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III. Application of Chemical Kinetics to Soil Solutions

# III. Application of Chemical Kinetics to Soil Solutions

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KINETICS OF IONIC REACTIONS

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 first-order, Elovich, parabolic diffusion, zero-order, secondorder, and two-constant 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 . First-Order 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 first-order kinetics.

For a miscible displacement technique, if the rate of adsorption of an ion

onto a colloidal surface follows first-order 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 first-order

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 First-Order Kinetics to Clay Minerals and Soils

a. Single First-Order Reaction. One should realize that the adsorption rate

coefficients (k, and kdrrespectively) determined from the previous first-order

equations are composed of numerous diffusional and chemical rate constants

(Jardine and Sparks, 1984a). It is appropriate to suggest that for the

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 rate-limiting parameter and will have the greatest

impact on the observed constant k,. Analogous expressions can be obtained

for the desorption process.

First-order 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 first-order reaction.

Sparks and Jardine (1984) studied the kinetics of potassium adsorption on

the standard clay minerals kaolinite, montmorillonite, and vermiculite. The

first-order 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. First-order 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 co-workers (Sparks et al., 1980b; Sparks and Rechcigl, 1982;

Jardine and Sparks, 1984a; Sparks and Jardine, 1984) and Talibudeen and

co-workers in Great Britain (Sivasubramaniam and Talibudeen, 1972) have

also found that potassium reactions between solution and exchangeable

phases in soil systems follow first-order kinetics. An illustration of potassium

adsorption in soils conforming to first-order 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 first-order rate

equation described potassium desorption for an average of 165 and 173 min

for the aluminum- and calcium-saturated samples, respectively, in the Ap, A2,

and B21t horizons, and for an average of 439 and 505 min for the aluminumand calcium-saturated samples, respectively, in the B22t horizon (Table 11).

These represented times when potassium desorption was virtually complete

in the respective soil horizons. The first-order rate equation described

potassium desorption well, with r values ranging from -0.993 to -0.998.

b. Multiple First-Order Reactions. Many researchers have found that the

kinetics of ionic adsorption and desorption conform to a single first-order

reaction. However, a number of workers have found that the kinetics of

reactions in clays and soils are characterized by multiple first-order 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 First-Order Rate Equation Described

Potassium Desorption for Dothan Soil from Greensville and Nottoway Counties"

~~~

~

KOb

(mol/lO- kg soil)

Saturation

treatment

Soilhorizon

Time of first-order

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

~

4-40

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 first-order 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. First-order kinetics for potassium adsorption at three temperatures on Evesboro soil; the inset shows the

initial 50 min of the first-order 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 first-order 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 first-order 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

first-order reaction. The authors then took the total phosphate concentration

in solution at intervening reaction times ( t = 0-10 min) and corrected for the

influence of the slow reactions. The corrected data were then plotted

according to a second-order 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 first-order dissolution of the phosphate mineral from the surface of

calcite. The much slower second reaction was also a first-order 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 first-order 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 Al-hydroxy

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 first-order

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 first-order 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 first-order 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 first-order 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 first-order

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 <2-pm clay fraction.

3. Elovich or Roginsky-Zeldovich Equation

The Elovich or Roginsky-Zeldovich 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

oxide-aqueous 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 first-order reactions for phosphate

sorption and release in soils. To describe phosphate sorption the modified

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 anion-exchange 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 10-3

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 aluminum-potassium exchange

on British soils which gave two distinct slopes, which the authors theorized

could be indicative of two simultaneous diffusion-controlled reactions. The

authors speculated that the rate-controlling 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 diffusion-controlled 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 mass-action 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.

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