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V. Exchange Equilibrium and the Kinetics of Potassium Exchange

V. Exchange Equilibrium and the Kinetics of Potassium Exchange

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between the activation enthalpy (AH$)

of an ion-exchange reaction and the

standard enthalpy of exchange for the solid phase (AW,),





is the standard enthalpy of exchange for the liquid (solution) phase. He

suggested that the best way to establish such a connection was through selfdiffusion studies. Unfortunately, no further publication has appeared clarifying

this idea or confirming the relationship. The paper was also somewhat puzzling

in that a direct relationship between chemical equilibria and kinetics of completely reversible reactions (such as ion exchange is taken to be) was established

long ago (e.g., see Glasstone et al., 1941; Laidler, 1965). Use of this was made

by Keay and Wild (1961), who calculated the enthalpy and entropy of Na+Mg2+ exchange on a vermiculite from measurements of the kinetics of Na+ -+

Mg2+ and Mg2+ + Na+ exchange. The relationship was applied for the first

time to K+-exchange studies by Sparks and Jardine (1981), who calculated

standard thermodynamic parameters of K -Ca2 exchange from measurements

of the kinetics of this reaction.

The apparent adsorption rate coefficient for K + adsorption (k:) is given by



lOg(1 - K,/K,)




and the apparent desorption rate coefficient for K + desorption (k> is given by





where K, is the amount of K + adsorbed on soil exchange sites at time t , K, the

amount at equilibrium, and KOthe amount at zero time. For a completely reversible reaction, the thermodynamic equilibrium constant is given by

K = k!JkA


(Glasstone et al., 1941). Thus the free energy of exchange is readily obtainable if

kk and ki can be measured.

The energy of activation for K + adsorption (E,) and desorption (Ed) is calculated from the Arrenhius equation

d In ki - Ei



where i = a or d. Then, following a similar argument, the standard enthalpy of

exchange is obtained from the difference between activation energies of the

forward and reverse reactions


= E, - Ed


The standard entropy of exchange can then be obtained from the Gibbs equation.

Sparks and Jardine (198 1) measured values of kL and kA at 3,25, and 40°C, and

accordingly calculated AGO, AHO, and A,!?. They also calcuated AGO, AW,and



Table II

Standard Free Energies (AG3, Enthalpies (AlT), and Entropies (W,of Caz+

Exchange on Soils, Calculated from Equilibrium and Kineticsa






Method of calculation




Kinetics data

Eyring's absolute reaction

rate theory

Equilibrium data (Deist

and Talibudeen, 1967b)

- 1.274






-1.05 to -3.42

-0.91 to -8.49

-2.89 to -22.10

'Data after Sparks and Jardiie (1981). Adapted from Soil Sci. SOC.Am.J . 45, 1097-1099. By permission of the

Soil Science Society of America Journal.

AS' using Eyring's absolute reaction rate theory (Laidler, 1965). In this method

free energies (AGS),enthalpies (AH$),

and entropies (ASS) of activation for

forward and reverse reactions are calculated, and the difference between them is

again AGO, A W , and W .These data are shown in Table 11, and the agreement

between the methods is excellent. Unfortunately the authors did not also calculate these quantities from exchange isotherm data and compare the results, which

would have been the best experimental verification of the theory. However,

values for Ca2+ + K + exchange on a silt loam soil at 25°C were of the same

order of magnitude as those found by Deist and Talibudeen (1967b) for a range

of soils at the same temperature (Table II).


The essential elements of a thermodynamic analysis of cation-exchange equilibria have been presented, and their application to potassium exchange in soils

and clays has been examined. It is true to say that useful applications have so far

been limited to characterizing the exchange properties of soils and establishing

the selectivity of various soils and clays for potassium. No thermodynamic

parameter has yet been found to predict crop yield or response to K + fertilizer

from soil K+ measurements; K+ potentials seem closest to being used in this

way. Perhaps the most promising new application suggested in this article,

although not directly related to agriculture, could have useful indirect benefits.

The use of calorimetrically measured enthalpies of exchange to detect small

amounts of impurities in formerly pure clay minerals may well aid the clay

mineralogist. It may also aid agriculturalists, because a more detailed picture of

soil mineralogy will help to explain and predict the reaction of a soil when K +



fertilizer is added. The differential enthalpies and entropies used for this purpose

certainly avoid Walsh’s complaint (see Section I) that thermodynamic functions

integrate variable quantities giving a (misleading) average value.

However, one of the most important areas of agricultural research at the

present time is the modeling of soil-plant processes (Cooke, 1979). Cation

exchange, and in particular the adsorption and release of nutrient K + , should be

part of this. As Cooke and Gething (1978) said, “Probably the practical returns

from the basic work resulting from applying thermodynamic principles to soil

systems, begun 25 years ago, will come from the development of models that

lead to more efficient fertiliser recommendations.” Empirical models have

proved unsatisfactory, but a basic set of thermodynamic equations describing

cation exchange is now available. Rigorous and theoretically sound, and being

based on no assumptions, it is universally applicable. Although the equations

were initially written to describe only simple systems (e.g., binary exchange an

“pure” exchangers), equations are now being developed to describe more closely the complicated situation found in multi-cation-exchange reactions in soils,

and they are thus relevant to agriculture. The equations may be initially disconcerting to the agronomist because of their complexity, but the availability of

computers makes them accessible and usable.


Activity of cation A

Unit cell area of a clay

Activity ratio

Layer charge of a clay

Differential free energy of exchange

Differential enthalpy of exchange

Differential entropy of exchange

Equivalent fraction of cation A

Energy of activation for K+ adsorption

Energy of activation for K + desorption

Activity coefficient of adsorbed cation A, calculated according to the Vanselow

convention up to the end of Section II,E and according to the Gaines and Thomas

convention thereafter

Activity coefficient of adsorbed cation A, calculated according to the Gaines and

Thomas convention up to the end of Section 1I.E

Ionic strength

“Intensity” of K + in the soil (Section IV,F)

Thermodynamic equilibrium constant

Gaines and Thomas selectivity coefficient

Uncorrected Gaines and Thomas selectivity coefficient

Gapon constant (or selectivity Coefficient)

Vanselow selectivity coefficient


Uncorrected Vanselow selectivity coefficient

Amount of K+ adsorbed at time t (Section V)

Amount of K+ adsorbed at zero time (Section V)

Amount of K+ adsorbed at equilibrium (Section V)

Apparent adsoption rate coefficient (Section V)

Apparent desorption rate coefficient (Section V)

Mean ionic molality [Eq. (30)]

Molarity of cation A

Mole fraction of cation A

Number of moles of water sorbed by the exchanger [Eq.(1 l)]

“quantity” of K+ in the soil (Section IV,F)

Gas constant

Anhydrous radius of an ion

Equivalent anionic radius of a clay (Section IV,E,2)

Absolute temperature


Valency of cation A

Valency of cation B

Fractional K saturation of exchange capacity

Ion charge (Section IV,B,l)

Activity coefficient of cation A in solution

Free energy term defined by Karamanos and Turner (1977) (Section IV,F)

Potassium potential with reference to calcium

Excess free energy of exchange

Standard Gibbs free energy of exchange

Activation energy

Excess enthalpy of exchange

Standard enthalpy of exchange

Integral enthalpy of exchange

Activation enthalpy

Standard enthalpy of exchange for liquid or solution phase

Standard enthalpy of exchange for solid phase

Change in adsorbed (exchangeable) K+ (Section IV,F)

Excess entropy of exchange

Standard entropy of exchange

Activation entropy

Chemical potential

Chemical potential in standard or reference state

Osmotic coefficient [Eq.(30)]



The critical assessment of this article by Dr.0.Talibudeen is gratefully acknowledged.


Addiscott, T. M., and Johnston, A. E. 1975. J . Agric. Sci. 84,513-524.

Addiscott, T. M., and Talibudeen, 0. 1%9. Potush Rev. (Berm) Subj. 4, 45th Suite, pp. 1-24.

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