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, MgSO4, etc.)

±KCl = [(K)(Cl)]1/2

Ksp= ±KCl2(mK+)(mCl-)

• MacInnes Convention K = Cl= ±KCl

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

Mean Ion Activity Coefficients versus Ionic Strength

Debye-Hückel

• 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=1.824928x10601/2(T)-3/2, B=50.3 (T)-1/2

Where is the dielectric constant of water and is the density

IBa

IAz

i

ii

1log

2

IAzii2log

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

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

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

Extended Debye-Hückel

IIBa

IAzAz

i

ii 3.0

1log

22

bIIBa

IAz

i

ii

1log

2

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)

I

I

IAzi 3.0

1log 2

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

k

i jmIjiDz )(),,()log( 2

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

ijk

kjijki

jijii mmEmIDfyz ...)(ln 2

Setchenow Equationlog i=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• 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.

e

ape log

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

24

2

eH

Mn

aa

aK

21

2

4

H

Mne Ka

aa

DEFINITION OF EhEh - 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

or

EhRT

pe303.2

Ehpe 9.16

peEh 059.0

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 Gr ??

Gr = - nE– 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

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

Qn

RTEEh log

303.20

100

3

4log8

0592.0

HNO

NH

aa

aEEh

Let’s assume that the concentrations of NO3- and

NH4+ have been measured to be 10-5 M and

310-7 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

n

GE

or0

volts88.0)42.96)(8(

9.6790 E

O2/H2O

C2HO

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.

20

21

2

1log

0592.0

HO apn

EEh

20

21

2

1log

2

0592.0

HO ap

EEh

As for the pe-pH diagram, we assume that pO2

= 1 atm. This results in

This yields a line with slope of -0.0592.

221

2log0296.023.1

HO apEh

pHpEh O 0592.0log0148.023.12

volts23.1)42.96)(2(

)1.237(00

n

GE r

pHEh 0592.023.1

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

pHEh 0592.0

H

H

a

pEEh

21

2log1

0592.00

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?

2

30 log

1

0592.0

Fe

H

a

aEEh

• How about:

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

Ksp=[H+]3/[Fe3+]

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 K2O-Al2O3-SiO2-H2O 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 a K+/a H+ vs. a H4SiO40.

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

log aH4SiO4

0

-6 -5 -4 -3 -2 -1

log

(aK

+/a

H+)

0

1

2

3

4

5

6

7

KaoliniteGibbsite

Muscovite

K-feldspar

Pyrophyllite

Qua

rtz

Am

orph

ous

silic

a

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.

2 3 4 5 6 7 8 9 10 11 12-10

-8

-6

-4

-2

0

2

pH

log

a A

l++

+

Al+++

Al(OH)2

+

Al(OH)4-

AlOH++

Gibbsite

25oC

Greg Mon Nov 01 2004

Dia

gram

Al+

++,

T =

25

C,

P

= 1

.013

bar

s, a

[H

2O

] =

1

Seeing this, what are the reactions these lines represent?