242
Anal. Chem. 1983, 55, 242-248
Kinetics of the Formation and Decomposition of 12-Molybdophosphate C. C. Klrcher and S. R. Crouch* Department of Chemistty, Michigan State University, East Lansing, Michlgan 48824
Stopped-flow kinetics studles of the formatlon and decomposltion of 12moiybdophosphate (12-MPA) in HNO, and HCIO, solutlons have been performed at 25 O C and 3.0 M ionlc strength as a functlon of solution pH and Mo(V1) concentratlon. The formatlon kinetics profile follows a sum-of-exponentlals equation indlcatlve of a consecutive reactlons mechanlsm. The maximum 12-MPA formatlon rate varies wlth H+ and Mo(V1) concentrations In a complex manner, even though the dependence of the rate on the phosphate concentration Is a simple first-order dependence. I n addltion the acidic and basic hydrolysis of 12-MPA and the reactlon of 12-MPA wlth additlonal phosphate to form the dlmerlc 9moiybdophosphate are reported here. Implications for reactlon-rate methods for determining phosphate are discussed.
Mo(V1) species concentrations present, much more so than upon CH or CM. The reaction orders with respect to [ H , M O ~ ~ ~range ~ ' ] from 0.5 to 3.5 as the solution [H+] is successively increased from 0.4 to 1.0 M. On the other hand, the H+ reaction orders range from -0.5 to -4.0 as the [HzMo2062+] is successively increased from 0.001 to 0.01 M. These complicated dependences are due, at least in part, to the overall kinetics profile observed for 12-MPA formation. The concentration of 12-MPA vs. time follows a sum-of-theexponentials equation, which indicates that Mo(V1) polymerizes around phosphate in a consecutive step mechanism. This finding implies that the single rate-determining step mechanism proposed in previous work (3) does not completely describe the molybdophosphate reaction.
EXPERIMENTAL SECTION Instrumentation. Stopped-flow measurements were performed automaticallywith the system described by Beckwith and In determining phosphate concentrations in various samCrouch (6) and modified by others (7-10). The quartz observation cell had a 1.87 f 0.02 cm optical path length and was interfaced ples, many laboratories employ the molybdophosphate by a quartz fiber optic (Schott Optical Co.) to a monochromator chemical reaction in which phosphate reacts with Mo(V1) (GCA/McPherson EU-700 series) with a tungsten light source species in strong acid solution (pH CO.9) to produce the yellow (GCA/McPherson). Spectrophotometric measurements were polyanion, 12-molybdophosphate (12-MPA). Many experimade at 430 nm (2 nm spectral band-pass). Spectrophotometric mental modifications involve reducing 12-MPA to an intensely scans of various molybdophosphate solutions (Cary 17 spectrocolored heteropoly blue for additional sensitivity or employing photometer) showed that 12-MPA is the only molybdophosphate a reaction-rate method to reduce the analysis time. However, species which absorbs 430-nm radiation. until recently, many features of the reaction, such as the The stopped-flow system was interfaced to a minicomputer stoichiometry of the various molybdophosphates present, were (Digital Equipment Corp. PDP 8/e). Programs written in not known because the speciation of Mo(V1) in strong acid FORTRAN/SABR (10) were used to operate the data-taking and valve-sequencingfunctions. For most of the kinetics experiments, solution was not known. Thus, previous results on the kinetics the absorbance was measured each millisecond after a stoppedand equilibria of 12-MPA and heteropoly blue formation (1-3) flow push and a 7-ms delay time. One hundred of these meawere reported in terms of analytical acid and molybdate surements were averaged per data point for improved precision, concentrations (CHand CM,respectively) instead of individual and one data point was recorded every 0.1 s over a 10 s total Mo(V1) species concentrations. Now that the equilibrium analysis time. An average absorbance at each time point was constants have been reported (4)for the predominant strong calculated for eight stopped-flowpushes and stored on floppy disk. acid molybdates, M o ( O H ) ~(also written as Moo3), MoPhotocurrents proportional to solution transmittance were ob(OH),(HzO)+ (HMoO,'), Mo20(OH)s(Hz0)z2+(HzM0206~+), tained with a photomultiplief tube (RCA IP-28A), converted to Mo20(OH),(H20)+ (HMozOe+),and M O ~ O ( O H ) ~ ( H ~ O ) ~voltage ~ + (Keithley 427 current amplifier), digitized (Date1DAS16-Ml2B 12-bit analog-to-digitalconverter), and acquired by the (H3M02063+),one readily observes that the pH of such soluminicomputer. An optointerruptor module (G.E. Model H13B1) tions depends upon C M as well as upon CH. Conversely, the attached to the stop syringe plunger provided a trigger signal to concentrations of the Mo(V1) species depend upon both C H the minicomputer at the end of the flow to initiate the data-taking and Cw sequence. Other peripherals included a terminal for controlling Stoichiometric and continuous variations experiments on the stopped-flow operations, a line printer for displaying the data the 12-MPA equilibrium were recently performed (5). Several numerically and an oscilloscope for a graphical display. The drive experimental observations and results from computer simusyringes, mixer, observation cell, and solution receptacles were lations implied the coexistence of two or more molybdothermostated at 25.0 f 0.0 "C. phosphates in strong acid equilibria. The chemical model Reagents. Working solutions were prepared from stock solutions of standardized 4.943 M HN03,4.960 M HClOI, 0.01000 which best matched the experimental data contained three M KH2P04,and 0.500 M Na2MoO4.2H2O.The ionic strength of molybdophosphates with 1:1, 1:12, and 2:18 phosphate-toeach solution was adjusted to 3.0 M with a stock 5.00 M NaC104 molybdate ratios. Their equilibrium constants were tabulated reagent. The solutions were stored in polyethylene bottles after in "OB, HC104, and HzSO4 acidic media in terms of [H+] preparation to reduce the contamination of any silicon from the and the analytical molybdate concentration CM. The anavolumetric glassware. The concentrations of each reagent in the lytical concentration was used because the relative amounts stopped-flow solutions required to give a particular solution pH of the predominant Mo(V1) species vary with CMand because upon mixing or a particular isopolymolybdateconcentration were the particular Mo(V1) species which reacts with phosphate calculated with the FORTRAN program MOLYB,which solves the was not known. equilibrium constant equations given by Cruywagen et al. (4),and The stopped-flow kinetics studies reported here reveal a the Mo(V1) and H+mass balance equations for [H'], the predominant Mo(V1) species concentrations, and the solution ionic complex dependence of the reaction rate upon the H+ and 0003-2700/83/0355-0242$0 1.50/0
0 1983 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 55,
strength without NaC104 present. The molybdophosphate solutions used in the decomposition studies were prepared both from phosphate added it0 an acidified molybdate solution and from solid 12-molybdophosplhoricacid (Climax Molybdenum Co., New York), dissolved in 0.1 M acid solutions (11). In the former case, the initial 12-MPA concentration was calculated through use of the HALTAFALLprogram (12) and the molybdophosphate equilibrium constant values. Certain experimental observations and conditions restrict the ranges of acid, molybdate, and phosphate concentrations that can be used in kinetics studies. First, the phoEiphate concentration must be small so that 12-MPA is the predominant molybdophosphate formed (5). Additionally, with large excesses of H+ and molybdate relative to phosphate, the solution pH and molybdate content do not change significantly as 12-MPA forms or decomposes, and only a negligible amount of dimeric 9-MPA is formed (5). The acid concentration has to be in a certain excess (lo-foldexcess in inost cases) of the total molybdate concentration to prevent precipitation of Mo(V1) as Mooa. On the other hand, the solution must not be so acidic that 12-MPA does not form. With these considerations inmind, the ranges of Cp, CM,and [H'] concentrations used in this study were 0.1-0.5 mM, 0.01-0.05 M, and 0.03-1.0 M, respectively. Data Analysis. The imaximum 12-MFA reaction rate was determined as the maximum slope obtained from linear regression analyses of the 430-nm absorbance-time data over any 0.5,s interval. All correlation coefficients obtained were greater than 0.9990. The maxiinum rate data were fit to various linear models in [HzMoz062+], IHMoOSf], and [ H + ] through a program in a HP-25C pocket cdculator. The best-fitting linear models and their rate constants were used as the equation form and initial estimates for the adjustable parameters in a general purpose r r program employs curve-fittingprogram, KIM^ (13). The m a Runge-Kutta procedure to fit the input experimental data to a rate equation of a specified form, first without and then with the use of statistical weights. Criteria for a suitable convergence of the equation with its parameters include a low sum-of-theresiduals-squared from KINIFIT,a low sum-of-the-residuals-squared from the calculator program, and a low relative standard deviation for the adjusted equation parameters obtained from KINFIT.
NO. 2,
FEBRUARY 1983
243
t
:;:I\ O.*
0.04
4
d
0.004
I
0.002 o'oaO
L
I
2 3 4 5 6 7 8 9 IO Time(sec)
A
Flgure 1. Semilogarlthmlc plots of 12-molybdophosphate formation , = 0.030 M, I = 3.0 M, T = 25.0 OC, X = 430 nm, reaction: C HCIO, medium, Cp = 0.10 mM; (a) [H+] = 0.46 M, (b) [ H ' ] = 0.20 M.
Table I. Some Reaction Schemes Consistent with Equation 2 (1)Consecutive Reactions (with or without a Reversible k
Step): A 4 B
+c k
k3
A = A , exp(-k,t)
RESULTS AND DISCUSSION 12-MPA Formation--Preliminary Observations. The absorbance (A,) at 430 nm was plotted vs. time for each kinetics run;afterward, mathematical modeling was performed to fit the data to a simple exponential equation Ax = O1 O2 exp(O,t) (1)
+
or a sum-of-the-exponentials equation Ax = O1 + O2 exp(8,t) f14 exp(O,t)
+
(2) The absorbance value a t equilibrium A , was obtained by extrapolation from data points between 6 s and 10 s of elapsed reaction time, and In ( A , -- A,) was plotted vs. time t. A linear plot over the entire time domain would indicate that the simple-exponential equation (eq 1) fits the experimental ki. netics data. However, most of these plots (see Figure 1) showed negative concavity in the first 2-4 s of the chemical reaction and linearity for all times thereafter. Mathematically, these observations indicated that the sum-of-the-exponentials equation (eq 2) would better describe the 12-MPA formation and that the parameters 02 and O4 would be opposite in sign (0284 < 0). All the experimental kinetics results fit this model well, as evidenced by the low sum-of-the-residuals squared; results satisfied the necessary condition that 4 = 0, 3- O4 if the background ahsorbance a t 430 nm is negligible. Though constant within a given experiment, the 0 parameters varied between data sets! as the molybdate and hydrogen ion concentrations were varied. Some chemical reaction schemes consistent with the mathematical model are shown in Table I. The suggestion of a consecutive reactions mechanism makes the analysis of 12-MPAkinetics difficult. For example, if a
+ e,-
where
l / , [ ( k , + k,
k 3 t k,) 'i) k 3 + h4)'- 4(k,k, t k3k, + k , k 3 ) ] and h , = t solution to above quadratic, and h , ,)-
(k, + k
h, =
- solution.
consecutive reaction mechanism is postulated, then the initial rate of 12-MPA formation is no longer synonymous with the maximum reaction rate because dC/dt = d[lP-MPA]/dt
244
ANALYTICAL CHEMISTRY, VOL. 55, NO. 2, FEBRUARY 1983
Table 11. Variation of Maximum Rate of Formation of C with Rate Constants for Consecutive Reactions Mechanismsa (1)Consecutive Reactions with One Reversible Step b , c
O.O+
6 r*)
0.04
v
rt
5
(!!L%!!L)(-k
?-
k ) / ( k +k g k
1
kl
-
0.02-
h
'6
IP) n a, 0.010+ N D 0.008r
v
(2) Consecutive Reactions with Two Reversible Stepsbsc
003M
0.02M
a Maximum rate found by differentiating equations for C in Table I and solving for rate( dC/dt) when d2C/dt2= C' = 0. It has been assumed that Bo = C, = 0. C corresponds to 12-MPA; A, corresponds to initial phosphate concentration. 0
-$
0.1
,
0.2
O
O I 0.4 0.60.81.0
L 2.0
004-
4 2 -
Flgure 3. 12-MPA formation rate as a function of acid concentration: A = 430 nm, I = 3.0 M, T = 25.0 OC, HCi04 media.
002-
J
0010-
0.3-0.5 M acidity range. Because the pH of these lower acidity solutions is close to the Mo(V1) isoelectric pH of 0.9, the amount of Mo(V1) species available to react with phosphate is reduced through Moo3 precipitation. The linear models, slopes, and Y intercepts that fit the experimental data over limited conditions are presented in Table 111. For the molybdate dependence in the low acid limit between 0.3 and 0.5 M H+, the linear equation m --1 +b (3) RATE [HzMo2062+]
'0
ar 0008-
h?
0006-
C
E
0004-
i
2 0002-
I
I
I I
I
l
l
I 1
I
I
l
l
gave the best fit of the experimental data. The other equations that were used, but did not fit as well, are listed in ref 14. Since eq 3 held over a fairly wide acidity range, the variation of rn and b with [H+] was tested. The constant b was found to be independent of [H+], while m varied with the square of [H+]. Equation 4 described the 12-MPA formation data better than eq 3 at [H+] = 0.750 M. m --1 +b (4) RATE [H2Moz062+]2 In 1.00 M H+solutions eq 5 fit the data slightly better than eq 6, but both expressions gave correlation coefficients greater than 0.9990.
(5)
- -1 RATE
-
m +b [HzMo2062+]4
(6)
For the analysis of how the maximum rate varies with acidity, eq 7 best fit the kinetics data for solutions low in molybdate concentration (CM5 0.02 M). 1/RATE = m[H+I8 + b (7) In the higher molybdate concentrations eq 8 provided the best fit, but this fit was poor compared to the other fits described above. 1/RATE = rn[H+] + b (8) There are at least two possible explanations for the observance of these different linear expressions. First, the linear equations reflect the predominant terms in the complex rate expressions shown in Table 11. The relative magnitude and significance of each term in the equations vary with changes
ANALYTICAL CHEMISTRY, VOL. 55, NO. 2, FEBRUARY 1983
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Q,
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Q,
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Q,
0
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Q,
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Q, Q, Q,
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+
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in either acid or molybdate concentrations. The second explanation suggests that a single consecutive-step reactions mechanism does not occur over the entire range of H+and Mo(V1) concentrations studied. Several mechanisms consistent with the linear equations for the 12-MPA formation are presented in ref 14. The molybdate linear model in the low acid limit (from eq 3) suggests an acidic dissociation by H3P04prior to combination with HzMoz062+(or HMoz06+ or H3Mo&3+) in the rate-determining step. The other mechanisms consistent with the linear equations suggest that either the number of molybdate species bound to phosphate changes in an intermediate which deprotonates or else that the rate-determining step switches to occur when a different molybdophosphate is formed. 12-MPAAcidic Decomposition Studies. Since hydrogen ions are released as 12-MPA forms, the reaction can be studied in the reverse direction by adding acid to 12-MPA. In this study, the 12-MPA, molybdate, and H+ concentration ranges employed were 0.02-0.20 mM, 0.01-0.05 M, and 0.75-2.00 M, respectively, after stopped-flow mixing. For solutions in which phosphate was added to acidified molybdate solutions ([H'] = 0.50 M prior to mixing with the acidic reagent), the general shape of the 12-MPA decomposition profile showed an exponential decay of the absorbance at 430 nm to the final value, as expected. In fact, a plot of In (A, - A,) vs. time was linear over the entire time range (10 s) with no initial curvature. Preliminary reaction-rate measurements revealed a first-order dependence of the decompasition rate upon the initial 12-MPA concentration, as expected. Surprisingly, no dependence upon the H+ concentration was observed, which means that the rate-determining step involves the dissociation of molybdate from a molybdophosphate and that the molybdate species accepts hydrogen ions in subsequent, rapid equilibrium steps. The general shape of the 12-MPA decomposition profile showed anomalous behavior for 12-MPA solutions prepared from the solid reagent. As a typical reaction began, the absorbance (430 nm) increased significantly before decreasing with time as expected. The time period over which the absorbance increase occurred decreased with decreasing [ 12MPA] and increasing [H+]. The absorbance increase interfered with initial decomposition rate measurements because a proportionately higher amount of 12-MPA was formed at the same time that the excess H+ decomposed the complex. No molybdate or acid was added to these 12-MPA solutions, only enough NaC104 to bring the ionic strength to 3.0 M. Subsequent experiments deduced that the initial absorbance was due to additional 12-MPA being formed in the strong acid solution; in experiments where [H+] was 1.0 M, the final absorbance was greater than the initial absorbance (the pH of the unmixed 12-MPA solutions was about 3). Because the absorbance increased and then decreased instead of decreasing for all time, the formation of 12-MPA occurred at a faster rate than the decomposition under these experimental conditions. In addition, subsequent experiments revealed that the 12MPA decomposition rate has reciprocal molybdate dependence. Thus, in these solutions the 12-MPA inhibits its own decomposition by increasing the amount of unbound molybdate in solution. Although the trend of a decreased decomposition rate with increased molybdate concentration was apparent, no linear equation such as eq 3 fit the experimental data very well. Several nonlinear models were tried. One unsuccessful candidate was RATE = k1[l2-MPA] - kl[HMo03+][11-MPA] which was derived with the 12-MPA 11-MPA equilibrium as the rate-determining step and 11-MPA present in significant concentration at the start of the reaction. One nonlinear expression, eq 9, reproduced all the measured decomposition rates to within 5% error:
-
2 n
+
m
h Y
245
246
ANALYTICAL CHEMISTRY, VOL. 55, NO. 2, FEBRUARY 1983
- d[ 12-MPAl
-
[HzMoz0,2+]
+ [HMo03+] Kz
Table IV. Rate Constants for 12-MPA Conversion to Dimeric 9-MPAa
)
-d[l2-MPA] dt -
[ 12-MPAl (9)
where K1 = (2.21 f 0.6) X M s-l and K z = (9.64 f 0.20) X M s-l for HNO, and K1 = (2.15 f 0.08) X M s-l and K 2 = (4.99 f 0.17) X M s-l for HCIO1. Because the decomposition rate equation involves two terms, a branched chemical mechanism is indicated. The mechanism P04(M003)123P04(M003)113MOO3
K1, e P04(M003)113- + MOO3
k, [ 12-MPA][H,PO, ] [HMoO,+] k,[HMoO,+] t ~ , [ H M O O , +t] ~[H,PO,][HMOO,+]~ HNO, solutions HClO, solutions k, = 0.215 i 0.024 s-' k , = 0.272 * 0.012 s-' k , = (2.8 ?: 2.4) X lo-' M4 k, = ( 3 i 3) x lo-'' M4 k , = (1.4 ?: 1.1)X lo4M-' k , = (4.3 * 0.6) X l o 4 M-' a
T = 25.0 'C, I = 3.0 M.
KlO
eP04(MO03)lo3- + MOO3
+ H" 2
2HM003'
& HzMoz02+ k
P04(Mo03)12-& products P04(Mo03)113-
- products ka
reproduces the above rate equation with K1 = klK1&11Ka2Kd and K z = k2Kl1K,. According to this mechanism the acid decomposition occurs through both 11-MPA and a 10-MPA species; equilibrium approximations on these intermediates reproduce the observed 12-MPA and molybdate dependencies. Since [MOO,] is a constant linear multiple of [HMoO,+], there is still no dependence upon [H+]. One implication from this mechanism is that 10-MPA coexists in equilibrium with 11MPA and 12-MPA in higher acidity solutions, even when molybdate is in 100-fold excess over phosphate. Even so, the molybdophosphate equilibrium constants determined in the lower acidity range 0.2 M I[H+] I0.5 M (5) should not be affected significantly. 12-MPA Decomposition with Excess Phosphate. In this set of kinetics studies, a large amount of phosphate was added to 12-MPA in order to convert 12-MPA to the dimeric 9-MPA. Preliminary measurements revealed a first-order dependence of the decomposition on the 12-MPA concentration; a second-order dependence would have been observed if the rate-determining step had involved a dimerization such as between two 9-MPA monomers. Also, plots of In ( A , - A,) vs. t were linear over the entire time range for most of these experiments with no curvature indicative of an initial reaction lag period. Equation 10 best fit the experimental data for the phosphate dependence - -1 -- m +b RATE [H,PO4] which indicates that the rate-determining step is not the initial dissociation of 12-MPA. The molybdate dependence was more complex. Equation 11 describes the data in low acidity so1/RATE = m[HMoO3+I3 + b
about the central phosphate in the 12-MPA structure (15,161 leads us to conclude that molybdate dissociation or association with molybdophosphates could occur three Mo atoms at a time. There was no acid dependence of the 12-MPA conversion of dimeric 9-MPA. Either the free molybdate species accept or donate hydrogen ions in rapid equilibrium steps not related to the rate-determining step or the errors associated with the initial rate measurements are so large that the acid dependence was not detected. From these experimental observations and linear equations, a consecutive step mechanism was formulated in which there is dissociation of molybdate, combination with the second phosphate, and then addition of molybdate to form the dimeric 9-MPA. With steady-state approximations on certain intermediates and equilibrium approximations on the others, various forms of the rate equation were derived for use during KINFIT execution, and the constants m and b from the linear equations were used as the initial estimates for the adjustable parameters. Only eq 13 converged during KINFIT execution with low relative standard deviations associated with the adjusted parameters. The constants K,, K2,and K3 are given - d[ 12-MPAI dt K1[12-MPA] [H,PO4] [HMo03+I6 Kz[HMo03+I3
+ K3[HMo03+I9 + [H3PO4] [HMo03+I6 (13)
in Table IV for HNO, and HC104 solutions. The residuals (differences between the theoretical and experimental initial reaction rates) appeared to be randomly distributed for the entire range of reaction rates measured. The chemical mechanism consistent with the rate equation is shown below with hydrogen and oxygen atoms omitted from the chemical speices for clarity. The rate equation derived MO + MO + M o ~
MO + M o == ~ Mo~
(11)
lutions (-0.4OO-O.5OO M H+),while eq 12 fits the data for 0.75
M H+ solutions. As was the case in the acid decomposition studies, Mo(V1) dimers couid be substituted for HMo03+in the equations with the exponents halved (i.e., 1/RATE = rn[HzMoz062+]3/2 + b ) and still give the same linearity. The observation of third-order molybdate dependencies and the appearance of trimeric molybdates arranged tetrahedrally
from this mechanism with steady-state approximations on intermediates PMog and PzMogand equilibrium approximations on Moz, M o ~PZMol5, , and PZMo12is identical with eq 13 with K1 = kl, K2 = k-lk-z/k2k3, and K 3 = k-l/kz. It should be noted that this mechanism does not elucidate whether 6
ANALYTICAL CHEMISTRY, VOL. 55, NO. 2, FEBRUARY 1983
mol of Mo(VI) monomeni, 3 mol of dimers, or any combination thereof combines with :PzMogin the step to form P2MoI5. 12-MPA Basic Decomposition Studies. A kinetics study of 12-MPA decompositioii in concentrated hydroxide solutions was attempted. In spite of the fastest stopped-flow settings (one data point per millisecond over a 0.100-s reaction time), the 12-MPA decomposition appeared to be complete by the time the solutions were mixed and delivered to the observation cell. Ionic Strength Effects on 12-MPA Formation. Ionic strength was found to influence the 12-MPA formation rate significantly over the range of I = 1.50-3.00 M. The maximum 12-MPA formation rate increased nearly linearly with increasing I over this range (rxy= 0.970). Plots of In ( A , - At) vs. time for different ionic strength solutions revealed no change in the overall kinetics profiles; a sum-of-the exponentials equation was followed at all ionic strengths studied. The variations in formation rate were significant enough to emphasize that ionic strength should be carefully controlled in reaction-rate determinations of phosphate. Rate Law and Mechanism for 12-MPA Formation. Various attempts to integrate the rate equations under limited experimental conditions and to fit all the kinetics data with this integrated equation were performed with KINFIT. The consecutive-step mechanism with only one rate-determining step, which was postulated previously (1,3, 7), may have been valid since their solution ionic strengths were 2 M and below. However, a consecutive reactions mechanism with two ratedetermining steps had to be postulated for the 12-MPA formation kinetics in 3.0 M ionic strength solutions. The equation relating the maximum reaction rate, the total phosphate concentration Ao, and the rate constants kl and kz for a consecutive reactions mechanism is
(g)
c"=o
[(
=--klkzAo
2)-k1/(kz-k1)
-
(
2)-kz/(k2-k1)]
kz - k l (14)
where k, and k2 each var:y with the acid and molybdate concentrations present. Although this equation may seem difficult to solve, the situation siimplifies considerably because the 12-MPA kinetics has first-order molybdate dependence and follows a simple-exponential equation when 0.3 M I[H+] I 0.5 M and CM I0.02 M. Since the reaction rate decreases with increased [H'] and 12-MPA formation releases H+into solution, kz must increaoe faster than k, so that the kinetics are controlled by the first rate-determining step in the low acid limit. Mathematically, as kz increases relative to kl, eq 14 becomes
which approaches klAo (1.0-0.0) = klAo as k2 approaches -1m.
When 0.3 M I[H+] I0.5 M and CM 1 0.02 M, the maximum reaction rate is proportional to [HMoO,+]/[H+]. This relationship is consistent with previous reporta of biomoleculm 12-MPA kinetics in the low acid limit (1,Z) if Mo(OH)~ is the species that combines with H3P04. Ifk1 = K,[HMoO,+]/[H+], ICl constant, the range of imeasured rates from 0.8Ao to 650Ao confines k, to values ranging from 10 to 650 and sets 1.0 a8 the lower bound for kz. The k z values that, along with the kl values, reproduced the measured maximum rates were observed to vary proportionally with [HMoO3+I9/[H+]lo. With k , = K1[HMoO3+]/[H+1 and kz = K,[HMo03+lg/ [H+]lO,eq 14 converged during KWFIT execution after only nine iterations. The final values for the adjusted parameters K , and K 2 are given in Tablle V. The residual errors between
247
Table V. Rate Law and Rate Constants for 12-MPA Formationa d[ 12-MPAI max. rate =( , dt )c"=o=
[ HMoO,+3
where k , = K ,
[H+1 HNO,
and k , = K ,
[HMoO,'] [,+]lo HClO,
K , = 10.3 k 0.2 s-' K , = (6.2 rt 0 . 7 ) X 10" M s-'
12.5 rt 0.3 sF1 (4.3 rt 1.4) X 10" M s-'
I = 3.0M, T = 25.0°C, A = 430 nm.
the measured reaction rates and the KINFIT-calculated rates were reduced by as much as 60% over the besbfitting equation derived from a single rate-determining step mechanism. Furthermore, the relative standard deviations associated with the adjusted parameter values decreased or stayed the same. The chemical mechanism given below reproduces the rate equation when equilibrium approximations were applied to MOO, (same as Mo(OH),), (HP04)(Mo03)2-,and (PO,)(M003)g3-.
HMo03+ + Moo3
H3P04+ Moo3
kl
+ H+
(HPO,)(MOO,)~-
+
fast
+ 2H+
(HP04)(Mo03)2- 8Mo03 + (PO4)(MoO3):(P04)(Mo03)93-+ Moo3
kZ
slow
+ H+
fast
products slow (Po,) (M003)10,11,1!23-
The proposed mechanism has the following support. First, with equilibrium approximations on the molybdophosphates following the second rate-determining step, the overall 12MPA reaction stoichiometry
+ 12HMo03+ + PMoI2O403-+ 15H+ is reproduced. Second, the second slow step (K2) is the same as the rate-determining step postulated from the 12-MPA acid decomposition rate equation. Third, the second slow step (K,) reflects the parallel mechanisms of monomeric 9-MPA coordinating another phosphate to form dimeric 9-MPA or coordinating more molybdate to become 12-MPA, depending upon the relative amounts of phosphate and molybdate present in solution. Finally, in the more acidic solutions ([H+] > 0.5 M), with equilibrium approximations on all intermediates prior to the k2 step and consideratioqof only the first term in the acid decomposition rate equation, the 12-MPA equilibrium constant is kf
klfk2f
kr
kdk2r
Keq=-=--
-
[12-MPA]
[H+]l3
[HMo03+I2[ H M O O ~ + ] ~ ~ [ H ~ P O ~ ]
All the stoichiometric coefficients are reproduced in the equilibrium constant exponents. The H+ exponent represents the weighted average of the number of H+ released from each Mo(V1) species during complexation rather than from HMo03+ or MooBonly. As pointed out by Truesdale et al. (17), knowledge of molybdate and heteropolymolybdate speciation in strong acid solution has revealed more definitive information about the chemistry and kinetics of 12-MPA formation, decomposition, and conversion to dimeric 9-MPA. Analytically, the experi-
248
Anal. Chem. 1983, 55, 248-253
mental considerations for reaction-rate phosphate determinations already published (2, 3) have amplified significance. Not only does performing the determinations at 0.3 M I [H+] 5 0.5 M and CMoI 0.02 M give the best sensitivity with faster reaction rates, but the 12-MPA formation kinetics are reasonably simple and well behaved under these conditions. In addition, the sensitivity is enhanced in perchloric acid solutions since the rate constants are proportionately larger in HC104 media. Reaction-rate determinations of phosphate should be carried out under conditions of constant acidity, constant molybdate concentration, and constant ionic strength. Registry No. 12-MPA, 12379-13-4;"Os, 7697-37-2;HC104, 7601-90-3.
LITERATURE CITED (1) Javler, A. C.; Crouch, S. R.; Malmstadt, H. V. Anal. Chem. 1988, 4 0 , 1922-1925. (2) Javler, A. C.; Crouch, S. R.; Malmstadt, H. V. Anal. Chem. W8g, 4 1 , 239-243. - .. - .-. (3) Beckwith, P. M.; Scheellne, A.; Crouch, S. R. Anal. Chem. 1975, 47, 1930-1 936. (4) Cruywagen, J. J.; Heyns, J. 6.6.; Rohwer, E. F. C. H. J. Inorg. Nucl. Chem. 1978, 4 0 , 53-59. (5) Klrcher, C. C.; Crouch, S. R. Anal. Chem. 1982, 5 4 , 879-684.
(6) Beckwith, P. M.; Crouch, S. R. Anal. Chem. 1972, 4 4 , 221-227. (7) Notz, P. K. Ph.D. Thesls, Michigan State Unlversity, East Lansing, MI, 1977. (8) Crouch, S. R.; Holler, F. J.; Notz, P. K.; Beckwith, P. M. Appl. Specfrosc. Rev. 1977, 13, 165-259. (9) Gall, R . S. Ph.D. Thesis, Michigan State Unlversity, East Lansing, MI, 1978. (10) Balciunas, R. Ph.D. Thesis, Michigan State University, East Lanslng, MI, 1981. (11) Kasprzak, M. S.; Crouch, S. R.; Leroi, G. E. Appl. Specfrosc. 1978, 32, 537-540. (12) Ingrl, N.; Kakolowisz, W.; Sillen, L. G.; Warnquist, 6.Talanta 1967, 14, 1261-1286. (13) Dye, J. L.; Nicely, V. A. J. Chem. Educ. 1971, 4 8 , 443-448. (14) Kircher, C. C. Ph.D. Thesis, Mlchlgan State Unlversity, East Lansing, MI, 1982. (15) Keggln, J. F. Nature (London) 1933, 131, 908. (16) Kasprzak, M. S.; Leroi, G. E.;Crouch, S. R. Appl. Spectrosc. 1982, 36, 285-289. (17) Truesdale, V. W.; Smith, P. J.; Smlth, C. J. Analyst (London) 1979, 104, 897-918.
RECEIVED for review July 12, 1982. Accepted November 1, 1982. The authors gratefully acknowledge the financial support of the National Science Foundation through a Graduate Fellowship (C.C.K.) and Grant No. 79-26490 (S.
R.C.).
Simultaneous Reaction-Rate Determinations of Phosphate and Si1icate C. C. Kircher and S. R. Crouch" Department of Chemistry, Mlchlgan State University, East Lansing, Michigan 48824
Phosphate and silicate were determined slmultaneously by stopped-flow reactlon-rate measurements at two different times durlng the formation of the 12-heteropolymolybdates. I n solutlons containing 0.40-0.50 M acld, the formatlon of 12-molybdophosphate reaches equlllbrlum rapidly, whlle p12-molybdosilicate is still being formed at its maxlmum rate. Both species can be determined in a 2 0 4 analysis time. The measured reactlon rates vary linearly with phosphate and slllcate concentrations over 0.001-1.00 mM and 0.050-1.00 mM ranges, respectively. The measured reactlon rates are reproduclble to wlthin a few percent relatlve standard devlation (RSD). The major sources of error are a synergistic effect of a faster reaction rate for one analyte increasing the reactlon rate for the other analyte (
