III. Derivation of Thermodynamic Relationships for Electron Activity in Soils
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R I C H M O N D J. B A R T L E T T A N D BRUCE R. JAMES
3. Then add H+ to balance the moles of H, usually on the left side of the
equation:
+ e- + 3H+
SO:- + 8e- + lOH+
FeOOH
+ 2H,O
= H,S + 4H2O
= Fez+
(5)
(6)
4. Check charge and mass balances for the equations.
These equations can now be used to develop expressions for electron
activity, based on equilibrium expressions and free energy data.
The principles for doing this can be understood by starting with a generic
reduction half-reaction:
A(ox)
+ B(e-) + C(H+) = D(red) + E( H,O)
(7)
where A, B, C, D, and E are reaction coefficients for the oxidized species
(ox), electron (e-), proton (H+), reduced species (red), and water (H,O),
respectively. The expression for the equilibrium constant, K, is equal to
K = (red)D( H Z O ) E / ( ~ ~
(e-)B
) A ( H+)c
(8)
Taking the log of both sides of the equation,
log K
+ log I / ( c ) ~ + log l/(H+)c
= log(red)D/log(ox)A
(9)
And,
log K = D log(red) - A log(ox)
+ B(pe) + C(pH)
(10)
where pe and pH are defined as -log of electron activity and hydrogen ion
activity, respectively.
The “p” notation means “power” and denotes the exponent for the
mol H+/kg water.
hydrogen ion activity, e.g., pH 4 is equivalent to
Therefore, H+ activity equals 10-PH. In contrast, the exponent for eactivity is not lO-w, because the electron activity is not defined in terms of
moles of e- per liter. Both pe and pH are analogous, though, if viewed as
related to the ability of e- and H+ to do thermodynamic work.
Further rearrangement of Eq. (10) yields a pe-pH relationship of the
following form:
[ 1/B log K - D/B log(red)
+ A/B log(ox)] - C/B(pH) = pe
( 1 1)
Equation ( 1 1 ) is the equation for a straight line with slope C/B and the
intercept is the terms in square brackets. It is clear that the intercept is a
function of log K for the half-reaction and the activities of the oxidized and
reduced species.
REDOX CHEMISTRY OF SOILS
157
For a one-electron transfer ( B = 1) coupled with one-proton consump
tion (C = I ) , and when D = A and (red) = (ox),
pe
+ pH = log K
(12)
And at pH = 0 ( 1 M H+ activity),
(13)
pe = log K
Knowledge of pe and pH is pertinent to describing the equilibrium
condition of a soil as defined by the master variables, pe and pH (Lindsay,
1979). The concepts of electron activity and hydrogen ion activity are
closely coupled and cannot be separated in assessing the oxidationreduction status of a soil system.
To relate the concept of log K to the Gibbs free energy change (AG,") for a
given half-reaction in the soil, the following expressions are pertinent:
AG,"= -RTln K
(13)
where AGIO is Gibbs free energy of reaction under standard conditions
(298.15 K and 100 kPa), R is the universal gas law constant (0.001987
kcal/mol/K), and T is absolute temperature (298. I5 K). Converting to
log,, (In K = 2.303 log K ) yields
AG,"/- I .364 = log K
(14)
Therefore, log K may be estimated simply from knowledge of free energies
of formation of H,O, red, and ox, because those of H+ and e- are zero, by
convention. To relate log K directly to pe as defined by Eq. ( I I ) for a
one-electron transfer, log K values must be divided by B.
Log K values also are related to measured electrochemical potentials, 8,,
according to the following expressions:
AGIO= - n98,
(15)
where n is the number of e- transferred/mole and 9 is the Faraday
constant (23.1 kcal/V/equivalent). Because both Eqs. ( 1 5) and ( 1 3) are
expressions for AG,",
-n98,
= -RTln
K
And,
RT 2.303 log K
= 0.059 log K
n9
ifh
=
If n
=
1 and (red) = (ox), pe
+ pH = log K [Eq. (12)], and
(16)
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R I C H M O N D J. BARTLETT A N D BRUCE R. JAMES
or,
(8,/0.059) - pH = pe
(19)
Therefore, calculating and interpreting 8, values rigorously requires a
knowledge of pH of the soil/water system. Equation ( 19)also demonstrates
the coupling of H+ and e- reactions in redox processes.
Applying these principles to the H,S-FeOOH problem above [Eqs. ( 5 )
and (6)], the following pe-pH expressions can be derived:
For FeOOH - Fez+,
pe = log KFe- log(Fs+)- 3pH
(20)
where activity of FeOOH is assumed to be 1, an assumption that may not
be valid for redox processes.
For SO, - H,S,
pe = (1/8)log Ks- (1/8)log pHS/(so4)- (5/4)pH
(21)
where KFeand Ks are the equilibrium constants for the FeOOH and SO,
reduction half-reactions. Calculating log K values and substituting into
Eqs. (20) and (21) yields
pe = 13.0 - log(Fe2+)- 3pH
(22)
pe = 5.2 I - (1/8)log PHS/(SO,) - (5/4)pH
(23)
and
where PHSis partial pressure of H,S in the soil atmosphere. At defined
activities for the ions and H,S, pe may be calculated, and if pe for Fe
reduction is greater than pe for SO, reduction, then FeOOH will be reduced and H,S will be oxidized at equilibrium. The same result will be
obtained if, after subtracting the log K value for S from that for Fe, a
positive answer is obtained.
IV. KINETIC DERIVATION OF THERMODYNAMIC
PARAMETERS FOR REDOX
This thermodynamic derivation of log K and its relationship to pe and
pH of soils is based on free energy of formation data for oxidants and
reductants, and it is not related to reaction mechanisms or rates. An
alternate procedure to obtain log K values is to use a kinetic approach
REDOX CHEMISTRY OF SOILS
1 S9
suggested by Sparks (1985) for cation exchange and by Harter and Smith
( 1981) for adsorption processes. This approach has been little used in redox
soil chemistry, probably due to difficulties in obtaining rate coefficients for
many electron transfer processes in soils. The application of such a p
proaches should, however, be appropriate for certain, reversible redox
reactions in soils, especially in situations where metastability and lack of
chemical equilibrium prevail, or when inaccurate free energy of formation
data are used for oxidants and reductants.
The equilibrium constant, K, for redox equilibria can be estimated from
the ratio of forward and reverse rate coefficients for a given reversible
reaction in soils:
K = kflk
(24)
where k, and k, are the rate constants for forward and reverse reactions,
respectively. The reliability of such an approach depends on the true
reversibility of the redox reactions, an assumption that is not true for many
redox couples, especially those mediated or catalyzed by microbial enzymes.
The Gibbs free energy for the reaction then could be calculated by
substituting In K into Eq. ( 13). Preparation of Arrhenius plots of the rate
coefficientsas a function of 1/T permits estimation of activation energies
for the forward and reverse reactions, and enthalpies for the reactions can
be calculated as
AH a = E:- E:
(25)
where AH' is enthalpy for the redox reaction and E: and E: are the
activation energies for the forward and reverse reactions. With a knowledge
of AG and AH ', entropy of the reaction can be calculated as
AS'
= AHo - AG"/T
(26)
Because the e- and H+ are key reactants, products, and ligands in the
thermodynamic sense, a refined knowledge of mechanisms for particular
redox reactions is needed because of our current limitations to understanding how similar or different e- and H+ reactions are in soils and
natural waters. A knowledge of thermodynamic stability for a redox system
does not necessarily predict kinetic lability, a concept directly pertinent to
reactivity of different types of complexes (Cotton and Wilkinson, 1980).
The kinetic lability of redox processes in soils has not been compared
systematically with predictions of stability based on thermodynamic data.
Such information could affirm the reliability of using thermodynamic data
to predict bioavailability of nutrients and pollutants, and to estimate true
reactivity of electron donors and acceptors in soil environments. The