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

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

3.  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

4. + Work State 1 State 2 Pure water

5. + Work State 1 State 2 Salt water + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + + - - + - + - + - + - + - + - - + - + - + + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + + - - + - + - + - + - + - + - - + - + - +

6. + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + + - - + - + - + - + - + - + - - + - + - + + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + + - - + - + - + - + - + - + - + - + - + - + Work State 1 State 2

7. + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + + - - + - + - + - + - + - + - + - + - + - + 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

8. Images that explain physically why activity coefficients are needed

9. Material Resistivity, R @20C ρ (Ω·m) Copper10 -8 Carbon steel (1010) 10 -7 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 c @20-25C 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

10. 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

11. 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

12. Na + Layer Inner layer radius Outer layer radius Layer volume H 2 O in layer H 2 O in sphere ÅÅnm3NN 10.83.50.266 23.56.20.82733 36.28.91.96497 48.911.63.6119216 511.614.35.7189405 614.317.08.3276681 717.019.711.43801061 819.722.415.05001561 922.425.119.16362197

14. Na + Cl - distance

15.  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

16. 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 -9 5792.78e101.000 1e -5 1e -8 2702.78e91.000 1e -4 1e -7 1252.78e81.000 1e -3 1e -6 582.78e70.999 1e -2 1e -5 272.78e60.996 1e -1 1e -4 12.52.78e50.988 10.0015.8277750.965 100.012.727780.902 1000.11.252780.771 100010.58280.644

18. * 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

19. 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

20. * 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. 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 = 1.825 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.

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

23.  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

24. 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

25. 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. 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. 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. 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. 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