
2
i
om
i
2
i
om
i
2
V [(  d   d )  b(  p   p )  b(  h   h ) ]/ RT
+
ln( i V / om V )  1 i V / om V
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
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
Where do you use this?
1.
Water solubility estimation
x wsat
2.

1
 wsat
Solvent-Water partitioning (Kow)
K sw
1
1 1
 sat
C w (1, L )  s Vs
Gas/Particle Partitioning
gas
Thermodynamic Equilibrium?
Kp
Cgas  particle  C part
Temperature
Humidity

particle
Compound
Particle type
What happens when a semivolatile organic
(SOC) encounters a particle??
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
Predictive Partitioning models

Pankow (1994) for absorptive
partitioning
7.501RTf om
Kp  9
i
0
10 MWom  om p L
fom- fraction extractable organic matter
i
om
- activity coefficient of SOC in om
MWom - molecular weight of om