III. Application of Chemical Kinetics to Soil Solutions
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239
where a and b indicate the reaction order for the individual constituents, [ A ]
and [B] are the concentrations of the reactants A and B, respectively, and k is
the rate constant.
One should realize that the rate law is determined by experimentation and
it cannot be inferred by simply examining the overall chemical reaction
equation. The rate law serves three primary purposes: (1) it permits the
prediction of the rate, given the composition of the mixture and the
experimental value of the rate constant or coefficient; (2) it enables one to
propose a mechanism for the reaction; and (3) it provides a means for
classifying reactions into various orders. The order of a reaction is the
summation of the powers to which the concentrations of the components are
raised in the rate law.
A number of equations have been employed to describe the kinetics of
reactions in clay minerals and soils (Chien et al., 1980; Sparks and Jardine,
1981,1984; Martin and Sparks, 1983; Jardine and Sparks, 1984a). These have
included the firstorder, Elovich, parabolic diffusion, zeroorder, secondorder, and twoconstant rate equations. Since a comprehensive article on
kinetics as applied to soil solutions has not been previously published,
complete derivations of most of these equations will be given. The final forms
of some of the above equations for adsorption kinetics using a miscible
displacement technique are given in Table I.
Table I
Equations Describing the Kinetics of Adsorption
Reactions in Clay Minerals and Soils Using a Miscible
Displacement Technique"
1. Elovich:
C, = a + b In t
2. Parabolic diffusion law:
C,/C,
=a
+ btIi2
3. First order:
log(1  C,/C,) = a  bt
4. Zero order:
(1  CJC,)
a
= a  bt
The terms in each equation are defined in the text.
240
DONALD L. SPARKS
B. BASICEQUATIONS
I . FirstOrder Equations
For a batch technique, if the rate of adsorption of an ion onto a colloidal
surface is proportional to the quantity of the ion remaining in solution, a firstorder equation is expressed as
d(C0  C)/dt = k,C
(5)
where C is the concentration of the ion in solution at time t, C , the initial
concentration of the ion added at time zero, and k, the adsorption rate
coefficient. The integrated form of Eq. (5) is
In C = In C ,  kt
(6)
Thus, plotting In C versus t will yield a straight line of slope  k and an
intercept of In C o if the data conform to firstorder kinetics.
For a miscible displacement technique, if the rate of adsorption of an ion
onto a colloidal surface follows firstorder kinetics, then
d(CJCm) = ka(Cm  Ct) dt
(7)
where C , is the amount of ion on the colloid at time t, C , is the amount of ion
on the colloid at equilibrium, and k, the adsorption rate coefficient. Separating variables and integrating results in
log( 1  CJC,) =  k, C , t
(8)
Since C , is a constant, one can call the product k,C, a constant. Thus,
log(1  C,/C,) = &t
(9)
where & = k,C,/2.303 and is an apparent adsorption rate coefficient.
Using miscible displacement, adsorption will vary only slightly with flow
rate, and since by definition the adsorption rate coefficient k, is constant, a
new term, the apparent adsorption rate coefficient k,, is defined for each flow
rate in the system (Sparks et al., 1980b).
If the rate of desorption of an ion from a colloidal surface follows firstorder kinetics, then for a batch technique
dC/dt = kd(C0  C )
(10)
where C is the amount of ion released at time t, Co the total amount of ion
that could be released at equilibrium, and kd the desorption rate coefficient.
For the initial condition of C = 0 at t = 0, the integrated form of Eq. (10)
becomes
ln(C,  C ) = In Co  kdt
(11)
KINETICS OF IONIC REACTIONS
24 1
If the rate of release of an ion from a colloidal surface follows firstorder
kinetics, then for a miscible displacement technique (for the solid surface
phase)
d(CJC0) = kdC, dt
(12)
where Co is the amount of ion on the exchange sites of the colloid at zero time
of desorption, C, the amount of ion on the exchange sites of the colloid at
time t, and kd the desorption rate coefficient. Integrating with appropriate
boundary conditions, one obtains
h(c,/co) = kdt
(13)
Expressed in terms of base 10 logarithms, we have
log(C/C,)
= &t
(14)
where kd = kJ2.303 and is an apparent desorption rate coeliic;en:.
2. Application of FirstOrder Kinetics to Clay Minerals and Soils
a. Single FirstOrder Reaction. One should realize that the adsorption rate
coefficients (k, and kdrrespectively) determined from the previous firstorder
equations are composed of numerous diffusional and chemical rate constants
(Jardine and Sparks, 1984a). It is appropriate to suggest that for the
adsorption process the rate of adsorption ra is
ra cc k,
+ k2 + k,
(1 5 )
where k, is the rate constant associated with the film diffusion process, k2 the
rate constant associated with intraparticle diffusion or, independently, the
rate constant associated with surface diffusion, and k 3 the chemical rate
constant.
Depending on the type of systems being studied, one or more of the above
rate constants (viz., k , , k2, k 3 ) may be negligible or absent. The rate constant
which is lowest will be the ratelimiting parameter and will have the greatest
impact on the observed constant k,. Analogous expressions can be obtained
for the desorption process.
Firstorder equations have been used by many soil chemists to describe the
kinetics of reactions in clay minerals and in soils. Sawhney (1966) described
the adsorption of cesium on vermiculite as a pseudo firstorder reaction.
Sparks and Jardine (1984) studied the kinetics of potassium adsorption on
the standard clay minerals kaolinite, montmorillonite, and vermiculite. The
firstorder equation described potassium adsorption on the clays extremely
well (Fig. 2).
TIME (rnin)
0
20
I
40
I
80
60
I
I
100
I
I
120
I
I
140
I
160
I
180
~
200
I
220
~
240
I
I
KAOLlNlTE
0 MONTMORILLONITE
A VERMICULITE
0
m
1.2

1.4

FIG. 2. Firstorder plots of potassium adsorption on clay minerals. (After Sparks and
Jardine, 1984.)
0
0.3 
Y
0.6 0.9 1.2 
?!
1.5
.
I
8
F
I
1.8 2.1 2.4
.
I
20
I
40
I
60
I
I
I
80
I
I
100
1~ 1
120
I
l
140
160
l
1
1
1
MATAPEAKE A
H MATAPEAKE B
A KENNANSVILLE A
A KENNANSVILLE B
0 DOWNERA
0
DOWNER B
I
~
I
~
243
KINETICS OF IONIC REACTIONS
Sparks and coworkers (Sparks et al., 1980b; Sparks and Rechcigl, 1982;
Jardine and Sparks, 1984a; Sparks and Jardine, 1984) and Talibudeen and
coworkers in Great Britain (Sivasubramaniam and Talibudeen, 1972) have
also found that potassium reactions between solution and exchangeable
phases in soil systems follow firstorder kinetics. An illustration of potassium
adsorption in soils conforming to firstorder kinetics (Sparks and Jardine,
1984) is shown in Fig. 3. Sparks et al. (1980b) investigated potassium
desorption kinetics in two Dothan soils from Virginia. The firstorder rate
equation described potassium desorption for an average of 165 and 173 min
for the aluminum and calciumsaturated samples, respectively, in the Ap, A2,
and B21t horizons, and for an average of 439 and 505 min for the aluminumand calciumsaturated samples, respectively, in the B22t horizon (Table 11).
These represented times when potassium desorption was virtually complete
in the respective soil horizons. The firstorder rate equation described
potassium desorption well, with r values ranging from 0.993 to 0.998.
b. Multiple FirstOrder Reactions. Many researchers have found that the
kinetics of ionic adsorption and desorption conform to a single firstorder
reaction. However, a number of workers have found that the kinetics of
reactions in clays and soils are characterized by multiple firstorder reactions
(Li et al., 1972; Griffin and Burau, 1974; Griffin and Jurinak, 1974; Jardine
Table I1
Values of K O and the Amount of Time the FirstOrder Rate Equation Described
Potassium Desorption for Dothan Soil from Greensville and Nottoway Counties"
~~~
~
KOb
(mol/lO kg soil)
Saturation
treatment
Soilhorizon
Time of firstorder
conformity'
(min)
Greensville
Nottoway
Greensville
Nottoway
5.64
6.44
6.10
6.64
6.39
6.80
7.95
9.00
5.77
6.64
6.21
6.77
6.44
7.00
8.00
9.28
152
163
161
166
162
169
438
500
160
170
175
183
177
186

~
AP
Al
Ca
A2
A1
Ca
B21t
Al
B22t
Ca
A1
Ca
~
440
5 10
~~
' Data after Sparks et al. (1980b). Used by permission of the SoiL Science Society of America
Journal.
Represents quantity of potassium on exchange sites at zero time of potassium desorption.
Represents time for which the firstorder rate equation described potassium desorption.
Time (rnin)
0
50
100
150
200
250
300
350
1
A=283 K
a = 298 K
=
. 313 K
8
0.6
$
0.8
Y
c
m
TIME b i n )
FIG.4. Firstorder kinetics for potassium adsorption at three temperatures on Evesboro soil; the inset shows the
initial 50 min of the firstorder plots at 298 and 313 K. (After Jardine and Sparks, 1984a.)
KINETICS OF IONIC REACTIONS
245
and Sparks, 1984a). Griffin and Jurinak (1974) investigated the adsorption of
phosphate on calcite and noted two simultaneous firstorder reactions. They
assumed the rates of the two reactions were independent and that the faster
reaction went to completion before the slower reaction began. The results
indicated a linear relationship existed for reaction times of about 10 min to
4 hr. The data for reaction times less than 10 min were curvilinear when
plotted according to firstorder kinetics. The linear portion of the plot was
then extrapolated back to zero time. The slope allowed the calculation of the
phosphate concentration in solution at any previous time due to the slower
firstorder reaction. The authors then took the total phosphate concentration
in solution at intervening reaction times ( t = 010 min) and corrected for the
influence of the slow reactions. The corrected data were then plotted
according to a secondorder kinetic expression which successfully described
the first 10 min of the reaction process.
Griffin and Jurinak (1974) found that the bulk of the phosphate was
desorbed during the more rapid reaction, which was dominant during the
initial 400 min of the reaction. The authors attributed the faster
the firstorder dissolution of the phosphate mineral from the surface of
calcite. The much slower second reaction was also a firstorder rate expression and was associated with the desorption of phosphate ions from the
surface of calcite.
Griffin and Burau (1974) investigated the kinetics of boron desorption
from soils and found that two separate pseudo firstorder reactions were
involved in addition to another very slow reaction. The authors postulated
that each of the three slopes represented a different site of boron retention.
The site corresponding to the line with the steepest slope was defined as site 1,
the intermediate reaction rate was attributed to site 2, and the very slow
reaction was attributed to site 3. Griffin and Burau (1974) found that about
90% of the boron desorbed from sites 1 and 2 could be attributed to site 1, the
most readily desorbed fraction, and about 10% could be attributed to site 2.
The authors noted that the relative constancy of the percentages associated
with sites 1 and 2 suggested that the desorption of the boron was from two
sites of retention on the same substance, possibly Fe, Mg, or Alhydroxy
compounds as suggested by others.
Jardine and Sparks (1984a) found that potassium adsorption (Fig. 4) and
desorption in an Evesboro soil from Delaware conformed to firstorder
kinetics at 283, 298, and 313 K. At 283 and 298 K, two simultaneous firstorder reactions existed, with the first slope containing both a rapid reaction
(reaction 1) and a slow reaction (reaction 2). The second slope described only
the slow reaction. The difference between the two slopes yielded the slope for
reaction 1. Reaction 1 conformed to firstorder kinetics for 8 min, at which
time it began to terminate, leaving reaction 2 to proceed for many hours. As
246
DONALD L. SPARKS
equilibrium was approached, deviation from firstorder kinetics occurred,
indicating that reaction 2 was nearing completion. Similar deviations were
noted by Boyd et al. (1947) and Sivasubramaniam and Talibudeen (1972)
when a relatively long time of contact was employed. Boyd et al. (1947)
suggested that deviations from firstorder kinetics as the equilibrium point
was approached may be related to irregularities in particle size of the solid
exchanger. Although this statement may have practical implications in soil
systems, it must be remembered that the region of near equilibrium (1  K J
K , < 0.2) is characterized by large experimental error (Boyd et al., 1947).
Jardine and Sparks (1984a) attributed the two simultaneous firstorder
reactions at 283 and 298 K in the Evesboro soil to exchange sites of varying
potassium reactivity. Reaction 1 was ascribed to external surface sites of the
organic and inorganic phases of the soil, which are readily accessible for
cation exchange, while reaction 2 was attributed to less accessible sites of
organic matter and interlayer sites of the 2: 1 clay minerals which predominated in the <2pm clay fraction.
3. Elovich or RoginskyZeldovich Equation
The Elovich or RoginskyZeldovich equation can be stated as follows
(Low, 1960):
dqldt = aeSq
(16)
where q is the amount of material adsorbed at time t and a and p are
constants during any one experiment. Assuming q = 0 at t = 0, Eq. (16)
becomes
+ apt)
(17)
+ to)  2.3/p log to
(18)
q = 2.3/p log(1
or
= 2.3/p log(t
where to = l/afl. If a volume of gas q is adsorbed instantaneously, and before
Eq. (16) begins to apply, the integrated form of the equation becomes
= 2.3/p log(t
where k
=
+ k )  2.3/8 log t o
(19)
to exp(DqO).If k is negligible in comparison with t, then
= 2.3/p log
t

2.3/p log to
(20)
or
= 2.318 log apt
(21)
247
KINETICS OF IONIC REACTIONS
Equation (2i) results directly from Eq. (17) if apt B 1, as shown by Chien
al. (1980). The Elovich equation was first developed to describe the kinetics
of chemisorption of gases on solid surfaces (Low, 1960). The equation
presumes a heterogeneous distribution of adsorption energies, where E,
increases linearly with surface coverage (Low, 1960).Parravano and Boudant
(1955) criticized using the Elovich equation to describe one unique mechanism because they found that it described a number of different processes, such
as bulk or surface diffusion and activation and deactivation of catalytic
surfaces. Recent theoretical studies with adsorption and desorption in
oxideaqueous solution systems showed that the applicability and method of
fitting kinetic data to the Elovich equation requires accurate data at short
reaction times (Aharoni and Ungarish, 1976). Ungarish and Aharoni (1981)
have also pointed out the inappropriateness of the Elovich equation at very
low and very high surface coverages (Atkinson et al., 1970; Sharpley, 1983).
These types of situations could often exist in soil and clay systems.
The Elovich equation has described the kinetics of heterogeneous isotopic
exchange reactions well on goethite (Atkinson et al., 1970), but not as well on
gibbsite surfaces (Kyle et al., 1975). All of the phosphorus adsorbed on
goethite was isotopically exchangeable and the data could be represented by
the Elovich equation (Atkinson et al., 1972). For gibbsite, the Elovich
equation fitted the data only after subtracting a very slowly exchangeable
component (Kyle et al., 1975). The proportion of the component varied and
was largest at low solution pH. Probert and Larsen (1972) reported that the
Elovich equation was not suitable for 32Pisotopic exchange data in soils.
More recently, a modified form of the Elovich equation was used by Chien
et al. (1980) to describe simultaneous firstorder reactions for phosphate
sorption and release in soils. To describe phosphate sorption the modified
Elovich equation would read
et
d(C,  C,)/dt = a exp[p(Co  C,)]
(22)
where C,  C, represents the net amount of phosphorus sorbed by the soil at
time t. After integration of Eq. (22) with the initial condition of C ,  C , = 0
at t = 0, Eq. (22) becomes
C ,  C , = (lip) ln(1
To simplify the Elovich equation, Chien
Thus Eq. (23) could be simplified as
et
+ apt)
al. (1980) assumed that
c, = c,  (1/8) In(@)  (1/P) In t
(23)
a/3t
9 1.
(24)
Calculated values of a and p from the modified Elovich equation for
phosphate release in some soils are shown in Table I11 (Chien et al., 1980).
The data show that the values of /3 varied widely with the soils, whereas the
248
DONALD L. SPARKS
changes in a values were relatively small. Chien et al. (1980) showed that a
was independent of soil type when the release of soil phosphorus was induced
by the same anionexchange resin, although it varied with soil type when
soluble phosphorus was added to different soils. In both cases /3 was a
function of type of phosphorus adsorbent and phosphorus source. The
calculated values of a and /3 (Table 111) show that the products of a and /3 are
much greater than one and thus support the assumption of apt 9 1 that was
used in deriving the Elovich equation. The constants ct and /3 also can be used
to compare the reaction rates of phosphate release in different soils. A
decrease of /3 and/or an increase of a should enhance the reaction rate. The
reaction rate for different soils is, however, questionable. The slopes of such
Table 111
Calculated Values of u and p of the Elovich
Equation for Phosphate Release in Three Soils”
u
Soil type
(pmol P/hr)
Waukegan silt loam
Fargo clay
Langdon loam
11.60
1.14
9.36
P
(pmol P)2.87 x 1 0  3
5.03 x 103
1.31 x
“Data taken from Chien et al. (1980). Used by
permission of the Soil Science Society of America
Journal.
plots vary with the level of addition of phosphate. The slope also varies with
the so1ution:soil ratio. Thus, the slope of these plots is not characteristic of
the soils, but depends on the conditions used. Sharpley (1983) concluded that
the modified Elovich equation was limited in modeling soluble phosphorus
transport in runoff unless these two parameters were included.
Since Chien et al. (1980) introduced a modified Elovich equation to study
ionic reactions in soils, some investigators have successfully employed the
equation (Ayodele and Agboola, 1981; Onken and Matheson, 1982). The
Elovich equation may reveal irregularities in data ordinarily overlooked by
other kinetic equations. It has been suggested that if it is characteristic of the
nature of the sites involved in the adsorption process, then any “breaks” in
the Elovich plot could indicate a changeover from one type of binding site to
another (Low, 1960; Atkinson et al., 1970; Chien et al., 1980). Such “breaks”
may not be artifacts of kinetic treatments (Low, 1960), but the nonlinear
Elovich plot may indicate a differing reactivity of sites for the adsorption of
ions on an irregular surface (Atkinson et al., 1970). Hingston (1981) notes
KINETICS OF IONIC REACTIONS
249
that the Elovich equation may be quite applicable to adsorption in soils and
sediments, where there is wide variation in activation energies because the
mixture of adsorption surfaces is so complex.
4. Parabolic Diffusion Law
A radical diffusion law can be expressed as (Crank, 1976)
C J C , = 4 / r ~ ' ~ ~ D t~Dt/rz
/~/r~
(25)
where C, is the quantity of ion adsorbed at time t, C , the amount of ion
adsorbed at equilibrium, r the average radius of soil or clay particle, t the
time, and D the diffusion coefficient. Eq. (25) can also be written as
(l/t)(CJC,) = ( 4 / r ~ " ~ ) ( D / r ~l/t'lZ)
) ' ~ ~ ( (D/rZ)
(26)
The parabolic diffusion equation can simply be expressed as
C J C , = Rt1I2
+ constant
(27)
where R is the overall diffusion coefficient.
Numerous researchers have found that the parabolic diffusion law describes the kinetics of adsorption and desorption of ions in clay minerals and
soils (Chute and Quirk, 1967; Quirk and Chute, 1968; Sivasubramaniam and
Talibudeen, 1972; Evans and Jurinak, 1976; Sparks et al., 1980b; Sparks and
Jardine, 1981, 1984). The parabolic diffusion law has been successful in
describing potassium release from clays, but it does not always adequately
describe the kinetics of exchange in soils (Evans and Jurinak, 1976; Sivasubramaniam and Talibudeen, 1972; Sparks et al., 1980b). One of the problems
in using the parabolic equation for soil systems may be ascribed to the
nebulous interpretation of the slope parameter. Sivasubramaniam and Talibudeen (1972) obtained parabolic plots for aluminumpotassium exchange
on British soils which gave two distinct slopes, which the authors theorized
could be indicative of two simultaneous diffusioncontrolled reactions. The
authors speculated that the ratecontrolling step in the adsorption of A13 +
and K + ions was the diffusion of the ions into the subsurface layers of the
solid. Sparks et al. (1980b) noted a nonlinearity with the parabolic diffusion
equation for the initial minutes of potassium desorption in soils (Fig. 5). They
attributed this deviation to film diffusioncontrolled exchange in the early
minutes of potassium exchange. Chute and Quick (1967) ascribed the lack of
conformity to the parabolic diffusion law of potassium release from micas in
the early minutes of release to massaction exchange. Evans and Jurinak
(1976) found that the parabolic diffusion law described the initial 16 min of
phosphorus release from a topsoil and subsoil sample at 284,298, and 3 13 K.