OCTOBER 1979 Vol. 104 No. 1243 The Analyst Kinetics of a- and P-Molybdosilicic Acid Formation Victor W. Truesdale Peter J. Smith and Christopher J. Smith Institute of Hydrology Maclean Building Crowmarsh Giflord Wallingford Oxfordshire 0x10 81113 The kinetics of formation of tl- and /3-molybdosilicic acid have been investi-gated and rate equations proposed together with possible reaction sequences involving individual molybdate species. The a-acid was found to form according to a sum of exponentials model [At = dl + d2exp( d32) + e,ex.p( e,t)] relating absorbance with time. In contrast the /3-acid formed according to a simpler exponential model [At = dl + d2exp(e,t)] which is consistent with pseudo-first-order behaviour with respect to silicate concentration. For the first time in a kinetic study of heteropolymolybdate formation the effects of changes in molybdate speciation upon the kinetics have been con-sidered in detail.The concentrations of individual molybdate species at pH between 0.5 and 5.0 and total molybdate concentrations up to 0.20 M as molybdenum have been calculated using equilibrium constants together with an existing computer program. The most interesting result of the study is that the observed a-acid kinetics demand the presence of a third silicon species in addition to the reactant silicate and the a-acid product. At this stage it is not known whether the third species should be another silicate condensation product or a molybdosilicic acid. The general application of the results of this study to silicate analysis is discussed.Methods of non-linear parameter estimation have been used extensively in fitting theoretical models to observed data. Keywords Molybdosilicic acids ; vnolybdate speciation ; kinetics ; modelling kinetics In this paper we describe the results of a study of the kinetics of formation of a- and /3-molybdosilicic acids. Although there were several reasons for conducting this study the most immediate need was to give a firm foundation to our earlier observations that at room temperature (20 “C) the P-acid forms between about pH 1.0 and 3.8 and the oc-acid between about pH 1.8 and 4.8.l These observations formed the basis for rejecting the earlier view that the conditions for molybdosilicic acid formation could be defined by the use of a variable known as “acid to molybdate ratio.”2 This study is part of a wider programme of research whose original objective was to try to verify the existence of a variety of silicate species in natural waters; this wider programme has already included an examination of the spectrophotometric characteristics of the molybdosilicic acids3 and a study of their transformation and decomposition.4 Detailed information about these systems is also needed to provide a firmer foundation to the widely used analytical methods for silicate that rely on the formation of the molybdosilicic acids.The molybdosilicic acids are also interesting because they are members of the large family of heteropolymolybdates ; conclusions reached about the behaviour of one member of the family can often have a bearing upon that of another.This study is novel in that it attempts to overcome the major problem that must be encounted with any study of heteropolymolybdate formation that of having an adequate description of the molybdate system under given conditions of pH total molybdate con-centration etc. Hence although it seems to be agreed that several molybdate species exist in solutions of pH between 0.0 and 5.0 little agreement exists as to which and how much of each of the individual species is present under given conditions.5 Indeed at one time or another molybdate systems have been defined using different combinations of molybdate condensation products containing 2 3 4 6 7 8 10 12 16 or 24 molybdenum atoms. Moreover in the aforementioned pH range the problem is aggravated by the fact that each condensation product can occur in several protonated forms.Presented with 89 898 TRUESDALE SMITH AND SMITH KINETICS OF Analyst Vol. 104 several alternative definitions of the molybdate system we found it impossible to choose the best one objectively. This was particularly so because the concentrations of the different species were not determined directly but were inferred from mathematical models that appeared to fit observed changes in relaxation or optical spectra etc. induced by changes in pH.5 These mathematical models contain different combinations of several of the ten molybdate condensation products (plus protonated forms) mentioned above. Accordingly, in this study where at best we can only hope to lay down a strategy for future studies we have found it worthwhile to consider only one of the molybdate speciation models.6 It is our intention to consider the alternatives later.The objective of this study then was to find the relationship between the rate of formation of the molybdosilicic acids and the concentrations of molybdate-molybdenum hydrogen ions and silicate-silicon. Earlier work suggested that there would be few problems of stoicheiometry. Moreover although separate pH regimens exist for the acids the /3-acid nevertheless transforms4 spontaneously into the oc-acid. Initially we chose to examine the rate of molybdosilicic acid formation using the approach in which the effects of changes in the concentration of a given variable are studied with the other variables being kept constant.Throughout the study all silicate solutions were undersaturated with respect to silica solu-bility and kinetic effects due to dissolution of solid phases' were avoided. Experimental Apparatus and Reagents Absorbance measurements were made on a Unicam SP 1800 spectrophotometer equipped with a constant-temperature housing for the cell. An EIL Model 23A direct-reading pH meter was used which was standardised at pH 4.0 by means of Soloid buffer tablets (Burroughs Wellcome & Co.) and at pH 1.0 by using a mixture of 25.0 ml of 0.20 M potassium chloride solution and 67.0 ml of 0.20 M hydrochloric acid.* Readings of pH relating to those solutions to which sodium chloride had not been added could be used directly to obtain the hydrogen-ion concentration.However when a background concentration of chloride (as sodium chloride) was maintained the observed pH had to be corrected for salt effect on the electrodes before it was converted into a hydrogen-ion concentration (in 1.0 M sodium chloride the observed pH values were 0.08 too low). AnalaR-grade sodium chloride, ammonium paramolybdate [ (NH,),Mo,02,.4H,0] and hydrochloric acid were used. A stock 1.000 g 1-1 silicate-silicon standard was prepared by fusing 2.1393 g of Specpure silica with approximately 10 g of AnalaR anhydrous sodium carbonate at 960 "C and dissolving the cooled pellet in distilled water to make 1.000 1 of solution. Method Mixtures (approximately 50.0 ml) of appropriate amounts of molybdate sodium chloride and acid in plastic bottles were brought to the required temperature (h0.05 "C).The mixture was inoculated with an appropriate amount of a silicate standard solution which was also at the same temperature and after mixing an aliquot was quickly transferredinto the spectrophotometer cell at the same temperature. For the more rapid reactions inocula-tion was performed directly into the cell. In these instances the contents of the cell were stirred continuously by mechanical means and afterwards the contents of the cell were established by weighing. In both instances the reaction mixture was retained until itspH had been measured. Molybdate Speciation As mentioned above a facility for quantifying changes within the molybdate system is a prerequisite of any satisfactory interpretation of the kinetics of molybdosilicic acid forma-tion.As the results obtained in this area are not only crucial to our work but also pervade it it is worthwhile to describe them now. The results were obtained by applying the computer program Haltafall,g which is designed to calculate the equilibrium concentrations of the species in mixtures of several components to existing equilibrium constants5t6 for the formation of the molybdate species under conditions of pH and total molybdate encountered in molybdosilicic acid formation (Table I). The prime consideration was to identify th October 1979 a- AND P-MOLYBDOSILICIC ACID FORMATION TABLE I VALVES OF THE EQUILIBRIUM CONSTAXTS ( @ q n ) FOR FORMATION O F THE MOLYBDATE SPECIES H,,(hfoO,),l'"' -'"- FROM p PROTONS AND 4 MOLYBDATE MONOMERS USED I N THIS STUDY 899 P 4 Loglo( isw) 1 1 3.53 2 1 7.26 8 7 52.80 9 7 57.46 10 7 60.84 12 8 71.56 form of the curves that relate the concentration of individual molybdate species to pH or total moljybdate concentration as this form can give a clue to the relationship between a given rate constant and the concentration of the relevant molybdate species.When discussing the effect that varying the total molybdate concentration has upon the concentra-tion of individual members of a family of species that are characterised by a given number of niolybdate atoms for example HM00,- and H,MoO, it is sufficient to consider 1 .o 0.8 0 I-4d 2 5 0.6 C s -a % 0.4 E .- - m 2 0.2 0 0.05 0.10 0.15 0.20 Total molybdate concentration/M as molybdenum Total molybdate concentration/M as molybdenum Total molybdate concentration/M as molybdenum Fig.1 . \'ariation i n the concentration of (a) Mo, ( b ) Mo; and (c) &lo8 species caused by change i l l total iiiolybdate concentration a t the three given pHs of ( A ) 1.0 (B) 5.0 and (C) 6.0 900 TRUESDALE SMITH AND SMITH KINETICS OF Analyst Vol. 104 only one of the family as they are linear multiples of one another. Moreover as ultimately it will be necessary to compare the behaviour of different families each of which can be present at markedly different concentrations it is convenient to normalise the concentration ranges . The graphs of normalised species concentration against total molybdate concentration (0.0-0.20 M molybdenum) for the families Mo, Mo and Mo are shown in Fig.1. This shows that the form of the variation of concentration accompanying change in total molybdate concentration is virtually the same at any pH between 0.6 and 5.0. The individual values for relative concentration at 0.050 M molybdate (Table 11) illustrate this well. Moreover these individual values together with Fig. 1 show that at pH between 5.0 and 6.0 the pattern of variation is very sensitive to changes of pH. The changes in the concentrations of Mo,O,,6- and Mo,02,4- species are very interesting (Fig. 1) because at pH below 5.0 for the Mo species and below 4.0 for the Mo species the graphs of relative species concentration vemm total molybdate concentr<ation are approximately linear. Above these pH limits as in the case already discussed the pattern of variation changes markedly with small changes in pH (Table 11 Fig.1 ) ; the linear approximation is lost. EFFECT OF pH ON THE SHAPES OF GRAPHS OF THE TYPE SHOWN IN FIG 1. Changes in shape are detected as a change in the concentration of the given molybdate species at 0.050 M total molybdate-molybdenum relative to that at 0.200 M total molybdate-molybdenum. PH 1.o-f 0.6 1.6 2.0 2.6 3.0 3.6 4.0 4.6 5.0t 5.6 6.0-f Relative concentration of individual molybdate species* Mo species M’o species Mo species 0.832 0.275 0.228 0.834 0.281 0.235 0.836 0.285 0.238 0.835 0.284 0.237 0.833 0.278 0.231 0.830 0.270 0.224 0.824 0.256 0.211 0.820 0.250 0.205 0.816 0.241 0.197 0.808 0.225 0.181 0.739 0.121 0.089 0.002 0.450 0.004 f A I * Concentration of species at 0.050 M total molybdate-molybdenum.Concentration of species at 0.200 M total molybdate-molybdenum. Full graph given in Fig. 1. ____ Fig. 2 shows how the partitioning of the molybdenum in solution at a given total molybdate concentration changes with pH. Thus at pH above 6.0 the predominant form is the unprotonated ion. As the pH decreases the Mo@246- HMo7?24‘- H2M0702,4-, Mo,O,,~- and H,MoO succeed the Moo4,- a s predominant forms. It is clear therefore, that any consideration of the effect of pH changes on the apparent formation rates of the molybdosilicic acids must take into account the substantial changes that pH induces in the molybdate speciation. Mathematical Modelling As in our earlier study of the transformation of the P-acid, it has been necessary to fit experimental data to mathematical models.In essence the numerical “optimisation” approach used here is similar to that explained earlier. However here it has been necessary to estimate the parameters of the sum of exponentials model A = 8 + 02exp(8,t) + B,exp(B,t). To overcome numerical problems inherent in the use of this model particularly that of preventing the Hessian matrix becoming ill-conditioned near the optimum solution, it has been necessary to reinforce our earlier numerical approach4 with an additional sub-routinelo that makes an approximation to the Hessian matrix that involves only first-orde October 1979 a- AND p-MOLYBDOSILICIC ACID FORMATION 901 1 .o 0 5 - m 0 ,,- 0.8 0 K 0 0 + w .- +.’ L Y- 0.6 u) m C .- CI 2 0.4 0, C 8 8 0.2 MOO 2-0 1 .o 2.0 3.0 4.0 5.0 6.0 PH Fig.2. Variation in the concentration of Pr400,~- HMo0,- H,MoO, M O ~ ~ , HMo,O,,~- H,Mo,O,,~- and Mo,O,,~- species with change in pH at a total molybdate concentration of 0.025 M as molybdenum. The curve for HMo04- is indistinguishable from the abscissa. derivatives (the Jacobian). The solving of the sum of exponentials models is acknowledged as being fundamentally difficult,ll and is encountered in many disciplines e.g. hydrology, nuclear physics and chemical kinetics. However our success results from the combined facts that the data are relatively free of “noise” and only two exponents are involved. Typical examples of fitting the sum of cxponentials model are given in Fig.3. As is demonstrated later in general the individual rate constants within any given over-all reaction scheme that gives the form characteristic of the sum of two exponentials model are related to the parameters Oi i = 1 . . . 5 of the model through non-linear functions. Moreover as there are fewer rate constants than parameters the set of equations are over-determined. Therefore to obtain estimates of the rate constants it has been necessary to perform a further “optimisation” stage. Results and Discussion During the formation of over 100 @-acid solutions at a wide variety of pHs and ionic strengths the increase in absorbance with time has been found to fit the simple exponential model A = + 6,exp(O3t) . . - - (1) with 8 = -O1 63 < 0 and where A is the absorbance at time t and el 6 and e3 are con-stants.As the molybdate-molybdenum concentration of each mixture was greatly in excess of that required for the formation of the ,&acid this behaviour is consistent with pseudo-first-order kinetics with respect to silicate-silicon that is, = K[Si] d W - MSAI - d[Si] --dt dt and therefore [/I - MSA] = [@ - MSA],(l - eckt) as C[Si] = [Si] + [/3 - MSA] where [p - MSA] and [Si] are the concentrations of the fi-acid and unreacted silicate-silicon a 902 TRUESDALE SMITH AND SMITH KINETICS OF Analyst Vol. 104 time t [/3 - MSA] is the final concentration of p-acid and k is an apparent rate constant. As in this instance absorbance is proportional to concentration, A = A (1 - e-kt) giving from equation (l) O1 = -02 = A and 8 = -K.Following accepted procedure,12 the sensitivity of the model parameters to changes in initial reactant concentration was tested to ensure that the adopted model was appropriate; where a model is inappropriate marked changes in the parameters generally occur in this test. The absence of marked changes in the parameters of equation (1) accompanying a five-fold change in initial concentration of silicate (Table 111) but under set conditions of pH and ionic strength supports the appropriateness of the simple exponential model for /3-acid formation. The values of the parameters obtained for the two initial silicate-silicon concentrations do differ slightly under some of the set conditions. However these differences are far too small to affect the appropriateness of the model; they can be attributed to experimental error.In fact these experiments are difficult to perform as they necessitate the use of extreme ranges of absorbance; also it is difficult to match the pHs of the final mixtures precisely because of the high carbonate content of the standard silicate solution. During the formation of a similar number of a-acid solutions the increase in absorbance with time was found to fit the sum of exponentials model A = + e2exp(8,t) + B,exp(e,t) . . TABLE I11 EFFECT OF CHANGING THE SILICATE-SILICON CONCENTRATION ON THE VALUE OF THE PARAMETERS ei (i = 1 . . . 5) I N EQUATIONS (1) AND (2) ( a ) Tests with solutions containing 0.050 M total molybdate-molybdenum and up to 0.1 M added sodium chloride at 22.0 "C-Conditions of test f A r -l PH Silicate-silicon 1.01 1.992 1.01 1.970 1.01 0.421 1.01 0.408 1.52 1.877 1.52 1.973 1.52 0.419 1.52 0.413 ( 0.03) concentration/mg 1-1 Product ,%Acid p-Acid I Parameters of equation (1) A 1 ell ( a b s o r b a n ~ e ) ~ ~ ~ 0 " ~ per mg l-l Si B,/min-I 0.430 - 1.58 0.430 - 1.59 0.444 - 1.60 0.446 - 1.65 0.430 - 2.50 0.428 - 2.48 0.439 - 2.58 0.438 - 2.63 0.358 -2.17 0.357 -2.12 0.372 - 2.30 0.372 - 2.27 0.278 - 1.50 0.277 - 1.48 u- + P-Acids U- + P-ACidS 0.298 - 1.62 0.299 - 1.55 0.267 - 0.65 0.267 - 0.65 0.264 - 0.60 0.265 -0.58 2.56 1.989 2.56 1.920 2.52 0.409 2.52 0.403 3.50 1.975 3.50 1.908 3.58 0.402 3.58 0.406 4.18 2.108 4.17 2.111 4.13 0.426 4.13 0.437 u- Acid October 1979 903 a- AND P-MOLYBDOSILICIC ACID FORMATION TABLE III(continued) ( b ) Tests with solutions containing 0.050 M total ~ o l ~ ~ b d a t e - ~ p z u l ~ ~ O d p n u m m i d 1 .0 bi added sodium cklorsde at 22 “C-Conditions of test --h----7 PH Silicate-silicon (k0.03) concentration/mg 1-1 1 .oo 1.883 1 .oo 1.958 1.00 0.393 1 .oo 0.413 1.55 1.862 1.55 1.965 1.51 0.383 1 i 1 1.51 0.380 J I 2.53 1.842 2.52 1.883 Product P-Acid P- Acid 1 a- + /I-Acids 2.49 0.386 2.49 0.385 3.52 1.894 3.52 1.868 3.48 0.381 ]i 3.49 0.398 a-Acid Parameters of equation (1) A 7 -7 el/ ( absorbance):!:l,:”’ per mg 1-1 Si 0.448 0.453 0.468 0.469 0.429 0.432 0.447 0.449 0.318 0.319 0.327 0.330 0.287 0.287 0.296 0.293 B,/mi 11 -- 1.70 - 1.65 - 1.70 - 1.67 - 2.50 - 2.48 - 2.62 -2.51 - 1.65 - 1.64 - 1.76 - 1.79 - 0.67 - 0.68 -0.78 -0.79 4.0 Fittings of experimental data to equation ( 1 ) inappropriate (G) l e s t s with solartions contaanang 0.200 M total n?ol~ibdatr-nzod,ibde~~i~na and 1 .O ni added s o d i i m chloride at 25.0 “C a% whach the a-aczd as fovrned accovdzng to the s w n of cxpoiirntzals n w d e -Conditions of tcst A I’arameters of cquation ( 2 ) - -7 -7 Silicate-silicon concentration/ - ___ PH mg 1-1 01 * e * 04* O,/min-’ O,/min-’ 3.41 3.227 0.269 -0.136 -0.138 -6.83 -1.09 3.39 1.308 0 266 -0.130 -0.139 -6.92 -1.17 * (Absorbance):9:l,P‘” per mg 1-1 Si. ~ i t h the parameters O, O, 8 and 8 all negative.[In extreme instances where 8 is much greater than 8, or vice versa and 6 and O4 are comparable a “rapid” initial increase in product is followed by a “slow” approach to equilibrium Fig. 3 (h).] The appropriateness of the model was confirmed in a similar test to that used with the ,&acid solutions where the initial silicate-silicon concentration was varied. This test showed (Table 111) that, within the hypothesis of marked changes the parameters 0 and 8 are independent of the initial silicate concentration but that the parameters 8, 0,. and 8 are linearly dependent upon it. Also the possibility that the goodness of the fitting of the model was due to a shortage of relevant molybdate species was rejected when it was observed that the addition of more silicate to a reaction mixture in a “slow” growth phase re-established a “rapid” growth phase.Experience has also shown that at low molybdate concentrations and low ionic strengths the sum of exponentials model for oc-acid foi-iiiation reduces to the single exponential model [equation (l)]. The approximation to pseudo-first-order kinetics at these low molybdate-molybclenuiii concentrations was shown to be satisfactory (Table 111) when the apparent rate constant under given conditions was found to be largely independent of the initial silicate-silicon concentration. More details of this will lie given below 904 TRVESDALE SMITH AX D SBIITH KINETICS OF . 4 t l 4 S t Yol. 701 0 1 2 3 4 5 6 0 12 24 36 48 60 Ti me/min Fig. 3. Typical examples of fitting experimental data points (x) taken directly froni the spcctro--1bsorbances a t 390 nm photomcter recortlcr chart to the sum of exponontials model jequation (2)].4-cm path length. Choice of Silicate Standard The results reported in this paper were obtained using working standard silicate solutions that contained 100.0 mg 1-1 or less of silicate-silicon. In preliminary work the silicon in stronger solutions was found to show different kinetic characteristics to that present in more dilute solutions. Thus in one experiment where the formation of the /3- ancl r.-acids was investigated at pH 1.03 ancl 4.09 at a molybdate-molybdenum concentration o f 0.050 M but without added sodium chloride pseudo-first-order rate constants for the formation of the acids were found to increase sympat1ietic;illy with a decrease in inoculum-silicate con-centration (Table 11').(By careful choice of pH the rate constants for both acids were TABLE IV EFFECT OF r m t i T I x G THE STOCK SILJCATI. IKOCULUM SOLUTIOK ox THE KIXETICS OF FORMATIOX OF Q- AND /~-ACII)S AT BOTH LOW IOXIC STRESGTH (0.050 11 J I O L I ~ ) E N ~ ~ J I ) ANT) AT PH 4.09 ,4m 2.03 RESPECTIVELY (0.0 BI EXCESS OF SODIUM CHLORIDE) AS11 LOW TOTAL MOLYBIIATE Tcnipcraturc not recorded but probably 17.0 "C. Concentration of silicate stock solution/ m g 1-' silicon 1000 500 400 200 100 50 20 First-ortlcr apparent rate constant requation (1) ; r-k,/niinrl 1.3s 1.53 1.55 1.57 1.61 1.67 1.64 1.36 1.49 1.58 1.55 1.63 1.64 1.67 made similar and indeed the two sets of results are almost identical.) At a given pH all of the reaction mixtures had tlie same composition; the volume of inoculum used was decreased as tlic silicate concentration was increased and compensating volumes of distilled water were added prior to inoculation.The results (Table IY) show that whereas there i October 1979 a- AND /~-MOLYBDOSILICIC ACID FORMATION 905 little or no change in the formation rate constant with up to 100.0 mg 1-1 of silicate-silicon, there is a marked change at higher concentrations. Despite its affecting the formation kinetics a change in the concentration of the inoculuni was founcl not to affect the yield of molybdosilicic acid. Thus in a separate experiment two sets of P-acid mixtures each of six replicates prepared in the aforementioned manner and using either 1000 or 100.0 mg 1-1 silicate-silicon standards yielded mean absorbances (390 nm 4 cm) of 0.800 and 0.801 after subtraction of a mean blank absorbance of 0.149 obtained in an identical fashion but without added silicate.As the standard deviations of these two sets of results is 0.005 in both instances the difference between the means is not significant. Although the changes referred to above in tlie rates of formation of the molybdosilicic acids are undoubtedly caused by silicate speciation it is not yet known how the change in speciation should be apportioned between the dilution and the concomitant change in pH of the silicate standard. Effect of Molybdate Concentration The effect that changing the total rnolybdate concentration has upon 13- and %-acid forma-tion was studied at pH 1.2 and 3.5 respectively in a background of 1.0 M sodium chloride.The sodium chloride was added to minimise the effect of ionic strength clianges that occur together with changes in total molybdate concentration. Under these conditions molvbdate salts crystallise within 1-2 h of mixing if the niolybctate concentration is in excess of approxi-mately 0.040 M as molybdenum; to circumvent this problem mixtures were prepared and tested within approximately 0.5 h. Blank determinations showed no change in absorbance within the test period. Although from earlier work1 it might appear that the pH of 3.5 is too low for a-acid production as will be explained later at these higher ionic strengths the pH regimen where only the a-acid f o r m is extended to this lower pH.,&Acid fownation At pH 1.2 the kinetics of @id formation are pseudo-first-order throughout the range 0.00-0.200 SI niolybdate-molybdenum. I'alues of tlie pseudo-first-order rate constants obtained at 25.0 "C are given in Table V. The data can be seen to fit the model well by the fact that the root-mean-square error is less than 1.99 x in all instances that is, less than 0.4o/b of the change in absorbance. For each instance tliis error measures the average deviation of all the data points from the fitted curve and is calculated by taking the square root of the quotient of the total sums of squared deviations and the number of data points. molybdate M O Q ~ ~ ~ - 3 1 0 ~ 0 ~ ~ ~ - or A 1 0 0 ~ ~ - concentratinns in any simple waj-.investigation showed however that the pseudo-first-order rate constant k of Preliminary investigations showed that the apparent rate constant is not related = k[Si] d[/3 - MSA] _ _ _ _ ~ ~ - ~ ~ dt where [#I - MSA; and [Si] are the concentrations of tively at time t is given by giving to total Further 13-acid and unreacted silicate respec- . . . . (3) 1 + n Of course an analogous equation in which SIo replaces Mo8 could have been developed. However because the Slo species is predominant in the pH range where tlw ,!?-acid forms (l5g. 3) it seems likely that the 110 species is the appropriate candidate 906 TRUESDALE SMITH AND SMITH KINETICS OF A~aaEyst Vol. 104 TABLE V APPARENT RATE CONSTANTS AND ROOT MEAN SQUARE ERRORS OBTAINEII BY FITTIKG EXPERIMENTAL DATA FOK P-ACID FORMATION TO THE EXPONENTIAL MODEL OF EQUATION (1) Total molybtlate concentration/ M molybdcnuin 0.200 0.175 0.150 0.125 0,100 0.090 0.080 0.070 0.060 0.050 0.040 0.030 0.020 0.010 Xpparcnt rate constant k b (- B,)/min-l 3.613 3.66 3.45 3.51 3.33 3.20 3.17 3.21 3.21 3.08 3.02 2.97 2.82 2.98 2.73 2.75 i"! 2.00 Root mean square error x lo4 9.73 19.2 13.5 11.9 6.94 6.83 13.1 11.5 19.9 11.6 18.8 18.9 0.81 14.2 8.8 12.9 0.6 12.0 1.0 14.7 14.0 In practice the curve of the graph of iMol]/k versus l/[Mog] was found to fit a linear model well (correlation coefficient 0.9870) with a gradient and intercept of 7.06 x 10-11 and 1.144 x respectively accompanied by standard errors of 0.25 x 10F1 and 0.013 x 10-8 respectively.These parameters yield values for a and h of 8.74 x lo7 1 g-ion-1 min-1 and 6.17 x lW3 g-ion 1-* respectively. As can be seen from Fig. 4 the experimental data lie close to the derived equation. Total moiybdate concentration/M as molybdenum Fig. 4. Comparison bctween observed (A) and predicted values [line derived from equation (3)] of the apparent rate constant for p-acid formation at pH 1.80 & 0.02 1.0 hi added sodium chloride, 25.0 "C and various total molybdate concentrations October 1979 u- AND /3-MOLYBDOSILICIC ACID FORMATION 907 This form of rate law [equation (3)] suggests that an equilibrium step occurs prior to the rate-determining step.A possible reaction path is k , k-1 Si(OH) + MOO^^- I + H 2 0 k 2 I + Mo8022- -+ products slow The rate-determining step is the reaction between the Mo species and the intermediate complex I which contains one silicon and one molybdenum atom. Assuming a steady-state condition for I this gives This equation is identical in form with the experimental rate equation and by comparison, values of 8.74 x 1071g-ion-lmin-l and 6.17 g-ionl-1 are obtained for k and the ratio k-,/k, respectively. As the concentration of the intermediate I could not be measured it was not possible to calculate the magnitude of the individual rate constants k- and k,. u-Acid formation At pH 3.5 the formation of a-acid followed the sum of exponentials model [equation (2)].Values of the parameters obtained at 25.0 "C at each of several molybdate-molybdenum concentrations are gven in Table VI. The data can be seen to fit the model well by the fact that the root mean square error is less than 1.83 x in all instances. Estimates of TABLE VI Total molybdate concentra-tion/M molyb-denum 0.20 0.150 0.100 0.090 0.080 0.070 0.060 0.050 0.040 0.030 0.020 0.010 0.005 PARAMETERS AND THEIR VARIANCE ESTIMATES OBTAINED FROM FITTING EXPERIMENTAL DATA RELATING TO DIFFERENT TOTAL MOLYBDATE CONCENTRATIONS TO THE SUM OF EXPONENTIALS MODEL Root mean 8 + square Var Var 8 O3 (.-kl)/ Var e3 Var O4 Ob(.-ks)/ Var Oh 8$ -k error 8 x lo6 8 x 109 min-1 x 106 O4 x 109 min-1 x 103 O4 x lo* 0.816 1.53 0.808 1.24 0.799 42.6 0.792 26.0 0.802 1.89 0.785 1.93 0.801 1.85 0.811 1.59 0.794 2.03 0.787 1.90 0.790 6.45 0.802 1.27 0.796 -0.789 -0.799 22.2 -0.388 - 0.366 - 0.371 - 0.348 -0.342 - 0.354 -0.331 - 0.303 -0.289 -0.272 -0.193 - 0.796 -0.789 -0.185 - 0.382 7.38 5.35 93.1 61.6 36.1 52.4 30.3 197 127 170 14 400 197 000 --2 20 - 6.89 - 7.11 - 5.54 - 3.36 - 3.22 - 2.84 - 2.51 - 1.95 -1.84 - 1.46 - 0.91 0 - 1.08 - 0.417 -0.217 - 0.034 8.50 6.62 8.57 3.48 4.18 1.92 3.66 2.30 1.82 43.7 15.9 10.3 --0.012 - 0.414 - 0.41 3 - 0.417 - 0.413 - 0.443 - 0.437 - 0.445 - 0.489 -0.516 - 0.613 - 0.481 - 0.500 - -- 0.572 6.84 4.55 44.3 60.7 30.9 48.1 25.7 189 111 152 14000 193000 - -336 - 0.887 - 0.898 - 0.780 - 0.623 -0.678 - 0.604 - 0.482 - 0.581 - 0.526 - 0.447 -0.471 - 0.495 - -- 0.144 0.138 0.014 7.73 0.109 0.013 7.19 1.8.5 0.016 18.3 0.249 0.008 14.5 0.145 0.011 6.62 0.07 0.002 7.62 0.193 -0.001 F.45 0.131 0.002 6.37 0.100 -0.002 6.09 1.93 0.002 7.88 0.27 -0.004 5.50 0.153 0.006 8.80 - - -- - -0.001 0.042 10.9 Time for 9876 formation/ min 3.7 3.6 4.2 5.2 4.9 5.5 6.9 5.8 6.5 11.0 r r I ./ 7 .4 9.4 51.0 18.0 the variance for each parameter are also given in Table VI. These values were obtained from the variance - covariance matrix which in turn is calculated from the inverse of the approximation to the Hessian matrix that involves first derivatives of the error function with respect to the parameters evaluated at their optimum values4 In all instances the variances are negligible when compared with the magnitude of the parameters.At 0.010 and 0.020 M molybdenum concentrations the sum of exponentials model collapsed to the simpler one, and estimates of the parameters 8 (-82) and e3 were obtained using the model represented by equation (1). At a 0 . 0 0 5 ~ molybdenum concentration however the data reverted to being better fitted to the sum of exponentials model [equation (a)]. Nevertheless the values of these parameters appear to be compatible with those obtained at the higher concentrations (Table VI). Within the limits of experimental and numerical method errors the paramete 908 TRUESDALE SMITH ASI SMITH KISETICS OF Aiialyst lJol.101 O1 was constant for all total molybdate concentrations and the sum of the parameters el 8, and 0 was equal to zero across the range of molybdenum concentrations studied; the para-meters 8 and 8 vary antipathetically. The results (Table VI) also show that notwith-standing the presence of “rapid” and “slow” growth phases tlie over-all rate of formation of tlie %-acid (to 98yA completeness) at a given pH increases with total molybdate concentra-t ion. Several cliemical reaction schemes that yield the sum of exponentials model can be written (Table VII) ~ ,411 involve combinations of first-order reactions that link tliree silicon species that are either synthesised after or introduced during inoculation of the reaction mixture with silicate-silicon ; this leads to the species Y having in the two instances an initial concentration (yo) o f zero or non-zero respectively.The equations that describe each system are gi\wi in full as in our experience several text-b~oks’~-l~ do not always express them correctly and their derivation is not simple. Further examples of schemes that give the sums of exponentials models can be obtained by making reversible any of the uni-directional reactions in Table VII. The identification of the actual scheme is probably best accoinplislied by combining chemical experimentation with nunierical modelling. Altlrougli a unique sclieine has not been identified using this approach it is worthwlde discussing further the applicability of schemes such as (i)-(v) (Table VII) as they assist in the appreciation of the chemistry involved.A s our results (Table VI) sliow that tlie parameters 8 and O4 of equation (2) are negative, reaction scheme (i) (Table VII) which requires 8 and 8 to have opposite signs (with yo and zo set to zero) can be rejected. Also all those scliemes involving the cx-acid in an cqui-librium with one or both reactants e - g . scheme (iv) in Table VII can also be rejected on stoicheiometric grounds as they are not consistent with the observation of full formation of %-acid across a wide pH range. This is supported by equation (9) where at equilibrium (t-+m) the concentration of x-acid z is only a fraction k 3 / ( k + k5) of tlie initial silicate concentration so; the remainder is present as the intermediate Y 1 equation @)I.The rate constants of schemes (ii) antl (iii) (Table 1’11) have been obtained (Table VIII) by using the values of the parameters (02-e5) found from the earlier fitting. This was not necessarjy with scheme (v) (Table VII) because there tlie reaction rate constants (k and K 3 ) can be considered to be the parameters 8 and O5 of the sum of exponentials model. As with the original fitting (Table i71) tlie magnitude of the root mean square errors given in Table l V I I I indicate that the fittings are satisfactory as they are less than 7.09 x 10-3, Similarly the magnitude of the variance associated with each estimate o f the rate constant is small when compared with the value of the respective rate constant itself.A surprising feature of the results in Table \;I11 is that the estimates of k, var k and root mean square error for schemes (ii) antl (iii) are tlie same for any given total molybdate concentration. Subsequent to observing this behaviour we have been able to show mathematically that this must be so at least for k,. Notwithstanding this we are pleased that the numerical approach adopted here lias demonstrated this fact as this has provided an unsolicited test of the methods used. The proof of this unexpectecl property of these reaction schemes will be given elsewliere. A clear understanding of the difference between schemes (ii) and (iii) (Table VII) can be obtaintd from Fig. 5 where for both cases tlie concentrations of reactant intermediate and product are plotted against time.The data for this exercise arise from a single kinetic run antl the result:; displayed in Fig. 5 are obtained by re-introducing the experimentally derived reaction rate constants into the relevant equations that clescribe the concentration of each species at an). time c.g. equations ( 5 ) (6) and (7) (Table WIT). The graphs show that although in both instances the concentration of the interniediate Y increases to a maximum after a short reaction period tlie maximum amount of interniediate formed differs in the two instances. After the maximum amount 11 as been attained this discrepancy is compen-sated for 1,)- an equal and opposite one in tlie concentration of unreacted silicate. It can be seen that in sclienic (iii) tlie unrcactcd silicate concentration is continuously maintained ltiglicr than that encountered in scheme (ii).To facilitate further interpretation of the results from Tables VI and VIII graphs of the rate constanti; w i w s total iiiolS.bdate-iiiolSrbdenuiii concentration have been plotted (Fig. 6). From these it can be seen that for all tliree models the rate constant k, appears to vary linearly with total niolybdatc concentration. It has already been shown (Fig. 1) that th TABLE VII EXAMPLES OF REACTION SCHEMES THAT FOR THE PRODUCT CONCENTRATION YIELD THE SUM Reactants or intermediates X and Y are converted into product 2 (molybdosilicic acid) in first-order steps ki (i = 1 . . . 5 ) . The concentrations of X Y and 2 a t any time t are denoted by x y and z respectively, yo and zo. in the present work demand that y o = zo = 0 in schemes (i)-(iv) and zo = 0 in Where appropriate terms hl and At which consist of combinations of k, k and k, have been Scheme Description Equations (i) Consecutive reactions __ x = x exp (- k,t) (v) Parallel reactions x = x o exp ( - k,t) [ k y = yo exp ( - k,t) z = x o + y o i- z - x o exp (- klt) - y o exp (- k,f) S k1 910 TRUESDALE SMITH AND SMITH KINETICS OF Analyst Vol.104 variation in the concentrations of both M o ~ O ~ ~ ~ - and Mo,O,,~- species is approximately linear with total molybdate concentration. Thus both observations are consistent with the rate constant K, being first order with respect to either Mo,O,,6- or Mo~O,,~- concentra-tion. Similarly in scheme (ii) (Table VII) the rate constant k, would also appear to be first order with respect to either of these species.The remaining rate constants are not related linearly to total molybdate (Fig. 6) and consequently their behaviour is more difficult to understand with the possibility of the otheir molybdate species being involved. TABLE VIII PARAMETERS (PSEUDO-FIRST-ORDER RATE CONSTANTS) AND VARIANCE ESTIMATES OBTAINED FROM FITTING THE PARAMETERS OF THE SUM OF EXPONENTIALS MODEL o, o, e AND e5 [EQUATION (2)1 TO TWO REACTION SCHEMES ( a ) The coinpetitzve - consecutive reactton scheme [the coej5caents and exponents of equation (6) scheme (ii), Table V 1 4 -Total molybdate concentration/ M molybdenum 0.200 0.150 0.100 0.090 0.080 0.070 0.060 0.050 0.040 0.030 0.005 k , 3.79 3.01 1.92 1.80 1.59 1.38 1.14 1.03 0.817 { ::% 0.116 0.090 Var 12 x lo3 138 186 8.82 6.47 0.371 0.219 0.074 4 0.905 0.593 1.95 1.76 7.56 15.2 k2 3.09 2.52 1.44 1.42 1.25 1.13 0.812 0.813 0.642 0.288 0.448 0.028 0.037 Var k x lo3 147 209 9.7 18.6 7.96 0.453 0.296 1.35 0.927 3.56 0.393 2.80 10.7 k , 0.887 0.780 0.623 0.678 0.604 0.482 0.581 0.526 0.447 0.470 0.495 0.034 0.012 Var k x lo3 265 362 16.7 28.6 12.3 0.713 0.418 1.68 1.08 3.06 2.83 0.132 13.9 Root mean square error 6.07 7.09 3.81 4.99 3.27 0.787 0.602 0.348 1.21 0.969 1.63 0.188 4.07 x 103 (b) The coiisecutivc reaction with reversible steps scheme [the coe8cients and exponents of equation (7) scheme ( i i i ) Table V11]-Total molybdate concentration/ M molybdennm 0.200 0.150 0.100 0.090 0.080 0.070 0.060 0.050 0.040 0.030 0.005 kl 3.79 3.01 1.92 1.80 1.59 1.38 1.14 1.03 0.817 { :::XE 0.116 0.090 Var k x loa 138 186 8.82 6.47 0.371 0.219 0.0744 0.905 0.593 1.95 1.76 7.56 15.2 k2 2.37 1.87 0.973 0.886 0.776 0.733 0.398 0.398 0.291 0.070 0.099 0.020 0.032 Var k x lo3 394 543 19.8 33.7 14.4 0.915 0.340 0.128 1.41 0.253 0.907 0.630 7.34 k4 1.61 1.44 1.09 1.21 1.08 0.877 0.995 0.941 0.798 0.688 0.844 0.043 0.016 Root mean square error Var k x lo3 x lo3 678 6.07 913 7.09 35.4 3.81 57.7 4.99 24.6 3.27 1.54 0.787 0.609 0.602 0.214 0.348 2.35 1.21 0.743 0.969 2.13 1.63 3.73 0.186 0.24 4.07 An important constraint upon the three sc‘hemes is that they should accommodate the observed collapse of the sum of exponentials niodel [equation (2)] to that of the simple one [equation (l)] at low total molybdate concentrations.This can be seen by re-writing the sum of exponentials model as where el& = 8 and = 04 and then comparing these coefficients and exponents with those of equations (6) (7) and (10) (Table VII) after yo and zo are set to zero where appro-priate and the equation is yre-multiplied by eZ the absorptivity of the cc-acid. Then for the individual schemes we have the following October 1979 a- AND P-MOLYBDOSILICIC ACID FORMATIOX 91 1 (a ) x + y + z 3 4 5 6 0 1 2 3 4 5 6 1 2 Time/min Fig.5. Variation with time of the concentrations of unreacted silicate intermediate and a-molvbtlosilicic acid when the a-acid forms by (a) the competitive - consecutive reaction jschcme ( i i ) Table \'LIj and by (b) consecutive reactions with a reversible step [scheme (iii) 'Table 1'11 j The coiiimon experimental data upon which these graphs are based arise from that shown in Fig. 3 ( a ) . (a) Schenze (ii). At low total molybdate concentration (Fig. 6) both k and k tend to k,. Therefore by comparing coefficients and letting k = k = k = k we obtain 1 - _ + == k 2 -k - k - k , Therefore A = eZxO(l - e-Xt) which is analogous to equation (1). k, k tends to zero.k = 0 we obtain ( b ) Sclzeme (iii). At low total molybdate concentration (Fig. 6 ) whereas k tends to Therefore by comparing coefficients and letting k = k = k and A = lim ( k + 6) = k and A = lim ( k - 6) = k 6+0 6+0 Therefore A = E ~ X ~ ( I - eckt) which is analagous to equation (1) 912 TRUESDALE SMITH AXD SMITH KINETICS OF Analyst Vol. 104 (c) Scherne (21). At low total molybdate concentration (Fig. 6) k tends to k,. Therefore, hy comparing coefficients and letting k = k == k we obtain 462 = - xo 0 = - k,= - k $4 = -Yo 0 5 - - - k 3 = - k Therefore A = eZ(xO -+- yo) (1 - e-kt) which is analogous to equation (1). 4.0 7 3.0 E E .-I w C C 2.0 2 2 1.0 a) c 0 0.05 0.10 0.15 0.20 Total molybdate concentration/M as molybdenum 0 0.05 0.10 0.15 0.20 Total molybdate concentration/M as molybdenum 8.0 C t 7 6.0 E 5 4.0 I-1 *.’ s 2 2.0 (u + 0 0.05 0.10 0.15 0.20 Total molybdate concentration/M as molybdenum \-dues of the individual rate constants obtained by fitting the parameters of the sum of exponentials model (Table VI) to three reaction schemes (Table 1’11) (a) competitive - consecutive reactions, schcme (ii) ; ( b ) consecutive reactions with a reversible step scheme (iii) ; and (c) parallel reactions scheme (v).In each graph a given symbol represents one curve; i t has no wider significance I;ig. 6. Xote that each curve is fitted “by eye” to its points. I t is not yet known why the fitting reverts to the sum of exponentials model at 0.005 M niolybdate-molybclenum.This observation is inconsistent with the analysis of the three models given above as they will remain as the simple exponential model. However it is suspected that the reversion is caused by there being insufficient total molybdate within the system a constraint that must be encountered if the total molybdate concentration is progressiveljr reduced. In this instance it is inevitable that the sum of exponentials model would offer the better fitting as it includes two extra parameters although it would not be the correct one. We believe therefore that the observation does not detract from the aforementioned analysis October 1979 a- AND P-MOLYBDOSILICIC ACID FORMATION 913 The possibility that the intermediate Y in scheme (ii) (Table VII) is P-molybdosilicic acid has also been considered.[A similar condition for scheme (iii) was rejected because of the unlikelihood of a reversible reaction between reactants and the @-acid.] In this instance the observed absorbance A t is composed of contributions from both acids Y and 2 which have absorptivities ey and eZ respectively Thus, A = Eyy + EzZ . . which from Table VII leads to Equation (12) is equivalent to the sum of exponentials model. However the inclusion of the P-acid in scheme (ii) (Table VII) yielded negative values for k at all but two of the thirteen total molybdate concentrations studied and consequently has been eliminated. As yet further elimination of the possible reaction schemes has not proved possible. Thus, schemes (ii) and (iii) (Table VII) cannot be ranked by any statistical method as the root mean square errors for each fitting within the sequence of total molybdate concentrations (Table VIII) are identical and moreover they both have the same number of parameters.Further a statistical comparison of these with scheme (v) (Tables VI and VII) is impracticable as the numerical method has not been taken to an equivalent state in the two instances; for scheme (v) it was possible to use the exponents of the more complex model directly (Table VI) . Effect of Hydrogen-ion Concentration on Formation of the Acids The greatest problem likely to be encountered in any study of the effect of variation in hydrogen-ion concentration on molybdosilicic acid formation is that of understanding what happens in the pH region where the acids form together.Outside this region the problems are only those of measuring and interpreting changes in the magnitude of pseudo-rate constants for the formation of a single product. Within the region of co-formation it is necessary to resolve the over-all change in reaction variables e.g. absorbance into a part due to the a-acid and a part due to the P-acid. Our experience suggests that this will be very difficult and perhaps impossible if the over-all change in absorbance during formation is determined by the combined effects of the a-acid formation following the sum of expo-nentials model and the P-acid formation following the single exponential model. Fortu-nately however some idea of the extent to which both a- and P-acid systems react to hydrogen-ion concentration can be gauged from studies at low total molybdate concentrations and low ionic strengths.There the sum of exponentials model for the a-acid collapses to the simpler one [equation ( l ) ] which is more amenable to treatment and moreover the over-all change in absorbance with time in the region of co-formation also follows the single exponential model [equation (l)] . The variation with pH of the apparent rate constant of formation was studied between pH 0.9 and 4.2. Various amounts of sodium chloride were added to each solution to main-tain a constant chloride concentration in all samples. Thus whereas samples with a low pH gained chloride from hydrochloric acid used in reaction with molybdate samples with higher pH required the addition of sodium chloride.In this way the ionic strength contri-buted by chloride salts to all of the solutions was kept within 1.00 0.01. The results show that the over-all apparent rate constant attains a maximum value at approximately pH 1.8 (Fig. 7). The variation in the final absorbance of each mixture as predicted by the model [S, equation (l)] is also shown in Fig. 7. This graph of absorbance against pH is similar to that which was originally usedl to de-limit the pH regimens of a- and P-acid formation but which was obtained by direct observation of solutions that had been allowed t o react for up to 30 min 914 TRUESDALE SMITH AND SMITH KINETICS OF Analyst Vol. 104 Further treatment of the data is dependent on the assumption of mechanisms for a- and /3-acid formation.Initially one likely possibility involved the competitive formation of the acids by independent first-order reactions : a:-acid 7 \ k b si\ /?-acid Also it had already been suggested2 that the formation of the P-acid could precede that of the a-acid in consecutive first-order reactions. Scheme (i) (Table VII) describes this mechanism when the intermediate Y is the P-acid and the product 2 the a-acid. Although both schemes can accommodate the observation that the "pure" solutions of a- and P-acids are formed according to the single exponential model the consecutive scheme is rejected as it does not represent the co-formation adequately. Moreover our earlier estimates4 of the rate constants of the second step of the consecutive reaction scheme (that is the trans-formation of the P-acid) are too low to support the observed over-all rates of a-acid formation.2.0 1.6 rl I C .-E \ tu 44 C 1.2 8 + 0.8 a w F 2! a C 8 0.L 0 - 0.8 - 0.6 -d E s- 0 0, W C - 0.4 5 2 2 n - 0.2 4 - 0.0 1 .o 2.0 3.0 4.0 PH Fig. 7. Variation with pH of the over-all apparent rate constant and the individual apparent rate constants for formation of a- and fi-acids a t low ionic strength and low total molybdate concentrations where the single exponential model [equation (l)] applies. The variation with pH of the final absorbance predicted by the model (6,) is also shown in the upper graph. The results were obtained a t 17 "C with 1 . 0 M sodium chloride and 0.025 M molybdate as molyb-denum present in the reaction mixture.In the lower graphs A represents the over-all (observed) apparent rate constant B that for the /I-acid and C that for the a-acid. Thus at 17.0 "C typical values for the transformation rate constant4 are only 0.05 times the over-all observed rate constant - O3 [equation (l)] for a-acid formation. Accordingly, the over-all apparent formation rate constant k(-0,) [equation (l)] has been divided int October 1979 a- AND /3-MOLYBDOSILICIC ACID FORMATION 915 separate rate constants k and kp appropriate to formation of a- and P-acids respectively. Under these circ~mstancesl~ the sum of the individual rate constants is equal to the over-all rate constant the ratio of k and kp is equal to the ratio of the final concentrations ceca and Pm respectively and the sum of concentrations of a- and P-acids at equilibrium is equal to the total initial silicate concentration so.Thus, k = k + kp . . . . (13) so = aca + p w - - . . . . . . (15) Substituting k from equation (14) into equation (13) and rearranging gives k p = ( a00 Bw + p m ) k = ( y ) k . . . . Also substituting am from equation (15) into equation (ll) which sums the absorbances contributed by the two acids with absorptivities E and E D gives or Thus from equations (16) and (17) we obtain Similarly, . . x k k . . . . (17) . . (18) . . . . (19) Through the use of equations (18) and (19) it has been found that the graphs of the apparent rate constant vemws pH for both a- and P-acid formation have an inverted U-shape.Although not drawn in Fig. 7 because of paucity of appropriate data these curves can be expected to approach the pH axis (zero rate constant) asymptotically. The apparent rate constants for a- and P-acid formation attain maximum values at pH 3.2 and 1.7 respectively. The changes in each of the two apparent rate constants caused by change in pH (Fig. 7) are much more difficult to interpret than the corresponding changes that resulted from variation in total molybdate concentration. Of course both interpretations are crucial to a complete understanding of the system. The difficulty that precludes full interpretation of Fig. 7 arises because a change in pH means not only a change in hydrogen-ion concentra-tion but also a concomittant change in the concentration of each of the several molybdate species present.Nevertheless it is possible to gain a semi-quantitative appreciation of Fig. 7 that substantiates the earlier interpretation of the effects of varying total molybdate at constant pH as well as indicating where future work might be most usefully directed. Thus there is a noticeable correlation between the shapes and positions of the graphs relating the apparent rate constants for formation of a- and P-acids to pH (Fig. 7) and the shapes and positions of the graphs relating the concentrations of H,Mo,O,,*- and Mo802,’-ions to pH (Fig. 2). These correlations are consistent with the mechanisms that were obtained from the total molybdate studies described earlier which include the ion H,Mo702:-as an important precursor of the a-acid and the ion Mo,O,,~- as a precursor of the P-acid.Moreover it would have been surprising if the variations in the apparent rate constants fo 91 6 Analyst Vol. 104 the formation of both a- and P-acids had been totally explained by changes in concentration of just these ions. Therefore any discrepancy between the observed variation in apparent rate constant and that predicted by the mechanisms can be attributed to the omission of other ions or species. With cc-acid formation which is now believed to be first order with respect to the concentration of H2M~70244- the inclusion of these other species would have to account for the small departure from linearity of the graph of apparent rate constant against H2Mo,02,4- concentration (Fig. 8). One plausible explanation for this is that a term for hydrogen-ion concentration should appear in the denominator of the rate equation, perhaps as a result of hydrogen ions being a product of one or more elementary reaction steps.With P-acid formation this approach is unsuitable because the relationship between the apparent formation rate constant and the concentrations of the identified Mo and Mo, precursors is not simple [equation (3)] and the appropriate graph cannot be drawn. As the description of the molybdate system used here is tenuous it would seem to be provident to reconsider the alternative descriptions of the molybdate system before attempting to force analvsis of Fig. 7 further. TRUESDALE SMITH AND SMITH KINETICS OF I 3.15 0 0.2 0.4 0.6 Concentration of H2 Mo 0244- ion Fig.8. Graph of the apparent rate constant for u-acid forination against the concentration of H,Mo,~,,~- ion expressed as a fraction of the total molybdate con-centration of 0.025 M as molybdenum. The solid line would be given by a perfect correla-tion. The dotted line indicates the sequence of the experimental points. General Discussion The results presented here have reinforced the view that the pH boundaries for a- and 13-molybdosilicic acid formation are kinetically determined and they depend on the time allowed for reaction. A practical implication of this is that observed boundaries for full formation can be in error if insufficient time is allowed; in our earlier work,l for example, we limited reaction times to 30 min. A further but less obvious implication is that owing to the instability of the P - a ~ i d ~ it is worthwhile to consider the state of the molybdosilicic acid system in both the short term (reaction times used in analysis) and the long term (infinite reaction time).In the short term the over-all pH boundaries for the formation of molybdosilicic acid arise from competition between formation and decomposition reactions. At low ionic strengths and room temperature these boundaries are approximately pH 1.0 and 5.0. Increases in ionic strength decrease the competitiveness of the formation reaction at the upper boundary October 1979 a- AND p-MOLYBDOSILICIC ACID FORMATION 91 7 thereby moving it to approximately pH 4.0 (not 3.5 as printed in an earlier paper4) at an ionic strength of approximately 1 .O.Within the over-all pH domain for molybdosilicic acid formation competition between separate formation reactions produces distinct pH regimes for a- and P-acid formation as well as a third where mixtures of the acids form. At low ionic strengths these pH regimes for exclusive a- and ,&acid formation are approximately 1.0-1.8 and 3.8-5.0 respective1y.l Increases in ionic strength increasa the competitiveness of the a-acid formation reaction so that at an ionic strength of approximately 1.0 only a-acid formation occurs at a pH as low as approximately 3.5. While this increas? in competitive-ness could be due to an increase in a-acid formation rate a decrease in /I-acid formation rate an increase in the rate of transformation of the P-acid4 or a combination of all of these changes the precise contribution that each makes to the over-all change is still unknown.In the long term at room temperature the pH boundaries for a-acid formation are as wide as the over-all pH boundaries applying in the short term. Thus a greatly increased reaction time allows all of the /I-acid to transform to the a-acid. Nevertheless there must be doubt about the composition of the product occurring at the lowx pH boundary where, under these circumstances decomposition also affects the a-acid. Under these conditions there remains the possibility of a complicated steady-state system existing which includes some /?-acid formed from the products of the decomposed oc-acid. As temperature also affects the rate of each reaction operating within the molybdosilicic acid system it also can affect the position of the pH boundaries for formation.Indeed the acceleration of the /3-acid transformation reaction is the most significant change brought by temperature increase. Thus the use of temperatures of approximately 100 "C during formation of molybdosilicic acid results in the oc-acid being the sole product at pHs where in the short term and at room temperature both of the acids would have been formed.16 As explained it has not been possible to identify a unique mechanism that accounts for the observed kinetics of a-acid formation. This must await further chemical information about the system because the numerical approach cannot offer further assistance. The nature of the species Y (Table VII) is of particular interest as it indicates the presznce of either an undiscovered silicate condensation product or an undiscovered molybdosilicic acid.Neither of these suggestions is inconsistent with any of the information obtained in this study. Thus whereas the observations concerning silicate speciation in strong stock silicate solutions offers some support to the former suggestion the fact that the individual rate constants obtained from fitting the various reaction schemes for a-acid formation to experi-mental data (Table VII) are dependent on the molybdate concentration supports the latter. One way of resolving these two possibilities is to examine the product(s) of the formation reaction as the reaction proceeds. Indeed exploratory experiments in which the absorption spectrum of the tin(I1) chloride reduction product1 was examined have shown that the a-acid is not the only product.In fact the observed progressive changes in the shape of the absorption spectrum were consistent with the presence of a small proportion of P-acid; however the amount was too low to conflict with the numerical approach's strong rejection of the P-acid being the intermediate Y (Table VII). Equally however these changes in the spectrum would be consistent with the presence of much larger amounts of another molybdosilicic acid that has a lower absorptivity. Notwithstanding this as these tests were conducted at pH 3.5 (in the presence of 0.10 M molybdate-molybdenum and 1.0 M sodium chloride) the effect could be due merely to entering inadvertently the /I-acid regime, proper.Our continuing work is directed towards solving this intriguing problem. The accuracy of our model for the kinetics of formation of the molybdosilicic acids can be questioned. However there seems to be no reason to question the logic of the over-all approach we have used in which the acid - total molybdate system is modelled separately and prior to the molybdosilicic acid system particularly as the acid - total molybdate system is known to equilibrate within a fraction of a second.6 This over-all approach is much superior to the earlier in which the total molybdate concentration was used as a variable in the rate equations and in which the apparent rate constant for formation was assumed to be dependent on a separable function of the hydrogen-ion concentration; that is a change in pH did not imply a concomitant change in molybdate speciation.We believe that the approach we have used would also be of value in studies of both the kinetics and stoicheio-metry of formation of other heteropolymolybdates where undoubtedly molybdate speciation should be taken into account.1 918 TRUESDALE SMITH AND SMITH Conclusions A proper understanding of the kinetics of molybdosilicic acid formation is entirely dependent upon having an accurate model for molybdate speciation; it is not justifiable to use total molybdate concentration as a variable in the rate equations. When examining the effect of changes in hydrogen-ion concentration on the kinetics of molybdosilicic acid formation it is essential to irecognise that concomitant changes occur in the concentration of the various molybdate speizies; indeed it is likely that the major part of any change resulting from pH variation is due to molybdate speciation.Silicate speciation in stock standard solutions affects the kinetics of formation of both molybdosilicic acids. During formation of the P-acid the absorbance increases according to a simple exponential model [equation (l)] ; this implies a reaction of which the velocity is first order with respect to unreacted silicate. During formation of the a-acid the absorbance increases according to a sums of exponen-tials model [equation (S)]; this implies a reaction scheme with first-order steps involving at least two other silicon species as well as the a-acid. The mechanism for p-acid formation is consistent with a rate-determining step in which an Mo species reacts with an intermediate the product of a pre-equilibrium involving monomeric silicate and molybdate species.The mechanism for formation of the a-acid directly from monomeric silicate species is consistent with a rate-determining step in which an Mo species reacts with a silicate monomer. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. References Truesdale V. W. and Smith C. J. Analyst 1975 100 203. Strickland J . D. H. J . Am. Chem. SOG. 1952 74 862 868 and 872. Truesdale V. W. and Smith C. J. Analyst 1975 100 797. Truesdale V. W. Smith C. J. and Smith P. J . Analyst 1977 102 73. Sillen L. G. and Martell A. E. “Stability Constants of Metal-ion Complexes,” Supplement No. I, Aveston J. Anacker E. W. and Johnson J . S. Inorg. Chem. 1964 3 735. Krauskopk K. B. Geochim. Cosmochim. A d a 1956 10 1. Weast R. C. Selby S. M. and Hodgman C. D. Editors “Handbook of Chemistry and Physics,” Ingri N. Kakolowicz W. Sillen L. G. and Warnquist B. Talanta 1967 14 1261. Gill I?. E. and Murray W. in “The National Physical Laboratory Optimisation Software Library,” Lanczos C. “Applied Analysis,” Prentice-Hall Englewood Cliffs N. J. 1956. Bunnett J . F. in Weissberger A. Editor “Techniques of Chemistry Volume VI Part I Investi-Wiberg I<. B. in Weissberger A. Editor “Techniques of Chemistry Volume VI Part I Investi-Frost A. A. and Pearson R. G. in “Kinetics and Mechanism,” John Wiley New York 1961, Szabo Z. G. in Bamford C. H. and Tipper C F. H. Editors “Chemical Kinetics,” Volume 2, Andersson L. H. Acta Chem. Scand. 1958 12 495. Hargis L. G. Analyt. Chem. 1970 42 1494. Backwith P. M. Scheeline A. and Crouch S . R. Analyt. Chem. 1975 47 1930. Special Publication No. 25 The Chemical Society London 1971. Forty-sixth Edition Chemical Rubber Co. Cleveland Ohio 1965-66 p. D73. National Physical Laboratory Teddington 1977. gation of Rates and Mechanisms of Reactions,” John Wiley New York 1974 pp. 129-209. gation of Rates and Mechanisms of Reactions,” John Wiley New York 1974 pp. 771-776. p. 176. Elsevier Amsterdam 1969 p. 30. Received January 23rd 1979 Accepted A p r i l 24th 197