Kinetics of cation adsorption on charged soil mineral as strong electrostatic force presence or absence

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1 J Sois Sediments () :53 6 DOI.7/s SOILS, SEC GLOBAL CHANGE, ENVIRON RISK ASSESS, SUSTAINABLE LAND USE RESEARCH ARTICLE Kinetics of cation adsorption on charged soi minera as strong eectrostatic force presence or absence Rui Li & Hang Li & Huain Zhu & Laosheng Wu Received: 7 May /Accepted: 9 August /Pubished onine: 8 August # The Author(s). This artice is pubished with open access at Springerink.com Responsibe editor: Huijun Zhao R. Li : H. Li (*) : H. Zhu Coege of Resources and Environment, Southwest Agricutura University, Chongqing 476, Peope s Repubic of China e-mai: hi@yahoo.com.cn L. Wu Department of Environmenta Sciences, University of Caifornia, Riverside, CA 95, USA Abstract Purpose The purposes of this research were to: () estabish new anaytica kinetic equation for describing the effect of strong eectrostatic force on adsorption; () experimentay determine if it is a strong or weak eectrostatic force adsorption process; and (3) evauate the adsorption energies of the strong and weak force adsorptions based on the proposed new theory of cation adsorption kinetics. Materias and methods The constanty charged materiamontmorionite was used in the experiment. The montmorionite was saturated with two Cation species: K + and Ca +, respectivey, using KNO 3 or Ca(NO 3 ) before it was used for the experiment. The miscibe dispacement technique under a steady fow condition was adopted to the kinetic studies of Mg + (Mg(NO 3 ) ) adsorption. In the experiment,.5 g of K + - or Ca + -saturated montmorionite was ayered on the exchange coumn, the thickness of sampe ayer was approximatey..3 mm, the cross-sectiona area of the coumn (sampe area) was 5 cm. The concentration of Mg + in the fowing iquid was 4 mo L. The fow veocity of the fowing iquid was. ml min. Effuent was coected at -min intervas. Resuts and discussion Firsty, new and exact rate modes for describing ion adsorption have been advanced. Secondy, based on the experiments of Mg + /K + and Mg + /Ca + exchange in K + and Ca + -saturated montmorionites, Mg + adsorption in K + -saturated sampe appeared zero-order kinetic process in the initia stage of the strong force adsorption for t= 45 min, and then the adsorption process converted to the first-order kinetics of the weak force adsorption, which agrees with the theoretica prediction. However, for the Ca + -saturated sampe, merey firstorder kinetic process appeared for Mg + adsorption. Either for Mg + /K + exchange or Mg + /Ca + exchange, the quantities of Mg + by weak force adsorption at equiibrium were amost the same. Thirdy, severa important dynamic and thermodynamic parameters can be theoreticay cacuated based on the new theory in describing cation adsorption. Concusions For Mg + /K + exchange, both strong and weak eectrostatic force adsorptions exist, but for Mg + /Ca + exchange ony the weak eectrostatic force adsorption occurs. The strong and weak force adsorption processes can be quantitativey described by the new anaytica kinetic equations of the zero- and the first-order kinetics, respectivey. Because each parameter in the anaytica kinetic equations has its definitive physica meaning, severa important dynamic and thermodynamic parameters in cation exchange can be theoreticay estimated. Keywords Cation adsorption. Charged soi. Diffusion ayer (DL). Strong eectrostatic force. Weak eectrostatic force Introduction The negative charges on soi partice surface, together with counter-ions in diffuse ayer (DL), form an eectrostatic

2 54 J Sois Sediments () :53 6 fied in the soi soution adjacent soi partice surface. Recenty, Li and Wu (7); Li et a. (); Li and Li () found that, cation exchange is essentiay a cation diffusion process in the eectrostatic fied of DL. Li et a. (); Li and Li () aso found that, the eectrostatic fied in the DL can substantiay speed up the cation diffusion or adsorption process. Besides, experimenta resuts reveaed that, when the soi sampes were Ca + - saturated, ony the first-order kinetic process was observed for Mg + adsorption in the Mg + /Ca + exchange experiment (Li et a. ). In contrast, as soi sampes were K + - saturated, both zero- and first-order kinetic processes appeared for Mg + adsorption in Mg + /K + exchange experiment (Li and Li ). The zero-order kinetic process was considered to be a strong eectrostatic force adsorption for Mg + (Li and Li ). Why did the zero-order process for Mg + adsorption appeared in Mg + /K + exchange but did not in Mg + /Ca + exchange experiment, and how coud we experimentay distinguish those two states of cation adsorptions? This research is to deveop new kinetic equations by considering the strength of the eectrostatic force from the charged partice surface, to determine if the adsorption is caused by the strong or weak eectrostatic force in one experiment, and to evauate the adsorption energies of the strong and weak adsorptions based on the new theory of cation adsorption kinetics. Theory. The fux equations for ion adsorption by considering the eectric fied from surface charges In the experiment of the miscibe dispacement, the adsorption process can be iustrated in Fig.. Under this experimenta condition, Li et a. () demonstrated that, in the domain of x=, the ionic dynamic distribution can be described by the foowing equation: f i ðx; tþ ¼ f i e Z i Fϕ ð x;t Þ RT ( X p 4 Dp i ðn þ Þp e 4 ðnþþ t sin ð xþ ðn þ Þ p ) n¼ ðþ where f i (x, t) is the concentration of ith ion species at the distance x away from partice surface and time t; 8(x, t) is the eectric potentia in the eectric fied of DL; Z i is the charges of the ion species; F is the Faraday constant; R is the gas constant; T is the absoute temperature; D pi is the apparent diffusion coefficient of ion ith species in soi, and for soi D pi ¼ qð= e Þ D i r (Kemper and Van Schaik 966), θ is the voumetric water content, (/ e ) is the tortuosity factor, D i is the diffusion coefficient of the ion; and r is a coefficient accounting for ionic interaction. Because the series in Eq. converges rapidy, at arge t (often arger than severa min depending on the vaue of π D/4 ), ony the term n= needs to be considered, Eq. becomes: f i ðx; tþ ¼ f i e Z i Fϕ ð x;t Þ RT 4 e p D pi 4 p t sin x p ðþ For a period of time t= t, the tota quantity of ith cation species diffusion/adsorption through the pane of x= into the space of x= shoud be expressed as: N i ðtþ ¼S Z f i ðx; tþdx ð3þ where N i (t) is the tota adsorption quantity (mo/g) from t= to t, S is the specific surface area of the sampe with unit dm g. According to Eqs. and 3, the fux of ith cation species diffusion/adsorption through the pane of x= into the space x= can be expressed as: j i ¼ dn iðtþ ¼ D pi Z pf i e pd 4 t e ZFϕðxÞ RT sin ð xþ p ð4þ According to the definition of the apparent concentration or activity by Li et a. (Li and Wu 4, 7; Li et a. ), Eq. is changed to: A i ðx; t Þ ¼ f i ðx; t ¼ f i Þe Z ifϕ ð x;t Þ RT 4 e p Dpi 4 p t sin x p dx ð5þ where A i (x, t) is the apparent concentration or activity of ith ion species at distance x away from partice surface and time t.introducing Eq. 5 into Eq. 4, there is j i ¼ dn iðtþ ¼ p D pi 4 Z e ZFϕðxÞ RT ½f i A i ðx; tþšdx ð6þ Equation 6 is the fux equation for describing ion diffusion/adsorption by considering the eectric fied from soi partice surface under the miscibe dispacement experiment. Equation 6 impies that, the eectric fied is

3 J Sois Sediments () : Fig. A conceptua diagram of ion adsorption process in the experiment of the miscibe dispacement through the average effect of the eectric fied from x= to to infuence cation diffusion/adsorption rate or the fux.. The adsorption kinetics as strong eectrostatic adsorption force presence The presence of strong eectrostatic adsorption force means a strong negative energy area or a deep potentia we near the soi partice surface (as shown in Fig. ). Once a cation fas into (adsorbed) the strong negative energy area or the potentia we, the cation wi be strongy adsorbed and thus it wi be very difficut to escape from the potentia we. Therefore, the strongy adsorbed cation can be treated as disappeared immediatey from the soution adjacent partice surface. The disappear of the absorbed cation means that the activity of the cation in the soution near soi partice surface (in the potentia we) is approximatey zero. Obviousy, this phenomenon wi ony appear in the initia stage of the cation exchange experiment. This process can be iustrated as in Fig.. Mathematicay, it can be expressed as: f i A i ðx; tþ f i ð7þ Introducing Eq. 7 into Eq. 6, we get the fux equation at the presence of strong adsorption force: j i ¼ dn iðtþ ¼ p D pi 4 potentia we f i Z RT dx ð8þ Fig. A conceptua diagram of ion adsorption as strong adsorption presence at partice surface Thus, the rate equation for describing ion adsorption can be expressed as: dn i ðtþ ¼ k ¼ k ½N i ðtþš ð9þ where k is the rate coefficient, and (Li et a. ): k ¼ p D pi 4 f i Z RT dx ðþ Because Eq. 9 is proportiona to the zero power of N i (t), it can be referred as the zero-order rate equation. Thus, the zero-order kinetic process in cation adsorption impies the presence of strong adsorption force. Li and Xue (996); Li et a. () suggested an approximate equation for describing zero-order kinetic process, in which the rate coefficient is assumed as: Z k ¼ D pi f i RT dx ðþ which is sighty different from the exact expression of Eq.. In order to convenienty using the experimenta data directy, Eq. 9 can be written to the discrete form: N i ðt mþ Þ N i ðt m Þ ¼ k N i t kþ= ðþ St ð mþ t m Þ where m ¼ ; ; ; 3;...; N i t m þ = ¼ Ni ðt m Þ þ :5½ N i ðt mþ Þ N i ðt m Þ. Equation impies that, using experimenta data obtained in the presence of strong adsorption forces to pot a reationship curve of ½N i ðt mþ Þ N i ðt m ÞŠ=Sðt mþ t m Þvs: N i t mþ=, a straight ine which is parae to abscissa axis can be expected, and the intercept of the straight ine wi be the rate coefficient k of ion adsorption for the strong force adsorption. However, when the strong adsorption sites are saturated by the adsorbed cations (strong force adsorption ceased or competed), then the driving force for cation adsorption wi shift to the weak eectrostatic force, which can be iustrated as Fig. 3. In the process of weak force adsorption, the distance of ion diffusion is shortened from - to - as shown in

4 56 J Sois Sediments () :53 6 Strong force adsorbed cations Weak force adsorbed cations where N i t mþ= ¼ Ni ðt m Þþ:5½N i ðt mþ Þ N i ðt m ÞŠ; m ¼ ; ; 3;...; and here k is: k ¼ p D pi 4 f i N is: Z RT dx ð7þ Fig. 3 A conceptua diagram of ion adsorption for the weak force adsorption Fig. 3, and the rate equation can be expressed as (Li et a. ): dn i ðtþ ¼ k N i ðtþ N ð3þ where N i ðtþ is the adsorbed quantity in the second stage for a period of time t= t (here t= means the initia time moment of the weak force adsorption); N is the weak force adsorbed quantity at equiibrium: N ¼ Sf i Z RT dx ð4þ Z k ¼ p D pi 4 f i RT dx ð5þ Equation 3 shows that ion diffusion/adsorption kinetics in the process of weak force adsorption is a first-order kinetic process. From the above anaysis, the adsorption process can be conceptuay described in two stages: zero- and firstorder processes. However, in the second stage (the weak force adsorption), the experimentay determined N i (t) vaues incude the contribution of the adsorbed cation quantities in the first stage, and it is difficut to determine how many cations are strongy adsorbed in advance in the first stage. In order to directy use the experimenta data of N i (t) and N to anayze the kinetic characteristics, the upper imit of the integra in Eqs. 4 or 5 can be repaced by, and correspondingy t= sti means the initia time moment of the strong force adsorption but not the initia time moment of the weak force adsorption. Nevertheess, we shoud note that in Eq. 5, the in π D pi /4 cannot be repaced by (see the derivation of k in our previous paper (Li and Li )). Thus the discrete form of Eq. 3 can be expressed as: N ¼ Sf i Z RT dx ð8þ Equation 6 impies that, if the weak force adsorption is present, using experimenta data to pot a reationship curve of [N i (t m+ ) N i (t m )]/S(t m+ t m ) vs. N i (t m+/ ), a straight ine can be obtained, and the straight ine bears three important characteristics: () the intercept of the straight ine is the rate coefficient k in the ion adsorption process; ()the sope of the straight ine is π D pi /4 S; and (3) the tota adsorption quantity N at the equiibrium can be obtained directy by using the intercept and the sope vaues. In summary, if a cation adsorption process in an experiment contains zero- and first-order kinetics, using experimenta data to pot the reationship curves of [N i (t m+ ) N i (t m )]/S(t m+ t m ) vs. N i (t m+/ ), two straight ines coud be obtained. In the initia stage of adsorption, the straight ine wi be parae to the abscissa axis, and the intercept of the straight ine wi be the rate coefficient k ; then, at a given time moment, that reationship curve wi transfer to another straight ine, the intercept and the sope of the straight ine wi be k and π D pi /4 S, respectivey. The turning point of the two straight ines corresponds to the time when the strong force adsorption transfer to the weak force adsorption. Considering <, there must be k >k (Comparing Eqs. and 7). Thus, if both strong and weak forces adsorption presence, the two straight ines coud be iustrate in Fig. 4. When the Eqs. and 6 were directy used to fit to the experimenta kinetic data, the vaue of N is not required in advance. In the past, it was common practice to use the [Ni(tm+)-Ni(tm)]/S(tm+-tm) k k N i (t k+/ ) N i ðt mþ Þ N i ðt m Þ St ð mþ t m Þ ¼ k k N i t mþ= N ð6þ Fig. 4 The adsorption rate curves containing both strong and weak force adsorptions; ( strong force adsorption, weak force adsorption)

5 J Sois Sediments () : integra form to describe the experiment data, and a straight ine of nð N i ðtþ=n Þ vs. t canbeobtainedwhenthere exists the first-order process (Sparks and Jardine 984; Aharoni et a. 99; Jardine and Sparks 984; LiandXue 996). But if the integra form is used, the N vaue must be known in advance. Unfortunatey, direct and accurate determination of the true N is usuay not possibe. From theoretica anaysis, the directy determined N within a finite time wi aways be ower than the true vaue, because the true vaue can ony be obtained when t=. When a ower than the true vaue of N is used for describing the experimenta data, the sope of the fitted curve of nð N i ðtþ=n Þ vs. t wi approach to infinite as time t increases, simiar to the dashed ine shown in Fig. 5. Considering the sope is the rate coefficient, π D pi /4 S,if the obtained sope is infinite, this resut must be absurd. 3 Materia and methods 3. Materia and sampe preparation The constanty charged materia-montmorionite was used as the experimenta materia. The montmorionite was bought from Chi-Feng WHTB mining Co., Ltd., China. The surface properties of the montmorionite determined by the method recenty deveoped by Li et a. (unpubished materia) are shown in Tabe. The montmorionite was pre-h + -saturated for the surface properties determination, and the method was based on the exchange equiibrium of Na + /Ca + at ph=7 for the montmorionite. Theoreticay speaking, the monovaent cations such as K +, Na + have weaker effect on shieding the negative eectrostatic fied from soi partice surface than that of the divaent cations such as Ca +, Mg +. Therefore, the eectric fied strength of the sampes saturated with monovaent cations is much greater than that of the divaent cations saturated with the same eectroyte concentration. This impies that, as the negative charges on partice surface are saturated by K +,a Tabe Determined surface properties of the montmorionite Sampe Specific surface area (S, m g ) stronger eectrostatic fied is potentiay created, and thus a stronger force adsorption may exist for Mg + adsorption in the Mg + /K + exchange experiment. On the other hand, as the negative charges on the partice surface are saturated by Ca +, no strong eectrostatic fied wi be created, and no strong force adsorption wi exist for Mg + adsorption in the Mg + / Ca + exchange experiment. Therefore, in the experiment the sampes were pre-saturated by K + and Ca +, respectivey, and the exchange cation was seected Mg Kinetic studies of adsorption Surface charge density (σ, mmo ( ) m ) Montmorionite Surface charge number (cmo ( ) kg ) The kinetics of Mg + (Mg(NO 3 ) ) adsorption in the K + -and Ca + -saturated montmorionites (Mg + /K + and Mg + /Ca + exchange) were investigated by the miscibe dispacement under a steady fow condition. The experiment was carried out as foows: approximatey.5 g of K + -orca + -saturated montmorionite was ayered on an exchange coumn (the mass of the sampes were determined at the end of each experiment finished). The thickness of the sampe ayer was very thin in the sampe chamber for the purpose of reducing the effect of ongitudina concentration gradient in the experiment, since our theoretica anaysis is based on the one-dimensiona case. In our experiments, the sampe ayers were in the..3 mm range (Li et a. ). The sampe cross-sectiona area was about 5 cm. The experimenta temperature was 98 K. The concentration of Mg + in the fowing iquid was 4 mol. The fow veocity of the soutions was. mmin. Effuent was coected at -min intervas. The quantity of the ion that fowed through the interface x= into the fixed iquid fim at x= for a period of time t= t was cacuated based on the difference of Mg + concentration in the infuent and the effuent. 4 Resuts and discussion 4. Mg + adsorption kinetic anayses for Mg + /K + and Mg + /Ca + exchange Fig. 5 The fitting resut for the kinetic data as using the integra form of the first-order kinetic equation; soid ine, as the adopted N to be the true vaue, dotted ine, as the adopted N ower than the true vaue The experimenta resuts of [N i (t m+ ) N i (t m )]/S(t m+ t m ) vs. N i (t m+/ ) for Mg + adsorption in Mg + /K + and Mg + /Ca + exchanges are shown in Figs. 6 and 7, respectivey. Figure 6 shows that: () The zero-order kinetic process was present in the initia stage of adsorption for t= 45 min, which is an indication of strong force

6 58 J Sois Sediments () :53 6 y ( -7 mmo min - m - ) y= y = -.48x R = x (mmo kg - ) Fig. 6 The experimenta resuts of the reationship between y ¼ ½N i ðt mþ Þ N i ðt m ÞŠ=Sðt mþ t m Þand x ¼ N i t mþ= of Mg + adsorption for Mg + /K + exchange; empty diamonds, experimenta data; soid ine, fitting ine adsorption for Mg + in the experiment of Mg + /K + exchange. Since there exists no strong chemica bond between Mg + and the partice surface for the sampe, the strong force adsorption can be attribute to the eectrostatic force; and () The experimenta resuts shown in Fig. 6 agree we with the theoretica prediction (see Fig. 4). After a period of time (about 45 min) in the experiment, the adsorption process shifted to the first-order kinetic process, and the turning point from the zero-order to the first-order was very distinct. However, for the Mg + adsorption in the Mg + /Ca + exchange, the zero-order kinetic process did not appeared with the theoretica prediction (see Fig. 7). Therefore, for Mg + adsorption in Mg + /Ca + exchange, ony the weak force adsorption process was present. It is we known that the eectrostatic fied strength around a partice in soution for Ca + -saturated sampe wi be much weaker than that for the K + -saturate sampe. Using the data shown in Tabe, the distribution of the potentia energy E(x)=ZF8(x) ofmg + in the space adjacent partice surface was cacuated (Li et a. 4). Figure 8 shows the distributions of the potentia energy of Mg + in the space y ( -7 mmo min - m - ) y = -.5x R = x (mmo kg ) Fig. 7 The experimenta resuts of the reationship between y ¼ ½N i ðt mþ Þ N i ðt m ÞŠ=Sðt mþ t m Þand x ¼ N i t mþ= of Mg + adsorption for Mg + /Ca + exchange; empty diamonds, experimenta data; soid ine, fitting ine adjacent to the partice surface for Ca + -andk + -saturated montmorionites, respectivey. In the cacuation, we used the initia concentration in buk soution of 5 mol for Ca + and K + in the suspensions of Ca + - and K + -saturated montmorionites. We know that, in the preparation of a basic cation species saturated sampe, the concentration of the basic cation species in buk soution cannot much ower than 5 mol. If the concentration is much ower than 5 mo/, the adsorption quantity of H + on partice surface cannot be negected because of H + /K + or H + /Ca + exchange even at ph=7. Figure 8 indicates that, at a given position in the space adjacent to partice surface, the potentia energy of Mg + - saturated in the K + -saturated sampe is much ower than that in the Ca + -saturated sampe. For exampe, at a distance of nm away from partice surface, the potentia energy of adsorbed Mg + is E Mg = 48,83 Jmo for the K + -saturated sampe; whie it is E Mg =,86 Jmo for the Ca + - saturated sampe. Therefore, Mg + can be adsorbed much stronger in the K + -saturated montmorionite than that in the Ca + -saturated montmorionite. On the other hand, at a distance of nm away from partice surface, the difference of potentia energy between E Mg and E Ca is ΔE Mg= Ca ¼ E Mg E Ca ¼ ; 86 þ ; 86 ¼ J mo for the Mg/Ca exchange; whie the difference of potentia energy between E Mg and E K was ΔE Mg= K ¼ E Mg E K ¼ 48; 83 þ 48; 83= ¼ 4; 4:5Jmo for the Mg/K exchange, which indicates that the Mg + exchange for K + is much easier than Mg + exchange for Ca +. Therefore, the Mg/K exchange exhibits the zero-order kinetic process, whie the Mg/Ca exchange does not. 4. Estimation of dynamic and thermodynamic parameters in cation exchange Because each parameter in the new kinetic equations has its definite physica meaning, some important dynamic and Potentia energy of adsorbed Mg + (- 4 J mo - ) K -saturated + Ca -saturated Distance away from partice surface x (nm) Fig. 8 The distributions of the potentia energy of Mg + in the space adjacent montmorionite partice surface for the Ca + - and K + - saturated sampes, respectivey

7 J Sois Sediments () : thermodynamic parameters in cation exchange can be cacuated based on the new theory for describing the cation adsorption. (). Estimation of the adsorbed quantities of Mg + by the strong and weak force adsorption at equiibrium From Eq. 6, using the sope and the intercept of the straight ine shown in Figs. 6 or 7, the tota adsorbed quantities: N can be cacuated as: N ¼ k jsopj ð9þ Since both strong and weak force adsorptions coexisted for the Mg + adsorption in the Mg/K exchange, from Fig. 6 the cacuated k =6.8 7 mmomin m, and sope =.48 7 kgmin m.usingthetwovaues,ineq.9, the cacuated the tota adsorbed quantities: N ¼ 44:4mmo kg. The adsorption quantity of Mg + at the turning point represents the maximum quantity of the strong force adsorption at equiibrium, whie the difference between the tota adsorption and the adsorption at the turning point is the weak force adsorption (see Fig. 6). According to the fitting equations shown in Fig. 6, at the turning point we had: 4:5476 ¼ :48x þ 6:8 ðþ where x is the adsorption quantity of Mg + at the turning point. From Eq. we can get the quantity of the strong force adsorption of Mg + at equiibrium, it was 7. mmokg. We can further cacuate the quantity of the weak force adsorption of Mg + at equiibrium, it was =37.3 mmokg. Considering the eectro-neutraity principe, for a constanty charged materia-montmorionite, the tota surface charge or the surface charge at the origin of the strong force adsorption ayer is 44.4 mmo ( ) kg =84.9 cmo ( ) kg. Correspondingy, the charge at the origin of the weak force ayer is 37.3 mmo ( ) kg =6.5 cmo ( ) kg, which is very cose to the determined surface charge number of 6.cmo ( ) kg,asshown in Tabe. Considering H + has a stronger adsorption force in a given eectrostatic fied than Ca +, this resuts might impy that, if a H + -saturated sampe were used in the cation exchange experiment for surface properties determination, a portion of the strongy adsorbed H + coud not be exchanged by a basic cation species, e.g., Ca + and Na + there. Since ony the weak force adsorption exists for Mg + adsorption in the Mg/Ca exchange, from Fig. 7 and Eq. 9, the cacuated adsorption quantity of Mg + at equiibrium is /.5=94.6 mmokg, which means that the surface charges at the origin of the weak force adsorption ayer is 58.9 cmo ( ) kg. This resut aso indicates that, because Ca + has a stronger adsorption force in a given eectrostatic fied than Mg +, a portion of the strongy adsorbed Ca + cannot be exchanged by Mg +. The above discussions shows that, either from the experiment of Mg/K exchange, Mg/Ca exchange, or Ca/H and Na/H exchange, the obtained surface charge numbers at the origin of the weak force adsorption ayer are amost the same, they are 6.5, 58.9 and 6. cmo ( ) kg, respectivey. The resuts seemingy impied, ony when the strong force adsorption exists for cation exchange, e.g., the Mg/K exchange, the tota surface charge number at the origin of the strong force adsorption ayer can be obtained. (). Estimation of the thicknesses of the strong and weak force adsorption ayers Based on the new theory, the average thicknesses of the strong and weak force adsorption ayers can be cacuated. From Eqs. and 7, we obtain: rffiffiffiffiffi ¼ k ðþ k In this exchange experiment, the sampe was amost water saturated, and the saturated water content in this experiment for the montmorionite was.44 mg approximatey. Therefore the average thickness of the fixed iquid fim was ¼ :44 6 ðm 3 g Þ= 65:5ðm g Þ ¼ 3:75nm. From Fig. 6, we had k = mmo min m, k =6.8 7 mmomin m,introducing them into Eq., we get /=.859. Therefore, the average thickness of the weak force adsorption ayer was =.859=3.9 nm, and the average thickness of the strong force adsorption ayer was.49=.559 nm in this experiment for the montmorionite. (3). Estimation of the apparent diffusion coefficient of cations From Eqs. 7 and 8, we have: D pi ¼ 4S p k N ðþ Introducing the corresponding vaues into Eq., we get the apparent diffusion coefficient of Mg + in the montmorionite: D pi ¼ 4 65:5 ð3:9 9 Þ 6:8 7 =3:4 :444 ¼ 3:979 m min. (4). Estimation of the average activity coefficient of the adsorbed cations

8 6 J Sois Sediments () :53 6 Low (98) defined an average activity coefficient of eectrostatic adsorbed ions: eg i ¼ R i FϕðxÞ e Z RT dx ð3þ where eg i is the average activity coefficient for a the eectrostatic adsorbed Mg +. Combining Eqs. 8 and 3 gives: eg i ¼ Sf i N ð4þ Aso, for the strong force adsorption ions, according to the definition of the average activity coefficient of adsorbed ions by Li and Wu (7), there is: eg i ðsþ ¼ f i e f i ðsþ ¼ f i N i ðsþ S ð Þ S ð ¼ N i ðsþ Þf i for the weak force adsorption ions, there is: ð5þ eg i ðwþ ¼ f i e f i ðwþ ¼ f i ¼ Sf i ð6þ N i ðwþ N i ðwþ S where eg i ðsþ; eg i ðwþ is the average activity coefficient of the adsorbed Mg + for the strong and weak force adsorptions, respectivey; e fi ðsþ; e f i ðwþ is the average concentration of the adsorbed Mg + at equiibrium for strong and weak force adsorptions, respectivey; N i (s), N i (w) are the quantities of the adsorbed Mg + at equiibrium for strong and weak force adsorptions, respectivey. Based on the above obtained vaues of N, N i (s) and N i (w), the average activity coefficients of the tota eectrostatic adsorption Mg +, the strong eectrostatic force adsorption Mg +, and the weak eectrostatic force adsorption Mg + can be cacuated, the resuts are isted in Tabe. The data in Tabe show that: () the average activity coefficient of Mg + is far smaer than due to the infuence of the eectrostatic fied from surface charges of soi partices; () the average activity coefficient of Mg + for the strong eectrostatic force adsorption is much ower than that for the weak eectrostatic force adsorption; (3) for the Mg/Ca exchange, a the adsorbed Mg + have high activity because ony weak force adsorption exists. The Debye characteristic ength, /κ was estimated to be 7.3 nm in this experiment, where κ is the Debye- Hucke parameter. The above discussions showed that, the average thickness of iquid fim in the montorionite was merey 3.75 nm in the experiment, thus <</κ, which impies that the diffuse ayer at the inner surface of the montmorionite was not fuy expanded. Considering the eectro-neutraity rue: Sf i Z k RT dx ¼ Sf i Z thus as </κ, there wi be: eg i ðþ ¼ R i FϕðxÞ e Z RT dx < eg ið=kþ ¼ k R k Z ifϕðxþ e RT dx RT dx ð7þ ð8þ Equation 8 shows that, as </κ, there must be eg i ðþeg i ð=kþ. Therefore, under the same temperature and eectroyte conditions for the same sampe, the activity of the adsorbed ions in a not fuy expanded diffuse ayer is much ower than that in a fuy expanded diffuse ayer. 5 Concusions By considering the eectric fied from partice surface, new exact kinetic equations for describing cation adsorption were deveoped. The new equations show that, when both strong force adsorption and weak force adsorption coexist in an exchange experiment, the initia stage of strong force Tabe The average activity coefficients of the adsorbed Mg + Treatments N a (mmokg ) N i (s) b (mmokg ) N i (w) c (mmokg ) egðsþ 4 d egðwþ 4 e eg 4 f Mg/K exchange Mg/Ca exchange a Totay adsorbed quantities of Mg + at equiibrium b Strong force adsorption quantities of Mg + at equiibrium c Weak force adsorption quantities of Mg + at equiibrium d Average activity coefficient of the strong force adsorption Mg + e Average activity coefficient of the weak force adsorption Mg + f Average activity coefficient for the whoe adsorbed Mg +

9 J Sois Sediments () : adsorption can be described by the zero-order kinetics. The adsorption then shifts to the first-order kinetics of the weak force adsorption. In this study, K + - and Ca + -saturated montmorionites were used in the Mg + /K + and Mg + /Ca + exchange to verify the theory. As predicted, because K + - saturated montmorionite can form a much stronger eectrostatic fied adjacent to the partice surface than the Ca + - saturated montmorionite, Mg + adsorption in K + -saturated sampe appeared to be a zero-order kinetic process in the initia stage (t= 45 min) of the strong force adsorption; the adsorption process then shifted to the first-order kinetics of the weak force adsorption. However, for the Ca + -saturated sampe, ony the first-order kinetic process was observed for Mg + adsorption, because ony weak eectrostatic fied presents in the system. Either for Mg + /K + exchange or Mg + /Ca + exchange, the quantities of Mg + weak force adsorption at equiibrium were approximatey the same (37.3 and 94.6 mmokg, respectivey). The new kinetic equations are mathematicay precise, and they were derived from the actua physica mechanism of cation adsorption. Thus, those new equations are fundamentay different from the cassic apparent kinetic equations. A the parameters in the new equations have definite physica significance. Therefore, using the new equations, some important dynamic and thermodynamic parameters in cation exchange can be estimated from the experimenta data. These parameters incude the tota adsorption quantity, the strong force adsorption quantity and the weak force adsorption quantity of cation at equiibrium; the thicknesses of the strong force adsorption ayer and the weak force adsorption ayer; the apparent diffusion coefficient of cation in cay; and the average activity coefficients of the adsorbed cations for the strong and weak force adsorptions. The study aso indicates that, ony when the strong force adsorption exists for cation exchange, e.g., the Mg/K exchange, the tota surface charge number at the origin of the strong force adsorption ayer can be determined through the cation exchange method. 6 Recommendations and perspectives The new theory unifies the diffusion process of cations in soution, the diffusion process of cations in diffuse ayer, and the exchange process of cations on surface, which opens a new window for future research of investigating the chemica adsorption mechanisms of heavy meta ions on partice surface. Currenty, for heavy meta ions adsorption, ony an overa process containing a types of diffusion processes, physica adsorption process and chemica adsorption process, can be detected in experiment. Acknowedgements This work was supported by the Nationa Natura Science Foundation of China (Grant No and ) and the Nationa Basic Research Program of China (Grant No. CB345). Open Access This artice is distributed under the terms of the Creative Commons Attribution Noncommercia License which permits any noncommercia use, distribution, and reproduction in any medium, provided the origina author(s) and source are credited. References Aharoni C, Sparks DL, Levinson S, Ravina I (99) Kinetics of soi chemica reactions: reationships between empirica equations and diffusion modes. Soi Sci Soc Am J 55:37 3 Jardine PM, Sparks DL (984) Potassium-cacium exchange in a mutireactive soi system: I. kinetics. Soi Sci Soc Am J 48:9 45 Kemper WD, Van Schaik JC (966) Diffusion of sats in cay-water systems. Soi Sci Soc Am Proc 3: Li H, Xue JH (996) Study on ion diffusion kinetics in soi systems. Acta Pedoogica Sinica 33(4):37 336, in Chinese Li H, Wu LS (4) On the reationship between therma diffusion and moecuar interaction energy in binary mixtures. J Phys Chem B 8: Li H, Wu LS (7) A new approach to estimate ion distribution between the exchanger and soution phases. Soi Sci Soc Am J 7: Li R, Li H () Comparison between Mg + -K + and Ca + -K + exchange kinetics as considering the eectric fieds at soid-iquid interface of soi. Acta Phys Chim Sin 6:55 56 Li H, Qing CL, Wei SQ, Jiang XJ (4) An approach to the method for determination of surface potentia on soid/iquid interface: theory. J Cooid Interf Sci 75:7 76 Li H, Li R, Zhu HL, Wu LS () The infuence of eectrostatic fied from soi partice surfaces on ion adsorption/diffusion. Soi Sci Soc Am J 74(4):9 38 Low PF (98) Soi physica chemistry. Agricutura, Beijing (in Chinese) Sparks DL, Jardine PM (984) Comparison of kinetics equations to describe K-Ca exchange in pure and in mixed systems. Soi Sci 38:5