A common laboratory exercise in the undergraduate study of chemical kinetics is the determination of the rate constant, k , of a first-order reaction A -products
This reaction may be described by the equation I - ni = exp (-kt;) (1) where ai is the fraction of reaction a t time ti, defined in terms of the concentration of the reactant by
ni = ([Ah - lAl;)/lAlo Clearly, n ranges from a = 0 when t = 0 to a = 1when t = m, (assuming [A,] = 0). Advantage is usually taken of the dimensionless nature of a to replace the concentration terms by some measurable physical property 4, such as pressure, titer, etc., which is a linear function of concentration; i.e. I9i - 9n-l = clAli where c is a constant and the modulus signs recognize that 4 may increase or decrease with time. Expressing a in terms of 6 gives = (90- 6.) (2) and, combining eqns. (1) and (2) and taking logarithms,
In (19; -9-1) = ln(l90-9-l)-kti (3) The rate constant k can then be determined from measured values of bi, i = 1, . . . , n, and 4, by plotting In (14; - @- 1) against ti, without the need to know $o. A problem arises, however, when an accurate measurement of 4, is not available, perhaps due to precipitation or further reactions of the products at high time, or when an estimate of the rate constant is desired before the completion of a very slow reaction. The methods ( 1 . 2 )which are usually advocated to solve this problem are based on Guggenheim's technique (3)which avoids the need to record 4, by taking measurements of 4; a t times separated by a fixed interval. This is not always convenient, however, and is in any case of little consolation when reliable data have been taken a t variable time intervnls and [he measurement oio. is clearly inconsistent. Some aurhors ( 4 ..5)have adopted as an alternative a computational approach in which the method of non-linear least squares is applied to data obtained at arbitrary time intervals. This technique is very efficient and also yields the standard deviations in k and the estimated value of 4,. However its successful application requires a sound knowledge of the mathematical background to prevent the iterative sequence frum divwging, or converging to a false solution. For this reason the methld is not espec~allysuitable for undergraduate work. The comprehensive scheme of Williams and Taylor (6) solves the problem by repeated application of the much simpler linear least squares analysis t o the points (ti, In (I@i4.. UJ 1 )I, i = 1,. . . ,n, where 4, ti) is one of a series of estimates of 4,. The best estimate of 4, is taken as that which minimizes the standard deviation in the least squares line. The present paper describes an altkrnative method which optimizes e, by a more efficient direct searchstrategy. It also
' To whom correspondence should he addressed. 426 1 Journal of ChemicalEducation
minimizes not the standard deviation in the line but the more fundamental weighted sum of squares of the residuals between the experimental and predicted values of G.As well as simplifying the experimental procedure, the method has the benefit of introducing students to a readily understood hut important technique which is a basis for more powerful optimization methods (7). Optlmlzatlon Procedure The speed and frequently the success of an optimizing procedure depend uoon the ~roximitvof the initial values of the optimization parameterb) to thk true value(s). In the present case an acceptable initial estimate, @,(OJ,for the infinity reading is readily obtained, either from the experimental value of 4,. or from the measurements 4; .. bv- a relationshio such as 9m'0'= 9" + (9. - $,)I5 For each of a series of estimates &'J, j = 0,1,. . . , a weighted least squares line
+
In (19; - 4-G)l) = aj bjti t eij (4) is fitted to the data points and the sum of squares of the residuals 6 ; ; is calculated from
where the weighting factors w;j are discussed later. When the estimate 4,' is found which minimizes Q, the intercept and slope of the line a' = ln (lh* -&*I)
and b* = -h*
provide the "best estimates" of the rate constant k and, if desired, the initial reading The optimization of 4, is now equivalent to a unidimensional search for the minimum of the function Q where the estimated infinity reading 4-ti) is the independent variable. The scheme of Williams and Taylor (6) searches for the minimum with a variant of the simple method of bisection. While this method is guaranteed to converge it can be notoriously slow if the minimum is required with precision since
Figure 1. Rogess of a typical D.S.C. seam applied to optimimtlon of he inf'blity reading. 4,'.
each step in the search entails a complete linear least squares analysis. An equally simple hut much more efficient approach is offered by the direct search algorithm of Davies, Swann, and Campey (DSC) (7). Figure 1shows the progress of a typical DSC search to optimize the infinity reading. (point I), and Q is evaluated at a single initial estimate 6, where 6 is an initial step size, typithen at @,I1) = @,(OJ cally
+
6 = 0.1 14, -
$11
If Q increases, the search is reversed hy setting 6 = -6. The prmess continues by doubling the intervals between successive estimated 4,'J; i.e. until Q increases [point 6 in the figure), indicating that the minimum has been traversed. The interval sire is then haked. the direction is reversed, and one further evaluation of Q is made to give four points (numbered 4, 5, 7, 6 in the figure) separated by a constant interval A ( = 86 in this case). Three of these points, 4,' - A, 4,' and 4,' A, are taken to define a parabola, where &' corresponds to the smallest calculated value of Q. The minimum of this parabola, evaluated from the interpolation formula (8)
+
&* = dm'+ ' h l [Q(&' - A) - Q(&'
+ A)]/[Q(&' - A) 2Q(&') + Q($-' + A)]
(6)
estimates the value of d, which corresoonds to the minimum of Q . The step-doubling process and polynomial approximation locate the minimum raoidlv. If desired. however, the estimate of +,* can he refinid h i restarting the process from 4,'O' = +,* with a reduced initial step size p6 and repeating the procedure until a termination condition, A < y, is satisfied where typically p = 0.1 and y = 10-26. A constraint is included in the present algorithm to ensure that (61- emO))($,- +-bl) > 0 i.e., the search is prevented from generating physically impossible values of 4-'J) which "enter" the range of the 4i data. A detailed study of the application of the direct search method to hoth experimental and simulated first-order reaction data (9) shows that Q has a single minimum under the inequality quoted above. Hence the proposed algorithm must always converge to a unique and correct solution. Least Squares Weighting Procedure
Several recent papers in this Journal (10-12) have drawn attention to the importance of correct weighting in least squares data analysis. In the present case, the weight factors wjj in eqn. (5) are related to the uncertainty in determining the corresponding values aij,where +ij =
In (16; - $-'jJl)
If aZ(4;) is the variance in & then, assuming no error in ti, the variance in ajj is given by (13)
02(@ij)= ~ ~ ( 4 i ) ( d + r j / d & ) ~ = s2(+i)/(C -
mmw
The weighting factors are inversely proportional to the variances in ajj, i.e.
The variances ai2(4;) are seldom known but in many kinetic experiments it is reasonable to assume that the measurements 4; have equal uncertainties, i.e. a2(+;) is constant for i = 1,. . . , n. Since constant terms do not affect the minimization, the correct weighting factors in eqn. (5) are therefore
Figure 2. Least squares plots of eqn. (3) for the acid datalyzed hydrolysis of
sucmse at 308.2'K. [ s u c r ~ s e= ] ~0.292 mol dm-3 [HGI] = 1.25 ma1 dm-=. 0:k ( A , - A - ) , observed A, = -3.66': 0 :In (A, - A.. '), optimized A; = 0.30'.
The weighting factors act as penalty functions to prevent from increasing indefinitely. Thus, if the magnitude of wi, is omitted from eqn. (5), the search alters 4, ti) such that 16, U J1 >> mi, and hence i = 1,. . . , n, +ij= ln(l#j - &'j'I)-ln I L ' j l I , i.e., the values are essentially constant. The least squares coefficients then approach the values and bj = 0
While these values reduce the residuals tij of eqn. (4) almost to zero, they are clearly wrong and the correct weighting factors must always he used. An indication of the uncertainty in h may he ohtained from the least squares analysis (14) as the variance in b*, the value of bj corresponding to @,*. As with the method of Williams and Taylor, however, the error analysis is only approximate since it takes no account of the (unknown) uncertainty in the optimized infinity reading. Results and Discussion
The direct search method has been tested by applying i t hoth to published research data and to student kinetic results. The performance is illustrated here by the analysis of data from a first-vear student exoeriment for which the techni~ne is now standard practice in our laboratory. The reaction, the pseudo-first-order hydrolysis of sucrose catalyzed by hydrochloric acid, is followed by polarimetry so that the measured parameter proportional to the extent of the reaction at time ti is the angle of rotation, Aj. A series of about twelve measurements of A;, separated by increasing time intervals, is taken between 2 mi; and 2 hiafter the start of the reaction. The solution is then kept for two or more days, after which the infinity reading can he recorded. This reading is often nnreliable because the students pay insufficient attention to preservjng the reaction mixture and to resetting the polarimeter. Typical student data, ohtained a t 308°K using sucrose and acid concentrations of 0.584 and 1.25 mol dm-" respectively, show that A; decreases from 13.96' to 0.70" over the ranee 2-86 min. ~ h e s data e have been analyzed, first using the &~erimentalinfinitv readine and then with the o~timizedvalue. i n Figure 2 the open circl& form a plot of In (A; - A , ) against Volume 54, Number 7, July 1977 / 427
ti using the experimental value A, = -3.66'. The slope and intercept of the weighted least-squares line yield the values
k ~ =+2.84 f 0.31 X 10-4 s-I
(kohl = k~+lH*l)
and Ao = 13.642 + 0.W5°
hut the obviously weak correlation between the lotted variables indicates that these estimates are poor approximations to the true values. However the smoothness of the curve which can be drawn through the data points suggests that these measurementsare reliable and that thevalue assigned to A , is resounsible for the ~ o o fit. r When the search method is applied to the data i t ylelds a value A,* = 0.30' which, as the solid circles in Fieure 2 show. is clearlv more consistent with the experimental"values A;. he goo&ess of fit and the low standard deviations in the estimates k ~ =+5.95 f 0.08 X 10-4~-1
ations of Q are required altogether. When the bisection-type search described hy Williams and Taylor (6)is applied to the minimization of Q using the same initial estimate and step size, 22 function evaluations are required to locate the minimum within a similar tolerance. The comparative execution times on a Texas Instruments 980B minicomputer are 6.1 s (bisection) and 2.7 s (DSC) demonstrating the greater efficiency of the DSC search. Similar results are ohtained from the analysis of published research data.2 The comoutational method described in this oaoer is an efficient teihnique for analyzing first-order kineti; data irrespective of the source when the data are ohtained a t arbitrary time intervals and a reliable infinity reading is unavailable. I t is sufficiently rigorous to make i t suitable for research purposes (16) while its heuristic basis ensures that i t is readily understood hv undergraduates. The a u ~ r o a c halso emphasizes the importance of weighting facio-rs in least squares analysis.
and
Acknowledgment
demonstrate the effectiveness of the search technique, the validity of which is confirmed by the excellent agreement between the optimized rate constant and the value
Literature Cited
k ~ =+5.79 f 0.08 X lo-( S-I calculated from Glew's formula (15). In the above analysis the search is started from A,(0) = -1.952" with the initial step size 6 = 1.326. A quadratic interpolation is performed, and the search is restarted from the inter~olatedminimum with 6 = 0.1326 i o = 0.1). A further quadratic interpolation yields the value A,* = 0.2991°, which differs from the true minimum of Q by O.OO1lO. Nine evalu?Typirnl test data (from ref. (2)) and results are supplied with a h r t r a n I\' program liilinp and explanatory notes available from the authors
The authors would like to thnnk Miss L. Hart and Mr. S. L. Porter who obtained the data for the sucrose hydrolysis. (1931).
111 Ruwvean, W.E.. J. A m m Chem. Sac., 53.1651 (2) Swinbaurne. E. S., "Anslyshol Kinetic Nelaon.London, 1971,c h a p 4. (31 Guggenhrim, E A.. Phil. Mag, 2.538 11926).
Data."
14) wentw0rth. W. E..J. CHEM. Enuc..42.96.162 (1965). 15) Muuro. P.. J Chrm.Soc. Forad.. 68.1890(1972). 161 Williams. R. C.. and Taylor. J. W.. J. CHEM. EDUC., 47.123 (19701. 171 Adby. P. R.. and Dernpatar, M . A. H.. "lntrduetion to optimization Mlthd8." Chapman and Hall, London, 1974. chap. 2. 18) Hirnmelblau. D. M., "Applied Nwlinear Pmgramming,"MeCra. Hill.Nm~Ymk,1912,
..
McCrsw Hill. New York.. I969.o. 182. 1141 ~ ~ i ~ g u ~ n . ~ ehap.6. ec.'r~n,
115) M
