Activity Coefficient Estimation Methods

Bharat ChandramouliFebruary 5, 2002

Activity Coefficient The activity coefficient is a measure

of the non-ideality of mixing

Two components, Enthalpic and Entropic

idealnonideal

lnln

esmix G

wG

ws RTxRTG

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

satsat x

1

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 i

om

calculation from solubility parameters UNIFAC calculation

UNIFAC the UNIversal Functional group Activity Coefficient model The activity coefficient is

calculated from two componentsRiCii lnlnln

Combinational

(V, SA)

Residual (interactions)

(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-OHMain Group.

Subgroup Rk (vol) Qk (SA) Amk

CH3 “CH3” 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

dARTV omiomiiom

i ,,)/(ln

Molar VolumeCohesive energy density

Size effect term

Enthalpy Entropy

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 solutionVVVVd iiomi /1)/ln(,

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

)()()( ydrogenhi

olarpi

ispersivedi

cohi EEEE

Solubility parameters are measures of cohesive energy

2/12/1 )/()( VEe icohcoh

solubility parametercoh. energy density

cohesive energy

Solubility parameter

Calculating solubility parameters

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

2/1,

2/12,

,

)/(

/)(

/

VE

VF

VF

ikhh

i

ikpp

i

ikdd

i

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”

2/1,

2/12,

,

]/)([

/)]([

/)(

omkhi

hom

omkpi

pom

omkdi

dom

VEx

VFx

VFx

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

222 )()()( hi

homi

pi

pomi

di

dom bb

Activity Coefficient Putting the two components

together

RTbbV hi

homi

pi

pomi

di

domi /])()()[( 222

VVVV omiomi /1)/ln(

+

ln iom =

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

2. Solvent-Water partitioning (Kow)

satw

satwx

1

sssatw

sw VLCK 11

),1(1

Gas/Particle Partitioning

What happens when a semivolatile organic (SOC) encounters a particle??

particle

gas

Particle typeCompound Temperature

Humidity

Thermodynamic Equilibrium?

partK

gas CparticleC p

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

0910501.7

Lomi

om

omp pMW

RTfK

fom- fraction extractable organic matterom - activity coefficient of SOC in om MWom - molecular weight of om