We’ve all heard of them; Why do they exist; how are they computed

 A work term – the amount of energy needed relative to a reference, or starting point 1000 kg Standard Energy Cost Real Energy Cost

 The Physical Chemist’s secret: the activity coefficient  A numerical correction applied to a model to predict properties when systems do not behave ideally  The ultimate fudge factor  An Activity Coefficients is a mathematical correction that describes the effects of the immediate environment on a dissolved ion.  The numerical value is the quantitative deviation from ideality - a value of 1 is ideal  What are the causes  Ideally -Ions behave independently, in dilute solutions because they are relatively far apart  Nonideally - At higher concentrations, an ion’s electric field affects other ions and changes their behavior

+ Work State 1 State 2 Pure water

+ Work State 1 State 2 Salt water

Work State 1 State 2

Ions are distributed in water. Simplify the system with a continuous dielectric medium with a swarm of charge no discrete charges + Reference ion Ion cloud with charge density  r Distance r from central ion r Volume element dV Electrostatic potential  r

Images that explain physically why activity coefficients are needed

Material Resistivity, ρ (Ω·m) Copper10 -8 Carbon steel (1010) Sea water10 -1 Drinking water10 3 Deionized water10 5 Glass10 12 Hard Rubber10 13 More Insulating of current Resistance of electrons to move at a fixed potential Material Dielectric constant, D Air1 Hard Rubber3 Asbestos4.8 Olive oil3.1 Acetone21 Ethanol24 Ethylene glycol37 Deionized water80 More shielding of electric field Resistance of electric field to transmit through substance

Electric field flux Attenuated flux Polarized H 2 O reacting to an Electric field Electron density shifts towards field A dielectric material is one that polarizes in the presence of an electric field electrons shift towards the positive charge, creating an internal electric field E H2O

M +z 3.5 Å. 0.7 Å M +z Slice it in half… A metal ion is a small “point charge” and produces an electric field M +z is relatively small in size, only six water can fit around it (in a sphere of course). H2O molecules orient “coordinate” around it

Na + Layer Inner layer radius Outer layer radius Layer volume H 2 O in layer H 2 O in sphere ÅÅnm3NN

Na + Cl - distance

 Start with 1m 3 deionized water  Add 0.1 mol NaCl  Calculate the ion population density  Invert to obtain the volume per ion  Calculate spherical radius of this volume

NaCl conc, mol/m3 NaCl conc, mol/l Inter-ion distance, nm H 2 O molecules per ion Act Coef,  00  1 1e -6 1e e e -5 1e e e -4 1e e e -3 1e e e -2 1e e e -1 1e e

* Ions can be visualized as point charge surrounded by water * Water shields the electric fields created by each ion * In dilute solutions, the ions are too far apart for the electric fields to overlap Na + Cl - δ-δ- =partial charge around the water sheath Sr +2 =the individual ion =the water sheath or the water oriented around the ion There is no interaction between the two ions, they are too far apart Distance=250 nm r=125 nm

Sr +2 δ+δ+ δ+δ+ δ+δ+ δ+δ+ δ+δ+ δ+δ+ SO 4 -2 δ-δ- δ-δ- δ-δ- δ-δ- δ-δ- δ-δ- Electric fields start to overlap SO 4- 2 δ-δ- δ-δ- δ-δ- δ-δ- δ-δ- δ-δ- Sr +2 δ+δ+ δ+δ+ δ+δ+ δ+δ+ δ+δ+ δ+δ+ When concentration increases, ions distance decreases. Electric fields start to overlap other ions and the water surrounding them This changes ion behavior

* This electric field (electrostatic) effect is quantified using an activity coefficient  : Sr +2 δ+δ+ δ+δ+ δ+δ+ δ+δ+ δ+δ+ δ+δ+ SO 4 -2 δ-δ- δ-δ- δ-δ- δ-δ- δ-δ- δ-δ- The charges are starting to interact SO 4- 2 δ-δ- δ-δ- δ-δ- δ-δ- δ-δ- δ-δ- Sr +2 δ+δ+ δ+δ+ δ+δ+ δ+δ+ δ+δ+ δ+δ+ Where: This is called the Debye Huckel model

21  Most common model for estimating activity coefficients  Assumes salts are completely ionized (limiting law)  Valid up to 0.01 mole/L salinity  Equation is -logγ± = Az+z-√I  where: o γ± = mean of the activity coefficients for the + and - ions; o A is a constant that depends on temperature and the dielectric constant ε (A = x 106(εT)-3/2 = 0.51 at 25◦ C in water); o z+ and z- are the + and - ion charges o I is the ionic strength.

* Activity coefficients becomes part of the equilibrium equation * In dilute solutions,  1, and

 Ionic strength is the sum of all the charges in the solution. where m i = concentration of an individual ion, mole/L z i = electrical charge of the ion

A 1.0 mol/kg solution of NaCl has 1.0 moles of Na +1 ion and 1.0 moles of Cl -1 ion per Kg H 2 O. The ionic strength is 1.0 molal. Ionic strength is a measure of the system’s electric field. Where m i = ion concentration & z i = ion charge Example

A 1.0 mol/kg solution of CaSO4 has 1.0 moles of Ca +2 ion and 1.0 moles of SO4 -2 ion per Kg H 2 O. The ionic strength is 4.0 mol

26  Single Ion Coefficient Model for Ion “z” -logγ z = Az 2 √I  Extended Debye-Hückel equation – valid for ionic strengths >0.01 – 0.1 mole/L -logγ ± = A │ z + z - │ √I 1+Ba√I  where: a is a hydrated size factor and B is a function of the temperature and dielectric constant  Example calculation in text

27  Debye-Huckel model (1922), most common and easiest  Good to about 0.01 mol/L concentration  Extended Debye-Huckel model – extends the concentration limits  Good to about 0.1 mol/L  Davies Equation (1938) – a further extension  Good to about 0.3 mol/L

28  Pitzer Equation  Bromley-Zematis Equation  Helgeson All are good to between 10 and 30 mol/kg H 2 O. The problem is trying to use them…

29  To get better activity coefficients, we need the more complex models  Complex models cannot be solved by hand (they are non-linear and need multiple iterations)  Thus freeware and commercial software products were developed