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ACTIVITY. Activity Coefficients. No direct way to measure the effect of a single ion in solution (charge balance) Mean Ion Activity Coefficients – determined for a salt (KCl, MgSO 4 , etc.) g ±KCl = [( g K )( g Cl )] 1/2 K sp = g ±KCl 2 (mK + )(mCl - )

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Activity coefficients
Activity Coefficients

  • No direct way to measure the effect of a single ion in solution (charge balance)

  • Mean Ion Activity Coefficients – determined for a salt (KCl, MgSO4, etc.)

    g±KCl = [(gK)(gCl)]1/2

    Ksp= g±KCl2(mK+)(mCl-)

  • MacInnes Convention  gK = gCl= g±KCl

    • Measure other salts in KCl electrolyte and substitute g±KCl in for one ion to measure the other ion w.r.t. g±KCl and g±salt

Debye h ckel

  • Assumes ions interact coulombically, ion size does not vary with ionic strength, and ions of same sign do not interact

  • A, B often presented as a constant, but:

    A=1.824928x106r01/2(T)-3/2, B=50.3 (T)-1/2

    Where  is the dielectric constant of water and r is the density

Higher ionic strengths
Higher Ionic Strengths

  • Activity coefficients decrease to minimal values around 1 - 10 m, then increase

    • the fraction of water molecules surrounding ions in hydration spheres becomes significant

    • Activity and dielectric constant of water decreases  in a 5 M NaCl solution, ~1/2 of the H2O is complexed, decreasing the activity to 0.8

    • Ion pairing increases, increasing the activity effects

Extended Debye-Hückel

  • Adds a correction term to account for increase of gi after certain ionic strength

  • Truesdell-Jones (proposed by Huckel in 1925) is similar:

Davies equation
Davies Equation

  • Lacks ion size parameter –only really accurate for monovalent ions

  • Often used for Ocean waters, working range up to 0.7 M (avg ocean water I)

Specific ion interaction theory
Specific Ion Interaction theory

  • Ion and electrolyte-specific approach for activity coefficients

  • Where z is charge, i, m(j) is the molality of major electrolyte ion j (of opposite charge to i). Interaction parameters, (i,j,I) describes interaction of ion and electrolyte ion

  • Limited data for these interactions and assumes there is no interaction with neutral species

Pitzer model
Pitzer Model

  • At ionic strengths above 2-3.5, get +/+, -/- and ternary complexes

  • Terms above describe binary term, fy describes interaction between same or opposite sign, terms to do this are called binary virial coefficients

  • Ternary terms and virial coefficients refine this for the activity coefficient

Setchenow equation
Setchenow Equation

log gi=KiI

  • For molecular species (uncharged) such as dissolved gases, weak acids, and organic species

  • Ki is determined for a number of important molecules, generally they are low, below 0.2  activity coefficients are higher, meaning mi values must decline if a reaction is at equilibrium  “salting out” effect

Half reactions
Half Reactions

  • Often split redox reactions in two:

    • oxidation half rxn  e- leaves left, goes right

      • Fe2+ Fe3+ + e-

    • Reduction half rxn  e- leaves left, goes right

      • O2 + 4 e-  2 H2O

  • SUM of the half reactions yields the total redox reaction

    4 Fe2+ 4 Fe3+ + 4 e-

    O2 + 4 e-  2 H2O

    4 Fe2+ + O2  4 Fe3+ + 2 H2O

Electron activity

  • Although no free electrons exist in solution, it is useful to define a quantity called the electron activity:

  • The pe indicates the tendency of a solution to donate or accept a proton.

  • If pe is low, there is a strong tendency for the solution to donate protons - the solution is reducing.

  • If pe is high, there is a strong tendency for the solution to accept protons - the solution is oxidizing.

The pe of a half reaction i

Consider the half reaction

MnO2(s) + 4H+ + 2e- Mn2+ + 2H2O(l)

The equilibrium constant is

Solving for the electron activity

Definition of eh

Eh - the potential of a solution relative to the SHE.

Both pe and Eh measure essentially the same thing. They may be converted via the relationship:

Where  = 96.42 kJ volt-1 eq-1 (Faraday’s constant).

At 25°C, this becomes


Free energy and electropotential
Free Energy and Electropotential

  • Talked about electropotential (aka emf, Eh)  driving force for e- transfer

  • How does this relate to driving force for any reaction defined by DGr ??

    DGr = - nE

    • Where n is the # of e-’s in the rxn,  is Faraday’s constant (23.06 cal V-1), and E is electropotential (V)

  • pe for an electron transfer between a redox couple analagous to pK between conjugate acid-base pair

Nernst equation
Nernst Equation

Consider the half reaction:

NO3- + 10H+ + 8e- NH4+ + 3H2O(l)

We can calculate the Eh if the activities of H+, NO3-, and NH4+ are known. The general Nernst equation is

The Nernst equation for this reaction at 25°C is

Let’s assume that the concentrations of NO3- and NH4+ have been measured to be 10-5 M and 310-7 M, respectively, and pH = 5. What are the Eh and pe of this water?

First, we must make use of the relationship

For the reaction of interest

rG° = 3(-237.1) + (-79.4) - (-110.8)

= -679.9 kJ mol-1



Upper stability limit of water eh ph

To determine the upper limit on an Eh-pH diagram, we start with the same reaction

1/2O2(g) + 2e- + 2H+ H2O

but now we employ the Nernst eq.

As for the pe-pH diagram, we assume that pO2 = 1 atm. This results in

This yields a line with slope of -0.0592.

Lower stability limit of water eh ph

Starting with

H+ + e- 1/2H2(g)

we write the Nernst equation

We set pH2 = 1 atm. Also, Gr° = 0, so E0 = 0. Thus, we have

Construction of these diagrams
Construction of these diagrams

  • For selected reactions:

    Fe2+ + 2 H2O  FeOOH + e- + 3 H+

    How would we describe this reaction on a 2-D diagram? What would we need to define or assume?

  • How about:

  • Fe3+ + 2 H2O  FeOOH(ferrihydrite) + 3 H+


    log K=3 pH – log[Fe3+]

    How would one put this on an Eh-pH diagram, could it go into any other type of diagram (what other factors affect this equilibrium description???)

Incongruent dissolution

  • Aluminosilicate minerals usually dissolve incongruently, e.g.,

    2KAlSi3O8 + 2H+ + 9H2O

     Al2Si2O5(OH)4 + 2K+ + 4H4SiO40

  • As a result of these factors, relations among solutions and aluminosilicate minerals are often depicted graphically on a type of mineral stability diagram called an activity diagram.

Activity diagrams the k 2 o al 2 o 3 sio 2 h 2 o system

We will now calculate an activity diagram for the following phases: gibbsite {Al(OH)3}, kaolinite {Al2Si2O5(OH)4}, pyrophyllite {Al2Si4O10(OH)2}, muscovite {KAl3Si3O10(OH)2}, and K-feldspar {KAlSi3O8}.

The axes will be aK+/aH+ vs. aH4SiO40.

The diagram is divided up into fields where only one of the above phases is stable, separated by straight line boundaries.

Activity diagram showing the stability relationships among some minerals in the system K2O-Al2O3-SiO2-H2O at 25°C. The dashed lines represent saturation with respect to quartz and amorphous silica.