Kinetic method that is insensitive to variables affecting rate constants

temperature that influence rate constants. The method uses a multiple-linear-regression program to compute values of the rate constant, k, initial abs...
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ANALYTICAL CHEMISTRY, VOL. 50, NO. 12, OCTOBER 1978

1611

Kinetic Method That Is Insensitive to Variables Affecting Rate Constants Glen E. Mieling and Harry L. Pardue”

Department of Chemistry, Purdue University, West Lafayette, Indiana 47907

This paper describes a new approach to first-order kinetic analyses that is insensitive to variables such as pH and temperature that influence rate constants. The method uses a multiple-linear-regressionprogram to compute values of the rate constant, k,initial absorbance, A,, and final absorbance, A,, that fit experimental data to a first-order model. Analyte A,. The Feconcentration Is computed from AA = A , (III)/SCN- system is used as a model reaction and dependencies of the method upon pH, temperature, ionic strength, Fe(II1) concentration, data range, number of data points, and initial estimates of k , A,, and A m are evaluated. When 248 A, vs. f points collected over four half-lives are used In the computation, within-run lmpreclslon Is less than 0.1 % for thiocyanate concentrations between 25 and 250 pmoi/L. When results for 278 pairs of results determined by the kinetic and a nearequlllbrlum method for different temperatures, pH, Ionic strengths, and Fe( 111) concentrations are compared, the least-squares equation for kinetic ( y ) vs. equilibrium ( x ) results is y = (1.006 f 0 . 0 0 0 6 ) ~ -0.0002 f 0.0001 with standard error of estimate of 0.001 and correlation coefficient of 0.9 9994.

-

Recent advances in instrumentation, data processing methods, and operating procedures have made kinetic methods of analysis competitive with the more common equilibrium methods in terms of simplicity, speed, and precision ( I , 2). However, it is generally recognized that kinetic methods tend to be more dependent upon experimental variables such as pH and temperature than most equilibrium methods. Variations in experimental variables among standards and samples are often translated into uncertainties in experimental results via variations in rate constants that are used explicitly or implicitly in computational steps. Procedures that could reduce the dependency of the analytical result upon variations in the rate constant could likely reduce the dependency of kinetic methods upon experimental variables. First-order reactions have the unique property that the first-order rate constant can be determined independent of any knowledge of the concentration of the rate-limiting species. It has already been noted that this property can be exploited by using the first-order rate constant as a diagnostic tool in the detection of errors in kinetic analyses based on first-order reactions ( 3 , 4 ) . Perhaps more importantly, this property should permit the independently determined rate constant for each sample to be used in a computational step for that sample. Such a procedure should minimize effects on analytical results of differences in experimental variables that influence the rate constant. This paper describes and evaluates a new approach to first-order kinetic analyses that takes advantage of this property of first-order reactions to reduce the dependency upon experimental variables. The method retains the advantages of selectivity and inherent blank correction normally associated with kinetic methods, 0003-2700/78/0350-1611$01 .OO/O

but it involves a sacrifice in speed compared to initial-rate methods. Simply stated, the new method uses a multiple-linearregression program to compute initial and equilibrium values of the signal and the first-order rate constant that represent the “best fit” of signal vs. time data to a first-order model. Analyte concentration is computed from the difference between initial and equilibrium signal values. The method derives its reduced dependency upon experimental variables from the facts that the total change in signal at equilibrium is less dependent upon the variables than are the kinetic data, and that the rate constant used to define the first-order process is determined independently for each sample while the analysis is in progress. A kinetic procedure for thiocyanate based on the Fe(III)/thiocyanate reaction was used as a model to evaluate important characteristics of the new approach. Data presented for temperature, pH, ionic strength, and Fe(II1) dependencies show that the characteristics of the method are more nearly those of an equilibrium approach than of a more conventional kinetic approach.

GENERAL CONSIDERATIONS The Fe(III)/SCN- reaction was monitored by measuring the absorbance of the product at 450 nm. Therefore, the treatment presented here is in terms of absorbance; however the treatment can be generalized to any signal system that changes linearly with concentration. Concept of t h e Method. Concepts involved in the new method are illustrat,ed in Figure 1. The open diamonds represent selected data points measured during a part of the reaction, and the solid diamonds represent near-equilibrium results that are used for comparison with kinetic results. The solid line represents a fit of the data, represented by open diamonds, to a linearized first-order model with a multiple-linear-regression method (5,6). The regression values pf the initial absorbance, A,, and equilibrium absorbaqce, a r t used to compute the total absorbance change, AA = A , - A,; and analyte concentration is evaluated from this absorbance change. Because the method computes the absorbance change that would be measured if the reactions were monitored to completion, the method should have characteristics more closely related to the equilibrium methods than to conventional kinetic methods provided the first-order rate constant used to fit the model is the correct one for the conditions existing for each individual sample. The multiple regression method satisfies this criterio? by detFrmining the value of the rate constant (as well as A. and A , ) that represents the “best fit” to the data for each individual sample. Mathematical Description. For a first-order reaction monitored via absorbance changes, the absorbance vs. time relationship is given by

4,

At = A , - ( A , - A,) e-kt

(1)

where Ao, A,, and A , are initial, intermediate, and final absorbances, and k and t are the first-oTdef rate coptant and time. The simultaneous estimation of A,, A,, and k is simpler 0 1978 American Chemical Society

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ANALYTICAL CHEMISTRY, VOL. 50, NO. 12, OCTOBER 1978

region where the x 2 hypersurface approximates a paraboloid. Poor initial estimates of k , Ao,and A, will result in the starting point being outside the region for rapid convergence. T o avoid this problem, an algorithm developed by Marquardt (7) was used to sense the condition and to use a procedure approximating the method of steepest descent to approach the approximate parabolic x 2 hypersurface where the regression method takes over and proceeds as described above. When this process is completed, then the projected change in absorbance is computed as

I 4

I T:V;

1

LA

is:

Figure 1. Illustration of the multiple-regression method with experimental

data in a linear model than the nonlinear model in Equation 1. An expansion of Equation 1 as functions of A,, A,, and k by a simplified Taylor’s series gives an equation that is linear in parameter increments as follows

=

lii, - i i O l

(5)

and analyte concenJration is computed with Beers law. The uncertainties of k , Ao,and A , are determined from the square root of the diagonal terms of the variance-covariance matrix multiplied by the:tandard error of estimate. The covariance between A. and A , is obtained in a similar manner fro? an off-diagonal term. Procedures for computing errors in AA are discussed later.

(4)

EXPERIMENTAL All measurements were made on an automated stopped-flow instrument system (8). In all studies reported below, the molar ratio of Fe3+to SCN- was at least 501 to ensure pseudo-first-order behavior and a pH range between 0.43 and 0.70 was used to minimize protonation of thiocyanate and formation of the Fe2(OH),4+dimer. Except for those studies involving ionic strength effects, all solutions were prepared to have an ionic strength of 1.0. For temperature dependency and error analysis studies, equal volumes of 0.040 mol/L Fe(C104)3in 0.80 mol/L HCIOl ( p = 1.0) and various concentrations of NaSCN ( p = 1.0) were mixed in the stopped-flow system. For ionic strength studies, 15.0 mL of 0.04 mol/L Fe(C104)3in 0.80 mol/L HC104was diluted to 31.2 mL with volumes of 1.5 mol/L NaC10, and water to give ionic strengths in the range from 0.5 to 1.0 and these solutions were reacted with equal volumes of 125 pmol/L thiocyanate solutions with the same ionic strength. For each solution studied, 248 absorbance values were measured at equal time intervals during three to four half-lives of the reaction, and then 30 points were measured between seven and eight half-lives (99.2 to 99.6% complete, see Figure 1). These data were then processed by three different methods for comparison purposes. The multiple-linear-regression method (henceforth called “regression-kinetic” or simply “regression” method) was applied to data collected over 0.5,1,2,3,and 4 half-lives to produce values of absorbance change, LA,that are related to concentration. A linear least-squares fit of nine points collected during the first 10% of the reaction time produced “initial-rate’’ results that are compared with the regression method results. Finally, the 30 data points collected between seven and eight half-lives were averaged t o obtain a near-equilibrium value of absorbance that is labeled A7.8in the remainder of this paper. It should be noted that this A7.8 value is expected to be only about 99.4% of the A , value generated by the multiple-linear-regression program.

where SA is the estimated standard deviation of absorbance measurements. No weighting function was used because the absorbance uncertainty is relatively constant throughout the range and because the Taylor’s series linearization does not require weighting. The function is minimized by setting the first derivatives of x 2 with respect to 6Ao,6A,, and 6h equal t o zero and solving the normal equations that result for the parameters 6Ao,6A,, and 6k. Because the truncated Taylor’s series in Equation 2 is only an apprqximatio? of the nonlinear model, first estimates of the A’, A,, and k will usually be in error and therfore successive iterations of the procedure described above are used to obtain best estimates of these variables. Iterations (typically 3 to 5) are continued until the change in x2 is less than 0.05% between successive iterations. One problem with the procedure described to this point is that it will converge rapidly toward a minimum x2 only in the

RESULTS AND DISCUSSION The principal objective of this study was to evaluate pertinent chaTacteristics of the proposed method and to compare these with characteristics of equilibrium and initial-rate procedures using the same system. All uncertainties are reported a t the one standard deviation unit (h 1s)and time information is reported in half-lives where each half-life is about 0.2 s for the Fe(III)/SCN- reaction. Regression Kinetic Method. Computed us. Experimental Data. The solid line in Figure 1 represents the fit of data represented by the open diamonds to the first-order model, and the closed diamonds represent measured values of absorbance, Ai.s, after seven half-lives of the reaction. The data show good agreement among the predicted and measured absorbances. Unless noted otherwise, regression-kinetic results reported below are based on fits of data collected during three to four half-lives of the reaction, and “equilibrium” results

aA,O aAto A, = A,’ + -6k + -6Ao

ak

aA0

aA,O + -&A, aA

(2)

where the term Atois an initial estimate of A, expressed in terms of initial estimates of A’, A,, and k . The partial derivatives are also derived from initial estimates of A,, A,, and 12 inserted into the following expressions

aAt0

-= t(A,

-

A,) e-kt

and

A multiple-linear-regression program is used to evaluate values of 6k, 6Ao, and 6A, which when a$ded toinitial estimates of k , A’, and A , will give values k , A,, and A,, which represent the best fit of the experimental A, vs. t data to the linearized first-order equation. The criterion used to obtain the best fit involves the minimization of the following function

ANALYTICAL CHEMISTRY, VOL. 50, NO. 12, OCTOBER 1978

1613

Table I. Linearity Studies' and Effects of Temperature on Regression, Equilibrium, and Initial Rate Methods slope i S, intercept f S, std error, ratio, slopelk L/pmol x l o 3 A A x io4 AA X lo4 regression method (regression of A f j b vs. CSCN) 5.2 f 2 6 24 3.23 i 0.02 1.616 i 0.001 1.622 i 0.002 -0.1 i 2 8 25 3.50 t 0.03 0.2 f 2 7 26 3.80 f 0.02 1.620 i 0.001 equilibrium method (regression of A,.,' vs. CSCN) 24 1.611 i 0.002 -6 i 3 9 25 1.614 i 0.001 -5 * 2 7 26 1.607 i 0.003 15i 5 18 initial rated method (regression of AA/At vs. CSCN) -21 i 21 70 1.51 24 3.23 * 0.02 4.88 t 0.01 25 3.50 i 0.03 5.22 * 0.02 9 i 26 85 1.49 26 3.80 * 0.02 5.68 t 0.01 -53 f 22 13 1.49 AA a Results based upon five measurements on each of ten solutions of NaSCN in the range from 25 to 250 pmol/L. value computed from 248 data points during about 4 half-lives. A,-, value computed from 30 data points at 7 to 8 halflives. Initial rates computed from 9 points during first 10% of the reaction; units for slope and intercept are L/pmol-s X l o 3 and s" x lo4, respectively. h

correspond to absorbances collected after seven to eight half-lives as represented by the closed diamonds in Figure 1 and corresponding to 99.2 to 99.6% completion. Analytical Performance. LINEARITY.The relationship between concentration and the computed absorbance changes was evaluated with ten solutions containing sodium thiocyanate in the range from 25 to 250 pmol/L in 25 kmol/L increments. Performance data for five runs on each solution at each of three temperatures are presented in the first three rows of Table I. The computed absorbance changes ranged from about 0.04 at 25 pmol/L to 0.40 at 250 pmol/L. Uncertainties in the slopes and intercepts are observed to be less than 0.270 and 0.5 milliabsorbance unit, respectively, and the standard error of estimate (S,) shows that the scatter about the least-squares line through the points is less than 1 milliabsorbance unit in all cases. Correlation coefficients were all greater than 0.9999. All of these statistics coqfirm a high degree of linearity between 1 A and CSCN- for AA computed from data collected during three to four half-lives. PRECISION. The bottom curve in Figure 2 illustrates the within-run imprecision of the regression method as a function of concentration where 0.170error at 25 pmol/L and 0.01% error at 250 pmol/L correspond to 0.00004 absorbance unit. The solid triangles represent between-run errors based on an average of five runs for each sample. The plot is approximately constant a t 0.270 and corresponds to about 0.00008 absorbance unit at 25 pmol/L and 0.0008 absorbance unit at 250 pmol/L. The decrease in relative within-run error with concentration suggests that the within-run error source is associated with the measurement/computational steps. The independence of the relative between-run errors with concentration suggests that the between-run error source is in the sample processing steps. Taken together, the data indicate that the measurement/computational errors are significantly less than the sample handling errors. Dependence on Experimental Variables. One of the expected advantages of the regression-kinetic method proposed here is its potential freedom from dependence upon some experimental variables that influence conventional kinetic methods. In order to evaluate this property of the method, dependencies upon temperature, pH, Fe(III), and ionic strength have been evaluated and are presented in this section. Temperature. Temperature represents one of the most critical variables in most conventional kinetic methods of analysis. Data presented in the first three rows of Table I show that the method is relatively independent of temperature.

--,

I

-

s? --i;C??USTE

-

.x

.j?

7?CEk-?TiO\

L

I

m &MOL/-;

L

hc

Figure 2. Relative experimental uncertainties for equilibrium, multiple regression, and initial-rate methods. 0 ,Regression, within-run ( R S A ~

= 100 X SA^^ + SA,' - 2S~0+p12)1'2 + LA). A,Regression, between-run, 5 replicates. 0, Equilibrium, within-run (re1 std dev of 30 points). +, Equilibrium, between-run, 5 re licates. H, Initial-rate, within-run (re1std dev of regression slope).,hoInitial-rate. between-run, 5 replicates When all of the results at 24 "C are regressed against results a t 25 and 26 "C, the resulting regression equations are AA24

+

= (0.996 f 0.001)AAZ6 (6 f 3) x

and ~ A 2 = 4

(1.002

S, = 9 x r > 0.9999

* o.oo~)AA,~- ( 5 * 2) x 10-4; S , = 7 x

r > 0.9999

When results at 25 "C are regressed against results a t 26 "C, the regression equation is

AA,,

= (0.999 f

o.oo1)aAZ6- (0.1f 3) x 1x

10-4;

s, =

r > 0.9999

Although the slope for the first equation is different from unity at the 95% confidence level, the difference is only 0.4% and these data demonstrate excellent immunity of the method to changes in temp>rature. To quantify a temperature coefficient, slopes (AA vs. CSCN)in the first three rows of Table I were regressed against temperature and the regression parameters are presented in the first row of the second column in Table 11. While the temperature coefficient appears to be about 0.12% per "C, the small value of the correlation coefficient (0.2) confirms a relatively small correlation between the result and temperature. It should be noted that any effect of temperature or other variable on the position of the

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ANALYTICAL CHEMISTRY, VOL. 50, NO. 12, OCTOBER 1978

Table 11. Effects of Experimental Variables on Regression, Equilibrium, and Initial-Rate Methods sensitivity factors and correlation coefficients regression equilibrium initial-rate range of k method, (factor method, (factor method, (factor parameter values, s - ' t s ) x 103 r * s ) x 103 r i s ) x 103 slopes (Table I ) regressed vs. temperature (24, 25, 26 " C ) 3.2-3.8 (-2 t 2 ) x 0.7 (2 * 3 ) x 0.6 (4 i 0.3) x 10.' Cpe, mmol/L response regressed vs. pH (eight values: 0.43 to 0.70) 5 1.8-2.3 1 i 1 0.2 0.8 i 1 0.1 96 * 9 12.5 2.8-3.7 2i- 1 0.3 42 2 0.5 343 i 14 17.5 3.6-4.5 0.7 f 2 0.1 0.4 i 2 0.04 363 i 61 response regressed vs. C F (mmol/L, ~ six values; 5 to 17.5 mmol/L) PH 0.43 1.8-3.6 3.1 i 0.2 0.98 3.1 i 0.2 0.98 1 9 * 0.5 0.52 1.9-3.9 3.2 i 0.2 0.97 3.2 i 0.2 0.97 20 * 0.6 0.70 2.3-4.5 3.3 i 0.2 0.98 3.3 i 0.2 0.97 25 i 1 response regressed vs. ionic strength (six values; 0.5 to 1.0) -16i 9 0.93 -17i 1 0.92 -158 L 8

I

i

Figure 3. Effects of CFeand pH on results obtained by three methods. A, Regression and equilibrium method. 0 , InitiaCrate method. Ordinates

\

r

0.997 0.91 0.98 0.80 0.99 0.98 0.98 0.96

\

Figure 4. Effects of ionic strength on results obtained by three methods. 0 , Regression. 0 ,Equilibrium. A , Initial-rate. Ordinates scaled to

scaled to illustrate relative deDendencies

illustrate relative dependencies

equilibrium or absorptivity of product will introduce a systematic error just as is the case with equilibrium methods. CFeand p H . The open triangles in Figure 3 represent AA values as functions of Fe(II1) concentrations for a 200 kmol/L thiocyanate solution and each triangle includes points for three p H values between 0.43 and 0.70. The p H and Fe(II1) dependencies are quantified in data rows 2-7 under "regression method" in Table 11. While the rate constant varies from 1.8 to 4.5 s-l for the range of parameters included in this study, the p H dependency coefficient varies from 0.7 to 2.6% per p H unit and the correlation coefficients are very small (0.1 to 0.3). The Fe(II1) dependency coefficient is much larger, being about 3.8% per mmol/L and the^ correlation coefficient shown a high correlation between AA and Cpe. Ionic Strength. Effects of ionic strength on the regression method are represented by the open diamonds in Figure 4 and the results are quantified in the last row of Table 11. The ionic strength coefficient is -10.6% per unit of ionic strength and the correlation coefficient confirms a definite negative correlation between AA and k . The effects of CFeand ionic strength on the results reflect the influence of these variables on the equilibrium constant for the reaction rather than kinetic effects. Comparison of Methods. The dependencies of the regression method noted above can be put into perspective by comparing them with similar dependencies for a near-equilibrium and an initial-rate methods. The same samples and data sets used to obtain the regression results were also treated and processed to obtain near-equilibrium and initial-rate results, and parameter dependencies for these results are included in Tables I and I1 and Figures 2 through 4. It should

be kept in mind that the near-equilibrium results were obtained for data between 7 and 8 half-lives where the reaction is 99.2 to 99.6% complete so that the regression results that are extrapolated to 100.0% completion should be about 0.6% higher than the "equilibrium" results. Temperature Dependence. Data in Tables I and I1 permit a comparison of temperature dependencies. The slopes of calibration data in Table I are virtually independent of temperature for the regression and near-equilibrium methods, but are dependent on temperature for the initial-rate method. Average values of slopes for the regression and near-equilibrium methods are 1.619 and 1.611, respectively, corresponding to an expected 0.5% higher value for the regression method. The ratios of calibration slopes to first-order rate constants in the last column show that the change in calibration slopes with temperature for the initial-rate method correspond to changes in the rate constant. Other parameters such as intercepts, standard deviations of slopes and intercepts, and standard errors also exhibit a closer correspondence of the regression-kinetic results to the equilibrium results than to the initial-rate results. The temperature dependencies were quantified by regressing the calibration slopes in Table I vs. temperature to give the dependency coefficients in the first data row of Table 11. The dependency coefficients and the correlation coefficient confirm a much smaller dependence of equilibrium and regression-kinetic results on temperature than for the initial-rate method. CFeand p H Dependencies. The open triangles in Figure 3 represent both regression kinetic and equilibrium results and the open diamonds represent initial rate results for three

ANALYTICAL CHEMISTRY, VOL. 50, NO. 12, OCTOBER 1978

pH values between 0.43 and 0.70 at each Fe(II1) concentration. It is apparent that the regression and equilibrium values are virtually independent of pH and show the same dependence on Fe(II1) while the initial-rate data are more dependent on both pH and Fe(II1) concentration. These dependencies are quantified in terms of relative dependency coefficients in data rows 2-7 in Table 11. The dependency coefficient and correlation coefficients show virtually identical behavior between the regression-kinetic and equilibrium methods and very dissimilar behavior between the regression-kinetic and initial-rate methods. Ionic S t r e n g t h Dependence. Ionic strength dependencies are presented graphically in Figure 4 and summarized quantitatively in the last row of Table 11. Again, the regression-kinetic method performs similarly to the equilibrium method and dissimilarly to the initial-rate method. Regression-Kinetic us. Equilibrium Results. T o further evaluate the agreement between results obtained with the regression-kinetic and the equilibrium methods, results obtained by the two methods for the different conditions discussed above were subjected to a least-squares fit. For 278 pairs of results obtained for different temperatures, pH values, ionic strengths, and Fe(II1) and SCN- concentrations, the regression equation and statistics were

The standard deviations of the slope and intercept, the standard error of estimate, and the correlation coefficient all confirm a high degree of correlation among the data. While the intercept is not different from zero at the 95% confidence level, the slope is greater than unity by 0.6%, a value near that expected because the near equilibrium data were collected when the reaction was only 99.2 to 99.6% complete. In our view, these results obtained for a wide range of conditions demonstrate a remarkable degree of agreement between the regression-kinetic and equilibrium methods. Any variable that influences the rate constant alone will have very little, if any, effect on the regression kinetic method; however any variable that influences either the equilibrium position of the reaction or the detector sensitivity factor (absorptivity for absorbance measurements) will have the same effect on the regression-kinetic method as on the equilibrium method. Precision. The open symbols in Figure 2 represent the within-run imprecision for the three methods and the closed symbols represent between-run imprecision based upon five runs for each sample. Within-run imprecision for the regression-kinetic method is lower than that for either of the other methods. The difference between the regression and equilibrium methods probably results from the difference numbers of data points used (248 vs. 30) and the fact that the absorbance changes by about 0.1% during the time between seven and eight half-lives when A7.8 is being evaluated. The difference between the regression-kinetic and initial-rate methods probably results from the different number of data points (248 vs. 9), the different absorbance ranges (0.04 to 0.40 vs. 0.004 to 0.04),and the greater dependence of the initial-rate method on variations in the rate constant during and among runs. It is interesting to note that the between-run errors are greater than the within-run errors for the regression and equilibrium methods, and less than the within-run errors for the initial-rate method. These relationships probably result from the sample-handling errors being sufficiently larger than the within-run errors for the regression and equilibrium methods to offset any advantage from replication and being sufficiently smaller than the within-run errors for the initial-rate method that some advantage is gained from replication.

dl

1615

1

-0.m.s

-o.zm

-2.3005

-0.0333

3.0035

:.::io

I YTEXEPT Figurt5. Joint confidence intervals of the slope and intercept for AAn vs. LA,. A, n = 1. A, n = 2 . A , n = 3. 0 , slope = 1.0, intercept

= 0.0 Effects of Data Selection. All regression kinetic results presented to this point were based on the same general procedure that included 248 data points collected during three to four half-lives of each reaction. Also, errors have been presented without discussion of their origins. Effects on the analytical results of the number of data points, the range of data, and initial estimates of regression parameters as well as origins of observed errors are discussed in this section. N u m b e r of Data Points. Th? effects of the number of data points on the uncertainty in PA were evaluated by processing different numbers (30 to 248) of the data points collected at fixed intervals over four-half lives of each reaction a t 26 "C. Least-squares slopes of log-log plots of error vs. the number of data points were -0.50 f 0.01 and 4 . 4 8 f 0.01, respectively, for the 25 and 250 fimol/L thiocyanate solution. These data coilfirm the expected square root dependence of the imprecision upon the number of data points used in each fit. Data Range. Effects of data range were evaluated by processing data during 3,2, 1, and 0.5 half-lives, and evaluating least-squares parameters for these vs. those obtained from four half-lives of data. The resulting least-squares equations are

A& = (1.0003 f 0 . 0 0 0 2 ) ~ A -~ (0.02 f 0.04) x 10-4; S , = 1.3 X r > 0.9999

AA,

= (1.0003

* 0 . 0 0 0 2 ) ~ A (0.5 ~ * 0.7) x 10-4; s,= -

2.2 x IO-4; r

AA,

> 0.9999

s, = io x

= (1.001 0 . 0 0 1 ) ~ A- ~(3.5 f 3) x

W4;r

> 0.9999

and

A&s = (0.97 f 0.02) AA4 - ( 2 3 f 52) X 17 X

S, = r > 0.99

where the subscripts refer to the numbers of half-lives of data included in each set of determinations. The 95% joint confidence intervals (9, 10) for the slopes and intercepts of the data for 3, 2, and 1 half-lives are presented in Figure 5 , along with the ideal 0.000, 1.000 coordinate (open diamond). Although the 3 half-lives data show a slight bias in the slope (