1828
ANALYTICAL CHEMISTRY, VOL. 51, NO. 11, SEPTEMBER 1979
Table IV. 0.1% k ,
+ 0.1% N o i s e into Second-Order Model
xt
Xt(Calcd)
0.908096 0.83233 0.766999 0.712005 0.663924 0.6217712 0.584866 0.552035 0.522359 0.496847 0.47 2 54 1 0.450320 0.431037 0.412812 0.396344 0.38086 0.36602 0.35271 0.339972 0.32886
0.910572 0.83316 0.76788 0.71208 0.66385 0.621737 0.584647 0.551733 0.52232 0.49589 0.47 201 0.450325 0.43054 0.41243 0.39578 0.38040 0.366192 0.353002 0.34073 0.32928
residual -0.002476 -0.00083 -0.000881 -0.000072 0.000074 0,0000342 0.000219 0.000302 0.000039 0.000957 0.000531 -0.000005
0.000497 0.000382 0.000564 0.00046 -0.00019 -0.000292 -0.0007328 -0.00042
xt-1
1.1012 1.2014 1.3038 1.4045 1.5062 1.6083 1.7098 1.8115 1.9144 2.0127 2.1162 2.2206 2.3199 2.4223 2.5231 2.626 2.7321 2.8352 2.9414 3.0408
and A. are very reasonable estimates of the input values (see Table I). A typical residual pattern for noisy second-order data is shown in Figure 1 for a system with 10% noise. In response to changing the number of data points (so long as at least 20 are used) while keeping the total time the same, almost no effect on the calculated values or goodness of fit to the model is observed. When noisy mixed first- and second-order data are evaluated, the acceptance or rejection, based on residual analysis, represents a balance between the magnitude of the noise and the magnitude of the first-order Contribution. The greater the noise, the more apt the program is not to detect a first-order contribution. This information is summarized in Table IV and is exemplified in Figure 2 in which data including 0.1 % of both noise and first-order contribution are graphed in the conventional manner. The program rejects these data as not being strictly second-order but the graph would be interpreted as giving excellent second-order results.
CONCLUSION
Investigation of Noisy Data. The most realistic case involves "noisy" data. The influence of noise was evaluated both on pure second-order data and on mixed first- and second-order data. Noise was added to a noiseless signal by converting absorbance to transmittance and adding a random amount of transmittance as shown in Equation 19 and reconverting to absorbance. This was done even while recognizing that it will not yield a true picture of the random errors which occur in actual spectrophotometric measurements.
This second-order noniterative technique is able to distinguish second-order data from non-second-order data, is able to verify very noise simulated data as being second-order, and yields accurate estimates of input rate constants and A. values. The technique is fast, requiring about 30 program steps in BASIC or about the same number on an SR-56 pocket calculator. All programs are available on request.
LITERATURE CITED
I
Tnoisy
=
+ T(R)
(19)
R is a random positive or negative number between zero and the maximum specified noise level (usually 0.005). When noisy second-order data are evaluated, the calculated values of k 2
(1) R. G. Corneii, Biometrics, 18, 104 (1962). (2) N. Draper and H. Smith, "Applied Regression Analysis", J. Wiiey and Sons, New York. 1966. OD 95-99. (3) Chemical Rubber Company, "Handbook for Probability and Statistics", 2nd ed., W. H. Beyer, Ed.. 1968, pp 414-424. (4) Paul B. Kelter and James D. Carr, Anal. Chern., Correspondence in this issue.
RECEIVED for review September 8, 1978. Accepted May 29, 1979.
Microcomputer Compatible Method of Resolving Rate Constants in Mixed First- and Second-Order Kinetic Rate Laws Paul B. Kelter and James D. Carr" Department of Chemistry, University of Nebraska, Lincoln, Nebraska 68588
A method is presented for treating kinetic data for systems in which first-order and second-order reactions proceed in parallel. Both the first- and second-order rate constants and absorbance at time f = 0 are calculated from absorbance vs. time data. The method Is, except for one minimization, a one-pass process and is easily used on a microcomputer or programmable hand-held calculator. The method will reject data not adhering to such a rate law even in the presence of reasonable noise levels. Suggestions for optimizing data collection are also presented. Using the method presented, the oxidation of formic acid by ferrate(V1) ion was investigated.
Rate constants for reactions which follow a rate law with both first-order and second-order components have been very difficult to resolve from spectrophotometric data. A typical such rate law is dt
= k , [ X ] + k?[X]2
(I)
0003-2700/79/0351-1828$01 O W 0
The two most common ways to solve for k l and k 2 are graphically, and via Fletcher-Powell minimization. To solve for k l and h2 graphically, one divides both sides of Equation 1 by [XI, giving Equation 2.
dX/[X] at
= k,
+ h2[X]
The quantity (dX/dt)/[Xj is then graphed vs. [XI to yield a slope of k 2 and an intercept of k l . Fletcher-Powell minimization is an iterative technique, in which differential equations are solved using "educated" initial guesses as t o the desired parameters ( 1 ) . It is an excellent minimization technique but requires the use of a macrocomputer, and as such is often prohibitively expensive. R. G. Cornell (2) introduced a technique in 1962 for evaluation of first-order kinetic rate constants. The technique was notable for its meed, simplicity, and ability to deal with ~. noise. It assumed that rate constants were constant from one absorbance point to the next and, as such, rate constants could C 1979 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 51, NO. 11, SEPTEMBER 1979
be determined by summing all absorbances and solving. This idea was extended to second-order systems by Kelter and C u r ( 3 ) . This paper is a further extension of this idea to mixed-order systems and is easily used with a microcomputer or programmable calculator.
THEORY Given the rate law in Equation 1, the concentration a t any time t l is (see appendix) klX0 xt = ( h , + h2Xo)eklt
-
k2Xo
(3)
1829
Equation 14 is transcendental-that is, it has no exact route to a solution. T o solve this equation, one must use a minimization technique. The one used here, Newton-Raphson minimization ( 4 ) ,works on the principle that the smaller the gradient of a given guess, the closer one is to the correct answer. Therefore, a derivative is subtracted until a constant answer is attained for two successive iterations. The general form of the minimization in solving for w is
From a profile of concentration vs. time, values of k , , h2, anc X,,(the concentration a t time t = 0) must be determined. Inverting Equation 3,
1 k2Xoeklt k,Xo _ - ( h , + h2Xo)eklt- h 2 X o = -hleklf +---
xt
hlX0
hlX0
h1Xo
klX0
Therefore,
(4) Therefore, for this application Equation 15 becomes
Let B = k 2 / k l . Then factoring, (5)
+
+
Let Y = 1/X, B . Let 6 t l = t , where 6 = time between points, and t l = time of the first available data point. Then
Let at = ekld;Equation 6 then becomes
We have found that a good blind guess for w is 1.2. A value for w can usually be determined in less than 10 iterations. In addition, one can note that if Q32 is less than 2, w will converge to 1,meaning by Equation 19 that k1 = 0. Looking a t the Qa2 value is therefore the first good diagnostic tool for a mixedorder system. Recalling that w = eklS,then
In - -w As in ref. 2 and 3, a summation is set up for each unknown to be solved for. The three unknowns here are hl, k2, and Xo. The summation has the general form
6
- kl
Then,
t =o
Let n = the total number of data points and let p = n/3. Let
and
Note that if t l = 6, then ekIrl= w. In addition, (1 - wp)(wp1) = -lap - 112. Recalling that B = k 2 / k l , and Y = l / X o B,
+
k2 = Bhl
x,=
(22)
1
Y-B
I _
To use absorbance data directly, Beer's law is introduced, and it can be shown that when both reactants and products absorb,
where A , = absorbance a t time t, A , = absorbance a t infinite time, At = molar absorptivity of reactants - molar absorptivity
1830
ANALYTICAL CHEMISTRY, VOL. 51, NO. 11, SEPTEMBER 1979
__ Table I. Noiseless Concentration vs. Time Data - -. input conditions k , , s" points tl, s 6,s 18
0.1
0.1
18 18 18
0.1 0.1 0.1 0.1 0.1
0.1
1 x 10.~ 1x 1 x 10-7
0.1
18 18 18
0.1
0.1 0.1 0.1 0.1
0.1
0.1
0.1
18 18
0.1
18 18
0.1
18
0.1
0.1 0.1 0.1 0.1 0.1 0.1
18 18 18
0.1
0.1
0.1 0.1 0.1
0.1 1 x 10-4 1 x 10-3
2.5 2.0 0.1
1 x 10-3 1 x 10-3
18 18
0.1 0.1
18 18 18
18 36
180
1x 1x 1x 1x 1x 1x 1 1 1 1 1 1 1 1x 1x 1x 1 1x 1x 1x 1x 1x 1x
0.1 0.05
0.1 0.1 0.1
72 144 1152
k,, M-ls-'
0.1
0.025 0.0125 0.0015625
0.1
0.01
1 1 1 1 1 1 1 1 1 1 1 x 10-3
10-5 10'~ 10-3
lo-,
1x 1x
1x
1 x lo-" 1 x 10-15 1 1 1 1 x 10-3 1 1 1
10-3 10-3
10-5 10-5
10-5 10-5
1 1 1
10-5
10-5
of products, and b = cell path length in cm. Equation 3 then becomes
At - A , A&
--
k,Ao
--
A,/Stb
[ k , f k2 ( A , - A,)/Atb]ekl:
-
h,[(A,,
-
A,)/&b]
(25)
Let
SI' =
c -A. , p
t=l
2p
i -
A,'
S2'= 2 t=pi-1
1 , S,' = A, - A m
*-._
c
1
3P
:=2p+l
-~ -
A,
Am
When one goes through the derivation taking absorbance of reactants and products into account. k l remains equal to In u/6 but
k 2 = k,BAtb
(26)
A =-- 1 (27) Y-B Effect of Knowing Xu.If X o is known, although there A,
-
are only two unknowns in Equation 7, three summations must still be taken, because Equation 14 must be the final result. Any discrepancies between true and erroneous X o values will therefore have no effect on hl, and depending upon the magnitude of the discrepancy, little effect on k p . Therefore, even if Xois known, it is felt that nothing is gained in time or accuracy of answers to justify inputting it as a known. In addition, calculating X, by the method serves as a good internal check with the known X o .
SIMULATED DATA Simulated data were generated and processed with either a Texas Instruments SR-58 programmable calculator equipped with a PC-100A printer, or with a North Star Horizon microcomputer equipped with 32K of 8-bit core. a Centronics
k , , s"
x 0
output conditions k,, Me's-'
-y0
0.8 0.8 0.8 0.8
1
1 1 1 1 1 1 1 1 1 1
1
1 x 10-3
1
1x
1
1
1.05 X
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3. 1 1
2.97 X 10.' 3.79 x 10-7 1.05 X
9.58 X 9.997 x 10-5 1 x 10-3 1 x 10-2 1 x 10-l 1 1
1x 1x
1 1 4.01
3.73 x 10." 3 x lo-" X
9.85 X 9.72 X 1 9.58
9.62 9.56 9.83
X X X X
lo-'
1 1
1 1 x 10-3 10.'
1.01x 10-5
9.62 x 10-5
1 1 1 1 1 1
0.9862 0.9998 1
I 1
I 1 1 1 1 I
1 1
1 0.9562 1.032 1.202 1
0.9975 0.9917
779 printer, and a Hazeltine 1500 Video Terminal. Programs for the Horizon were written in BASIC and are available upori request. Simulations of kinetic data described herein cover a broad range of conditions for both noiseless and noisy data. We present limitations of the method, and suggestions for optimizing the method. Also! there are observed differences depending on whether 8 or 13 significant figures are used in the calculations. Noiseless Data. As shown in Table I, K l cannot be detected below 1 x lo4 s-l when k 2 = 1.0 M-l s-l. This is due to the lack of detectable change in X,due to the iogarithmic nature of calculating k l (See Equation 19). J20w values of .k2, on the other hand, are clearly detectable down to 1 x 10 l o M-' s-' when k l = 1.0 s-'. There is no logarithmic operation occurring in t,he calculation of k,, so significant figures are not lost to the same degree as with hl. Incorrect values of A, also occur under conditions in which poor values of k , are obtained but values of k2 remain correct even in these situations. This is best explained by examining Equations 20 and 21. Equation 20 leads to a value for Xo.This value is dependent on the calculated value of kl. If klc,, is in error, X, will also be in error. Equation 21 leads to a value for h2. This value is dependent upon both k , and Xu. Therefore, the deviations in k , and X, balance in Equation 21 to lead to an accurate value of k S . When k l = 0, Q32must be less than 2. If Q3? is !ess thar, or equal to 2, minimization must lead to a value for w = 1 , Using Equation 19, k l therefore must equal zero. 'Therefore. whenever a Q32 value of less than 2 is calculated. h! = 0 and k 2 must be evaluated by another method 133. The value of Q3? is then a diagnostic for a rate law which does not involve a significant first-order component. When h z = 0 , there is no restriction on the value of Q3?,although it must he 2 2 . R wil! be equal to 0 when k 2 = 0. Taking more points in the same tutal time did not appreciably improve output parameters, and taking noiseless data for both a small percent of the first half-life and a s n d percent of the fifth half-life showed good agreement with inpu! values.
ANALYTICAL CHEMISTRY, VOL. 51, NO. 11, SEPTEMBER 1979
Table 11. Concentration vs. Time Data, 0.5% Noise Case ( A ) : input conditions: k , = 1 s - I ; k , = 0.05 M-' s - ' ; X , = t , = 0.1s
Table 111. Concentration vs. Time Data, Varying the Total Number of Pointsa
h , out, s - '
h , out, M-I s - '
X, out
points
18 0.100 0.98143085
0.083601273 0.013143915 0.051608236 0.064435882 0.063926146 0.06866388 0.031922662
1.0022554 0.99325066 0.99987921 1.0031728 1.0016643 1.0020465 0.99632158
18
0.1
24 36 36b 54 72 90
0.075 0.05 0.05 0.0333 0.025 0.02
points 24 36 54 72 90
6,
s
0.075 0.050 0.033 0.025 0.020
1.0152681 0.99954298 0.99508387 0.9936668 0.9910096 180 0.010 1.0062067
Case ( B ) : input conditions: h , = 0.05 s ' l ; h z = 1 M - ' s - ' ; X , = 1;t , = 0.1s 18 0.100 0.036408476 1.0227182
24 36 54 72 90 180
0.075 0.050 0.033 0.025 0.020
0.065654376 0.052572624 0.043491228 0.046724056 0.045959599 0.010 0.060719169
0.96906719 0.99516339 1.0143367 1.0066955 1.0072079 0.97772344
1.0011924 0.99444503 0.99862789 1.0029446 1.0005553 1.0002601 0.99277163
Case (C): input conditions: h , = 1 s - ' ; h , = 1 M - ' s - ' ;X , = 1;t , = 0.1s 18 0.100 0.97145
24 36 54 72 90 180
0.075 0.050 0.033 0.025 0.020
1.0229 0.998354 0.994226 0.99083 0.98599 0.010 1.0080718
1.07299 0.91982 1.00934 1.02763 1.030383 1.04358 0.96276
1.00751 0.98333 1.001525 1.00736 1.00493 1.00677 0.9919
Noisy Data. Random noise was added to theoretical concentration vs. time curves by Equation 28.
R is a random positive or negative number between zero and the maximum specified noise level (usually 0.005). This is a fairly simple way of considering noise to be a constant relative error in transmittance. For comparison, a more realistic noise profile was imposed on the system by utilizing Pardue's expression for spectrophotometric error in concentration data ( 5 ) which was combined with random noise to yield simulated concentration vs. time data. It was found that results were the same whether the simplistic noise profile or the more sophisticated profile was used. The results shown in Table I1 reflect 0.5% noise imposed upon theoretical concentration vs. time data. A t this noise ( h 2 being = 1.0 M-' s-l), level, when k , is less than about k , is not detected with acceptable accuracy. In almost all cases k 2 and X o output agree well with h2 and X o input. In Equation 26, it is shown that h2 is linearly related to A t b , whereas hl is unrelated to I t b . Therefore, k , will be accurately measurable when the ratio of k 2 to h i is less than lo2 S t b .
OPTIMIZING DATA COLLECTION Table I11 shows a series of typical simulated concentration vs. time data. As the change in concentration from the first point taken at time tl to the next point at time ti + 6 decreases. the accuracy of the output data vis-a-vis the input data reaches a maximum and then decreases. This is depicted in Table 111. This result indicates that there is a trade-off between the number of points taken and the change in absorbance between adjacent points. This trade-off is best explained by the nature of the calculation of h l , k,, and X o . From Equations 11-13, the values of D2,, D?,, and Q32 are all dependent upon the number of significant figures that can
1831
180
6
0.01
k , in, ( ~ A b s ) , 's - '
0.084 0.065 0.045 0.045 0.031 0.024 0.020 0.011
0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
k , out s-'
mk, o u t k , inl, s - '
0.08068 0.11421 0.10225 0.09829 0.09321 0.09291 0.09192 0.10829
0.01932 0.01421 0.00225 0.00171 0.00679 0.00709 0.00808
0.00829
a For the above simulations, h , = 1 M - ' s - ' , X,= 1, t , = 0.1 s and 0.5% noise level. Each of the 36 points is the average of the 11 central points taken from the mid-range of 36 intervals of 100 points each.
be extracted from concentration v!j. time data which can indicate a trend in the rate. For example, A
13
&, * 6
xtI
0.9073 0.8262
A X6
0.0811
0.9073 0.9071 0.0002
The data in set A have the same number of significant figures as the data in set B. However, when the difference in concentration, AX6 is calculated, set A has three significant figures, whereas set B has only one significant figure. Therefore, if data points are taken too often, significance is lost. However, if not enough data points are taken, the rate trend is lost. The trade-off between the number of points taken on a conventional spectrophotometer and the absorbance change between adjacent points is summarized by the following empirical observations: (1) d must be long enough to allow a minimum absorbance change of rt0.004 to occur (Le., about twice the readability of the spectrophotometer). (2) The time required for a constant absorbance change increases as the reaction proceeds. (3) Generally: (AAbs/G),, = lO(AAbs/6),,+,, for ca. 2.5 half-lives. Therefore, choose d such that (AAbs),, L 0.04. Obviously, if I A t , L 0.04, the larger the entire absorbance change. A. - A,, the more points can be taken in an individual kinetic run and still meet this criterion. If data are collected on-line and many hundreds of absorbance values spaced much more closely than 0.004 absorbance unit are available, the data can be averaged in many ways. For example, we find that when 3600 data points are taken in 36 groups of 100 points and the central 11 points of each group are averaged, the 36 resultant points show lkl(o,,t, - klc,J of 0.00171, an improvement from the 0.00225 obtained on 36 unaveraged points (Table 111) Effect of an Incorrect A,. The effect of incorrect values of A, was examined through a series of simulated absorbance vs. time data, as summarized in Table IV. One can see that when h2 is less than or equal to h l , the determined value of k 2 can actually become negative if A , is chosen incorrectly. A negative value of hz is, of course, a physical impossibility, but not a mathematical one. A negative kl is bcth a physical and mathematical impossibility, because h l is calculated via Equation 19, and neither In LLI nor 6 can be negative. Therefore, when a negative h , is encountered there are three possibilities: (1) The value of A , is incorrect. (2) The proposed rate law is incorrect. (3) h2 is essentially zero, and the calculated value is within error limits of zero. Comparison of 8 a n d 13 Significant Figures. In a method in which the calculations are dependent on so many factors, the number of significant figures retained by the
1832
ANALYTICAL CHEMISTRY, VOL. 51, NO. 11, SEPTEMBER 1979
Table IV. The Effect of an Incorrect A, input conditions points 72
k , , s-' 1 x 10-3
1 x 10-3
36
k 2 , M" s - ' 1 x 10-4
1 x 10-4
output conditions
A,- A , 1
1
6, s
40
40
t,, s 40
40
36
1x
1 x 10-3
0.5
30
30
72
1 x 10.~
1 x 10-3
1
40
40
+ 0.006
- 0.002 -0.006 +0.002 t0.006 t 0.020 -0.002 -0.006 -0.020 t0.002 +0.006 +0.038 -0.002 -0.006 -0.038 +0.002 0.006 -0.002 -0.006 +0.002 -0.002 I
1x
36
io-2
1 x 10-3
1
20
20
Table V. Comparison of 8 and 13 Significant Figures" Horizon BASIC TI-SR-58 (8 significant figures) (13 significant figures)b S! s2
s3
D,, D2i 9 3 2
W
calcd
'
calcd
Zcalcd Xocalcd
16.513304 23.722408 30.94758 14.434154 7.209104 2.0022119 1.0003157 6.3136884 X 0.99078553 0.99896108
16.513303605 23.7224078728 30.947456423 14.434152849 7.2091042679 2.0022116913 1.00018415484 3.6820796 X 0.9949970866 1,001256 1945
s - ' , h , = 1 M-' s - ' , X,= 1, Conditions: h , = 1 X noise = 0.5%, t , = 0.1 s, 6 = 0.05 s, n 36 points. b Only the 1 0 most significant figures are shown. a
calculating device affects the output parameters. T o test the importance of this effect, simulated data with k l = 0.001, h2 = 1, Xo= 1, noise = 0.5%, 6 = 0.05, t l = 0.1, with 36 points was generated by the Horizon BASIC, which carries 8 significant figures in its calculations. Data were not stimulated by the SR-58, because with 0.5% noise, the 9th-13th figures on a given X,are surely not significant. Data were processed by Horizon and by the SR-58, which carries 13 significant figures in its calculation. The results are summarized in Table V. One can see that for this simulation, the values for S1, S2, Ss, D,,, D,,, and Qs2 are the same to the 8th significant figure. It is in the value for w where a difference appears. The difference is probably due to round-off errors in BASIC encountered while performing software exponentiation during the minimization step. Here the difference between the two final answers is not of real significance. The major difference in the two calculations is the number of iterations necessary. Several hundred iterations (requiring 1-3 min) are required with the Horizon and 10-20 iterations (requiring ca. 30 s) are required with the SR-58. Again, round-off and number storage format in the Horizon BASIC may have the most to do with the differences encountered. The second feature is that although there are differences in the calculated value of w , the hl values obtained calculating with 8 and 13 significant figures are easily within acceptable
h i (s-l)x
AA, +0.002
lo3
k, (M-'
1.025 1.078 0.9759 0.9301
s-l)
__.
x lo3 A , - A ,
0.0 4 8 9 4 -0.06242 0.1482 0.2373 0.08982 0.06884 -0.1173
1.010
1.030 1.104 0,9902 0.9709 0.9064 1.022 1.067 1.508 0.9785 0.9367 0.6510 0.1051 0.1156 0.09496 0.08492 10.43 9.598
0.9897 0.9682 1.010 1.029 0.9976 0,9928 0.9757 1.002 1.007 1.024 0.4978 0.4932 0.4550 0.5022 0.5067 0.5412 0.9979 0.9935 1.002 1.006 0.9672 1.038
0.1100
0.1293 0.1914 0.9632 0.8848 -0.04881 1.035 1.102 1.481 0.9986 0.9953 1.001 1.004 -0.2507 2.131
error limits if k l is comparable in size to k p . Notice that when k l is small, there is a large relative difference between the two sets of data. This is explained in the following manner:
~.oxxx~xxx WB = ~~ooox~xxx -
WA
(A)
=
(B)
If 6 for A and B are the same, A represents a rate law with a large k l component, and B represents a rate law with a small kl component. If in a given calculation the Horizon BASSICand the SR-58 differ in the 6th significant figure, (labeled "L") In
WA
= 0.OXXXLXXX -
In WB = O.OOOXL_XXX The difference in klA will be on the order of ppt, while the difference in k l will ~ be on the order of 10%. Therefore, when k , is suspected to be very small, the use of 13 or more significant figures is critical to calculation accuracy, while the situation is not as important with a large h,.
REAL DATA The reaction of aqueous formic acid and ferrate(V1) to form products 2Fe042-
+ 3HCOOH + 2H20
-+
3C02
2Fe(III)
+ lOOH-
(29)
has been carried out under pseudo-first-order conditions (6). The absorbance vs. time kinetic data were processed by graphical means and by Fletcher-Powell minimization as well as the method described herein. Goodness of fit to the mixed-order model was done by inspection of the residual map, taking into account the number of runs, and positive and negative residues (7). A typical result is as follows: new method no. of points k , x 104(s-1) k,(M-' s - ' ) A , - A,
runsItresiduals
69 15.571 1.1592 1.001 32137
FletcherPowell
graphical
71
15.761 1.1148 ,999 33135
16.0 0.897
Table VI gives the points and residuals, and a residual map
ANALYTICAL CHEMISTRY, VOL. 51, NO. 11, SEPTEMBER 1979
1833
Table VI. Values for the Reaction of Formic Acid with Ferrate(V1) Absactuala
A bscalcd
0.975181 0.949329 0.925543 0.901758 0.884178 0.860393 0.842813 0.825233 0.806618 0.788004 0.767322 0.754912 0.13423 0.71 87 1 8 0.701 1 3 8 0.689762 0.672182 0.65667 0.641 158 0.628749 0.6101 34 0.601861 0.589452 0.579111 0.563599 0.553257 0.542916 0.532575 0.522234 0.506722 0.498449 0.486039 0.477766 0.461425 0.460186
0.975726 0.9511441 0.928563 0.906146 0.884459 0.863469 0.843146 0.823460 0.804384 0.185893 0.761962 0.750568 0.733690 0.111307 0.101399 0.685941 0.670935 0.656345 0.642162 0.628310 0.614955 0.601904 0.589203 0.516840 0.564802 0.553080 0.541661 0.530536 0.519695 0.5091 27 0.498825 0.488780 0.418982 0.469425 0.460101
Normalized so that A ,
a,
-
residual
-0.0545 - 0.241 5 1 - 0.302 - 0.4388 -0.0281 - 0.3076 -0.0333 0.1173 0.2234 0.2111 - 0.064 0.4344 0.054 0.1411 -0.0261 0.3815 0.1247 0.0325 -0.1004 0.0379 - 0.4821 - 0.0043 0.0249 0.2211 - 0.1203 0.0177 0.1255 0.2039 0.2539 -0.2405 - 0.0376 -0.2741 - 0 . 1 216 -0.2 0.0085
1
,004
1
Absactual
A bscalcd
0.44984 5 0.444674 0.434333 0.42606 0.415719 0.40848 0.401241 0.392968 0.384695 0.311456 0.372285 0.364012 0.359816 0.3 5 1603 0.34333 0.339193 0.33092 0.322641 0.320579 0.310238 0.30817 0.301965 0.2941 26 0.289555 0.284385 0.279214 0.274043 0.268873 0.266805 0.260600 0.2 5853 2 0.250259 0.248 19 0.24302
0.451002 0.4421 2 1 0.433452 0.424988 0.416723 0.408651 0.400767 0.393064 0.385537 0.378182 0.370994 0.363966 0.357096 0.350379 0.343810 0.337385 0.331101 0.324954 0.318939 0.313054 0.307294 0.301657 0.296140 0.290738 0.285450 0.280272 0.275202 0.270237 0.265374 0.260610 0.255944 0.251372 0.246893 0.24 2 505
residual x l o 2 - 0 . 1 157
0.2 553 0.0881 0.1072 - (1.1004 - Cl.0 17 1 0.0474 -Cl.0096 - Ci.084 2 - Cl.0726 0.1291 0.0046 Cl.278 0.1224 - 0.048 Cl . 18 08 -0.0181 - 0.2307 0.1658 -0.2816 0.0876 0.0308 -0.1414 -0.1183 - 0.1065 -0.1058 -0.1159 -0.1364 0.1431 -0.0010 0.2588 -0,1113 0.1297 0.0515
A, = 1.000 (true A , = 1.037, true A, = 0.070).
I
I ,006
X 10'
APPENDIX A Solution of Parallel First- a n d Second-Order Rate Constants. Given the reactions
h'
x-P 2x
k2
P
where the rate law is
& -.006
-
10
20
30
40
50
60
ax
-=
dt
k1
+ X + kpX'
(30)
variables are separated:
POlhT NUMBER
Figure 1. Residual map of mixed order Fe(V1) oxidation of formic acid
demonstrating random, nonweighted behavior is presented in Figure 1.
The right hand side of Equation 32 is a standard integral of the form d X / X ( a + b X ) , where a = k l and b = ha. The solution of the integral is
CONCLUSION The method described herein for calculation of kl, k p ,and absorbance at zero time is the method of choice for use with small computer systems. It is fast and reliable. It has limitations when k l is very small, or when the change in absorbance from one reading to the next is too small. Calculation of parameters for real systems is in good agreement with established methods.
'Og
kl
+ kzXo
1834
ANALYTICAL CHEMISTRY, VOL. 51, NO. 11, SEPTEMBER 1979
10196 ! 102W ! HMINIIIIZHTION I N PR13GRESS.. . . . " 10210 LET 12=0 10226 LET W=1.2 10230 LET Wl=W^(2*P)-Q3MP+Q3-1 10240 LET ~ 4 = 2 * c P l i b l " ( ~ P - l } - P * Q 3 ~ ( P - l ) 18258 LET bl=WlM 10260 LET @ W - W l 10278 IF W=bQ THEN 11000 18288 LET 12=12+1 102% IF I2)288 T E N 11886 10380 LET W=W2 10358 GOT0 10230 11080 ! nM#=u, w 11810 ! " f l I N I M I H D IN", 12, ITEmTIONS" 11826 LET K5=LOG(bl)/T2 11030 LET Yl=D2*(l-W) 11840 LET Y=EXP( K5*T1 j *(l-#T ( WT-1) 11845 IF +e THEN 2 m 110% LET Y=YlN 11068 LET B=Y*ExP (K5*T1) (l-GTP)! (1-W )-51 11070 LET 6=WP 11080 LET K6=K5*B 11898 LET XS=l/(Y-B) l l l e e ! nKl=n, K5 l I l f 0 ! "K2=@, K6 ll126!'FtBs AT T I E ZERO=",X5 19588 ! " 2 < 2 , n IWM !HTTHEREFORE, m=a IN MIXED-ORDER. 19780 END REFlDY
Exponentiating both sides,
+ k z X o ) / X o .Multiplying both sides by X, XBeklt = k , + k z X , and h , = X(Bekl' h,)
where B = ( h ,
~
Solving for X a t any time, t ,
(33) Substituting back for B in Equation 33,
XO and multiplying top and bottom by Xo,
k1Xo x, = ( k , + k2XO)eklt k,Xo
*
(34)
-
*
APPENDIX B The essential steps in the mixed-order program following entry of data as a series of concentrations of reactant or solution absorbance.
10m DIM X(1000)t A(1880) 18888 LET P=INT(N/3) 18804 LET 52=0 i0#S LET 53=0 1 0 ~ LET 6 si=@ 18810 FOR 1=3 TO N+2 18820 LET A ( I > = l I X ( I ) 18830 NEXT I 10050 LET fll=INT(N/3) 18868 LET N=#1*3 10070 FOR I=3 TO P+2 18888 LET Sl=Sl+A(I) 10090 LET S2=S2+A(I+P) 10188 LET S3=S3+A(I+BP) 10110 NEXT I 10126 ! 10130 ! Sl,S2,S3 10150 LET D3=53-51 10160 LET D2=S2-S1 10170 LET Q3=03Al2 181% !D3,02, Q3 10191 ! 18192 I F Q3C? THEN i9W 10195!
LITERATURE CITED R. Fletcher and M. J. D. Powell, Computer J . , 6 , 163 (1963). R. G. Cornell, Biometrics, 18, 104 (1962). P. B. Keker and J. D. Carr, Anal. Chem.. preceding paper in this issue. , G. S. G. Beveridge and R. S. Schechter, "Optimization: Theory and Practice", McGraw-Hill, New York, 1970, pp 56-60. (5) H. L. Pardue, T. E. Hewitt, and M. J. Milano, Clin. Chem. ( Winston-Salem, N.C.), 20, 1028 (1974). (6) A. R. Tabatabai, Department of Chemistry, University of Nebraska, Lincoln, Neb., unpublished data, 1978 (7) N. Draper and H. Smitt7, "Applied Regression Anaiysis", Wiley-Interscience, New York, 1966. pp 95-99 ~
RECEIVED for review September 8, 1978. Accepted May 29, 1979. The work upon which this publication is based was supported in part by funds provided by the Office of Water Research and Technology (OWRT-A-053),US.Department of the Interior, Washington, D.C., as authorized by the Water Research and Development Act of 1978.
