The Initial Rate Method in Chemical Kinetics Evaluation and Experimental Illustration Julio Casado' Departamento de Quimica Fisica, Facuitad de Quimica, Universidad, E 37008-Saiamanca, Spain M. Arturo L6pez-Quintela Departamento de Quimica Fisica, Facultad de Quimica. Universidad, E-Santiago de Compostela, Spain Francisco M. Lorenzo-Banal Escuela Universitaria de Profesorado, Universidad, E-Santiago de Compostela, Spain In textbooks and educational articles i t is not uncommon to find comparative discussions of the relative merits of the integral and initial rate methods of kinetic analysis. The chief disadvantage of the latter, the difficulty in approximating the taneent to the concentration-time curve. often appears to be exaggerated, and quantitative estimation uf the valup and limitations of thr method is eenerails absent. In this article we present aquantitative argument showing that the initial rate method (IRM) is not only perfectly compatible with normal standards of experimental accuracy, but in certain by no means exceptional circumstances, offers decided advantages over the integral method. We illustrate these advantages with a practical example which is recommended for a basic course in physical chemistry. Accuracy of the Nlathod Let us first calculate the error really involved in approximating the tangent to the concentration-time curve. In Figure 1,which shows the initial segment of a typical curve, the measurable quantity y is assumed to he linearly related to the concentration of the substance being studied, and the line yoT is the true tangent to the curve a t t = 0. In experimental practice a straight line y$ is fitted to the data for a small percentage of the reaction and the difference between y s a n d yoT is assumed to he negligible. If the last reading is taken at time t,, then in Figure 1 yaP will lie between the true tangent yuTand the chord yoQ. The latter may thus he taken to represent the worst approximation possible, and we shall therefore calculate the error involved in taking the gradient of yoQ for that of yoT. Suppose the kinetic curve is given analytically by y = f(t), whose second-order Taylor expansion a t the origin is f(t) = f(0) + tf'(0) + tZf"(0)/2+ R,(t)
(1)
We shall assume that Rz(t) is negligible compared with t2f"(0)12, i.e., that all the error involved in using yoQ instead of yoT is due t o the quadratic term in eq 1. The line YUQ is then given by the equation yChord(t) = f(0)+ Wt,)
+
= f(0) t[f'(O)
- f(O)l/t,
+ t,f"(O)/Zl
Since the true tangent yoT is given by v,..(t) -
+ tf'(0)
= f(0)
(3)
then e ( t A the percentage error involved in taking the w d i eut of the chord for that of the true tangent, is e(t,)
-
100t,f"(0)/2f(O)
' Correspondence may be addressed to any of the authors. 450
Journal of Chemical Education
(4)
I
i 1
1,
Figure 1. Typical concentration-time curve.
In other words, if we require that the error should not exceed n% then the last reading must he taken no later than a t time If Y is the total variation in y during the whole reaction, then substituting eq 5 in eq 1 shows that the percentage reaction followed must not exceed For a first-order reaction for which y = CeWkt,eq 6 means that the reaction may he followed to rather more than 2n%, whereas for a second-order reaction given by l l y = llyo kt, the permissible percentage is little more than n%. At times i t may therefore he advantageous t o use working conditions that force first.order in order to increase the percentage of reaction available for measurement. In any case, the above considerations lead to the following conclusion: By restricting the percentage of reaction followed to 5% or less, the error introduced by the IRMis kept well within the accepted limits for kinetic studies.
+
Advantages of the Method One of the chief nractical advantaees of the IRM. its saving in time, is often regarded as of little importance since even quite long reactions can he studied in a period of time by the integral method if, as is common practice (I), no more than 80% of the reaction is in f a d followed. We shall show here, by means of a numerical example, that the validity of so reducing the duration of the experiment from 100% of the reaction t o 80% should by no means he taken for granted.
1
2
3 1 0 - ~t
4
5
(S)
Figure 3. Plot of the date from Table 2 using the IRM
Figure 2,PlDt of the data from Table 1 using the integral method.
Table 1. Concentratlon-TheValues Let us consider a complex reaction whose true rate equation involves hoth first- and second-order terms with respect to some substance C: u = k,[C]
+ k,[CI2
0.0905
0.0653
1
5
0.0482 10
0.0312 20
0.0227 30
0.0177 40
(7)
where kt = 5 X 10-I s-' and k2 = 10-'M-'8-I. If 80% of this reaction is followed, the initial concentration of C heing 0.1 M, then the concentration-time data appearing in Tahle 1 are obtained. On anolvine the inteeral method to these results and d o t ting [ ~ i "a g a h t, theresulting straight line (Fig. 2) would he taken to indicate a second-order reaction. The first-order term kl[C] would be overlooked2. However, if the IRM is applied and uo/[C]o plotted against [C]o (Table 2), then the line now obtained (Fig. 3) has an ordinate a t the origin significantly different from zero by Student's t-test, so that hoth the first- and the second-order terms are recognized. The IRM can therefore he more sensitive to kinetic terms which, though quantitatively small, may qualitatively he very important for deducing a reaction mechanism (3). Naturally, the failure of the integral method to detect the first-order term in the above example was due to only 80%of the reaction heing followed. Had the experiment been continued to nearly 100%of the reaction, then hoth terms would have shown uo in the results. However. continuine" un. to completion of'the reaction would involve accentuating the disadvantages inherent in the integral method: 1) The risk of such complications as the autodecomposition of
the reagents (e.g., nitrous acid in nitrosation reactions) or the presence of competitive reactions, hothof which are considerably less important when the IRM is employed (4,5). 2) The sheer increase in the time necessary for each experiment, which may become prohibitive in systematic studies with many experiements involving very slow reactions. A Laboratory Experiment T o illustrate the use of IRM hv means of a reaction for inclusion in a set of physical chemistry experiments, we sueeest the diazotization of D-anisidinein aoueous oerchloric media. (Caution: Perchloric acid in contact with comhustihle material may cause fire. Prevent contact with eyes and skin (6).)The diazonium salt produced is perfectly stable, and its UV absorption peak a t 312 nm has a molar ahsorption coefficient of 2.5 X lo2 m2mol-', making UV spectrophotometry ideal for studying the kinetics of the reaction.
"-
[C] (MI t10-4irl
Table 2. Concentratlon-lnltlal Rate Values Corresponding to Eq (7)
At 25 "C and with concentrations of HClO4 and amine of 0.1 M and 3 X 10-4 M, respectively, by varying the initial M and concentration of nitrite over the range 3-10 X plotting the results as in Figure 3, we deduce the rate equation uo = k,[nitrite],
+ k,[nitrite]i
where kl = 2.6 X 10-5 s-1 and kz = 5.4 X M-I s-'. The time required to follow these reactions u p to P 5 % ranges between 10 and 30 minutes, with a total rise in absorbance of 0.2-0.3 units a t 312 nm. When studying the influence of the acidity of the medium on kl and k2, the following equations can he found: k, = a[Ht]l(b
k,
+ [Ht]) + [H'])'
= e[H+]l(b
witha = 2.6 X 10-5s-1, b = 1.0 X 10-3M-l, andc = 5.4 X 10-4 M-28-1. I t may be observed that in less acidic media, the secondorder term hecomes much greater than the first-order term, and the situation described ahove in relation toeq 7 appears. Conclusion The ahove analysis of the IRM shows that its accuracy is perfectly in accord with the usual standards of chemical kinetics, and that in certain circumstances i t is qualitatively superior to integral methods. However, we by no means wish to imply that the integral methodmay hediscarded; in many cases hoth methods are acceptahle and may be used to corroborate each other's results. Which method is to be used in Of course, the integral method could detect the first-urder term if we carried out more experimentsdecreasing the initial concentration of C (cf. 2)and assuming that the experimental method allows the reaction to be followed under these conditions.
Volume 63 Number 5 May 1986
45 1
a given case should therefore be decided on the basis of careful examination of the nature and conditions of that experiment. Acknowledgment
The authors wish t o thank the Spanish CAICYT for financid support and a for his kind and valuable ments on the original draft of the manuscript.
452
Journal of Chemical Education
Literature Clted 11, Espenson, J. H..-chemieei ~ i ~and ~b a ati o ni ~ ~ ~~ ~MCGX~W-H~II: h ~N*W ~ yolk, 1981. (21 Smbb. 2. G. In "Comprehensive Chcmiral Kinetics"; Bamf0rd.C. H.;Tipper, C. F. H.. Eds.: Elsevier: Amsterdam, 1969; Val 2, Chap 1. (31 Caaado, J.: c a s r o , A,; ~ 6 p e z - ~ u i n t e iM. a . A,; ~adriguez-~rieto. M. F. Z.phys chow. Nave F o l g ~1979.118.43. (41 Leidier. K. J. '"Chemical Kinetics"; MeGrsw-Hill: N w York, 1965. 151 Wfikinwn. F."Chemical Kinetics and Reaction Mechs~!sms": Van Nastrand: Berkshim (UKi, 19RO. 16) MU^, L. A. J. c h e m . ~ d u c .1 ~ 2 . 4 9 , ~ i 6 3 .
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