K. J. Hall, T. I. Quickenden, and D. W. Watts University of Western Australia Nedlands. 6009. W.A.. Australia
I I
Rate Constants from Initial Concentration Data
Little use has been made of the initial rate method in determining rate constants and mechanisms, although the method can have advantages over the traditional one of following reactions over several half-lives and fitting an integrated rate equation (1). Firstlv. when initial data' are used, the results are more often unromplicated by the effect of the reverse reaction. or by side reactions. Serondlv, traditional methods are often difficult to interpret whe; ionic strength and p H changes arcompanyinp s rpaction lcad to signiiirant changes in reaction conditions which are not easily specified hv the smirhi(metry of the reaction. I'nwided the initial data can be ohrained and treated for say the first 1Wo of reaction, all of these effects are minimized. Finally, provided good initialdata are a\~ailablefor different initial concentrati~msof reactants. the order of the reaction with respect to each separate reactant can he obtained in addition to the rate constant, and hence the data is amenable to normal mechanistic treatment (2). Despite these obvious advantages, the method of initial rates is used infrequently and receives only a brief mention in texts (1-3) because of two restrictions to its ao~lication. Firstly, the initial rate of a chemical reaction canbe determined only if very small amounts of the product or small losses in reactant can he measured accurately ( I ). Secondly the use of initial data requires that the slope of the tangent to the concentration-time data at zero time can he determined with accuracy (3).Numerical differentiation is unsuitable at the "time zero" point because such procedures require data for points before and after the point of concern ( 4 ) . The initial slope is normally obtained by visual estimation of the tangential gradient using a straight edge (5) or by using a mechanical device such as a front surfaced plane mirror (3) or a movable protractor fitted with a prism at the center (5). Disadvantages inherent in these methods include: problems
' In this paper, "initial data" refers to data collected within the first 1W of reaction.
of smoothing normal experimental scatter in the critical first few percent of reaction; heavy reliance on the earliest data which often suffer from errors such as temperature variation; uncertainties in the time zero due to problems in mixing; and difficulties in the estimation of error in the finally determined slope. We have found that the following technique overcomes these disadvantages, requires no special drawing apparatus, and is simple to use. A least squares fitting procedure is used to express the product concentration [PI as a polynomial in time, t [PI = n + bt + ct2 + . . . The initial rate is then given by
[TIt=,,+ =
[ ( b 2ct
+ . ..)
(1)
(2)
=b
This technique of representing data as a polynomial has been used in other areas of chemistry, such as gas properties (the virial equation) (6) and high pressure kinetics in solution (7). although usually without any theoretical basis. The application of this technique to initial rate measurements appear to be novel and is essentially the method described here. However, compared with mostcases of polynomial data fitting used in chemistry, in the case of initial rates there are two distinct advantages. Firstly, a theoretical justification of the polyuomial relationship can be given, and secondly, it is possible to simplify the mathematics involved substantially. Justification of the Polynomial Relationship Consider a general first-order reaction with the stoichiometry R-P and with the integrated rate equation (8)
(3)
[R] = [R]oeckt (4) where [R] is the concentration of the reactant a t time t, [RIo
Volume 53, Number 8, August 1976 / 493
Hypothetical Concentration versus Time Data for a Firrt-Order Reaction IPl/lRl,
IRI/lRI.
value of b and its associated standard error can be determined conveniently from a linear least squares fit of [Pllt versus t . The use of this technique can be illustrated by considering the first-order reaction shown in eqn. (3). Equation (4) can be expressed as
T i m~. et
Concenbation plot with initial tangent
In ([R]/[R]o) = -kt
its initial concentration, and k the rate constant. The product concentration, [PI, is given by [PI = [R]o - [R] = [R]&
- eckL)
(5)
and expanding e c k t as a power seiies gives (kt)% (kt)3 [PI = [R]o(kt - - -2! 9! -
+
...)
(6)
~
In the case of second-, third., and fractional-order reactions. providing a single reactant is i&olved or providing the initial concentrations of all reactants are equal, the rate equation can be expressed in the form (8) 1 -[R]n-I
(7)
where n is the order of reaction. Rearrangement of this expression followed by binomial expansion yields [PI = IRIO[ ( n - l)[(n - N R I ~ " - ~ ( ~~)I (n - l ) ( n - 2) [(n - l)[R]o"-'(kt)12 2! (n - l ) ( n - 2)(n - 3) [(n - l)[R]o"-l(kt)13 - . . .] (8) 3!
+
Thus in each case, the product concentration at time t, can be expressed as a polynomial in t
+ et2 + dt3 + . ..
(13)
Equations (5) and (13) can be used to produce concentration versus time data during the first 10% of reaction. The results of these calculations are shown in the table. Note that all numbers in this table are dimensionless and independent of [R]o and k. A conventional first-order rate wlot eives the rate constant k. However, if the [Pllt versus t da'ta is h t e d to a least squares line of best fit. it is found that: Intercent = initial rate = (0.9998 o.oo'os)~[R]~where the error'quoted is the 95% confidence interval. Clearly the initial rate constant thus obtained (kin)is related to the first-order rate constant through
*
kin = (0.9998)k
(n - 'Ikt
&=
[PI = bt
IPl/(tkIRl,)
kt
(9)
where b, c, and d, etc. are constants. This series can be truncated after the second term if only the first 10% of reaction is followed, i.e. [PI = bt + ctZ (10) The validity of this truncated expression can be tested by calculating the time required for 10% of reaction from the integrated rate equation and substituting in eqn. (10) to yield [PI which should be O.l[R]o. The value obtained from eqn. (10) is 0.2% low for a first-order reaction, 1.2%low for a secondorder process, and 3.3% for a third-order reaction and, of course, the agreement is better earlier in the reaction.
Thus the method yields a rate constant which is only 0.02% lower than that obtained from a conventional lot of the same data. For second-order data the value of hi, is b.2%lower than k, and for third-order data the value of k;. is 0.4% low. nrovided only data for the first 1% of reactionare included.'i'hus the e r n m due to the t r u n h i o n 01the ~olvnominlseries and the adoption of eqn. (10) are negligibie compared with the other errors normally associated with the accumulation of kinetic data. The present initial rate procedure can also be applied very successfullyto the data given by Frost and Pearson (9)for the total pressure, P, during the first-order decomposition of dit-butyl peroxide. A conventional plot of log (P, - P ) versus t yields a rate constant of (3.48 f 0.02) X s-' from this data. In the awwlication of our initial rate nrocedure lnroductllt is plotted against t. Here a plot of ( P Po)I(P, 1:Po)t versus t has an intercept of (initial rate/[RIo) or kin. The rate s-I. The constant by this method is (3.56 f 0.06) X ameement is remarkable when it is considered that all the data over 36% of reaction have been used while the method is designed to cover only the first 10%. 1 t is clear that the suggested method of obtaining initial rates leads to a simple and accurate method of determinine rate constants. The method is particularlyvaluable when side reactions complicate a mechanism and thus limit the use of an integrated rate equation.
-
Literature Cited
Simplification to a Linear Plot
It can be seen from the figure that the equation of the initial tangent to the concentration plot is
.
[PI = b t
(11)
where b is the initial rate. If eqn. (10) is divided by t it becomes [Pllt = b
+ ct
(12)
Thus a plot of [Pllt versus t must be linear with an intercept of b equal to the initial rate, as long as eqn. (10) holds. The 494 / Journal of Chemical Education
(31 M
York. 1961.2nd Ed.. 0.45
~ ~ D.,~..Combhensive ~ V ~ Chemical ~ . Kinetier:. IEditor: Barnford, C. H., and T i p p r , C. F. H I , Elrevier. New York. 1969. Val. 1. p. 356. 141 Msreenau. H., and Murphy, G. M..'The MethematicsofPhyrinand Chemistry."Ven Nostrand, New York, 1959.2od Ed.. Vol I, p 472. 151 Miekby, H. S.,Sherwd,T.K.. and Reed,C. R.."Applied Mathemeties in Chemical Engineering..'MeCmw-Hill, New York, 1957.2nd Ed.,p. 24. I61 Barrow. G.M. "Physical Chemistry," McCraw-Hill. New York, 1966.2nd Ed.. p. 12. 171 EckerGC. A..Ann.Reu. Phyl. Chem., 23.242 119721. 18) Reference (2). D. 13: 191 Reference 121. b. 31
