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Vol. 61
NOTES A FUNCTION TO AID I N T H E FITTING OF KINETIC DATA TO A RATE EQUATION BYJOSEPH H. FLYNN National Bureau of Standards, Washington 66,D . C. Received M a y Y , 1966
A chemical reaction in which the rate is proportional to powers of the concentrations of a single reactant or several reactants present in stoichiometric concentrations may be reduced to the equation
;;
- =
k(l
- 2)"
where x is the degree of advancement or extent of reaction, k is a constant t,hat may include stoichiometric relationships and initial concentrations, and n is defined as the order. A large number of chemical reactions that give a changing value for n (and k ) when represented by equation 1 may better be represented by
g
= K(1 -
5)P
Hence the parameters p, v, K and 01 can be determined from the limiting characteristics of n, and log dxldt. The function e
defines a one-parametric family of curves which is useful in the determination of a, 1.1 and Y. A set from this family is represented in Fig. 1. Such a set of 6' vs. x curves may be used to estimate values of a,p and v to be used as guides for more precise curve fitting.
(1 - ax)y
where K, a, p , and v may be considered in general as arbitrary parameters. A few examples of reactions that are well represented by equation 2 are (a) first-order reversible reactions ( p = 0, Y = 1, a > 1) (b) parallel reactions: one zero order and one first order ( p = 0 , Y = 1 , l > C Y > 0) (c) parallel reactions: one first order and one second order = 1, Y = 1 , 1 > CY > 0) (d) hydrogen bromide formation ( p = 6 / 2 , Y = -1, CY > 0 ) (e) . . manv catalvtic reactions ( Y > 0 for Dositivelv catalyzed"reactiok, and Y < 0 for negatively'catalyzld reactions. If a reaction is catalyzed by a reactant, a > 0; if catalyzed by an intermediate or a product, CY < 0) ( f ) reactions in which more than one reactant contributes t.o the order in propbrtion to a power of its concentration and one of these reactants is present in a non-equivalent concentration.
(r
This paper describes an approximate method for estimating p , v, K and a of equation 2 when inconstant values for n and IC are obtained from the differential rneth0d.l A plot of log dx/dt with respect t o log (1 - x) for data from a reaction whose kinetics follow equation 2 yields a curve with the slope, n, given by dx d log dt = nr = -(1 log (I - 2)
+ QV(1
- 2)
1 - ax
(3)
F r o m equation 3 it f o l l o ~ that s
and from equation 2 that (1) For a description of Letort's extension of the differential method of van't Hoff, see K. J. Laidler "Chemical Kinetics," McGraw-Hill Book Co., Inc., New York, N. Y., 1960,pp. 15-18. (2) The limit in equation 5 has no physical significance for LI > 1 as d d d t = 0 when x = l / a , 6 . 0 . ~ a reversible reaction.
-6-
0
'
.I
.2
I
I
I
I
.3
A
.5
.6
I 7
I
I
.8
.9
1.0
X.
c
Fig. 1 . A us. z for various a.
hu example to which this technique may be applied is the decomposition of NaCIOz in acetic acidsodium acetate buffer (pH 3.52, p = 0.11) at 40". This reaction has been described as second order.a Careful reinvestigation of the kinetics of this reaction4 indicated a curvature of the log dx/dt us. log (1 - r ) plot from which values for n, were obtained for x = 0 to 0.75. If it is assumed that 0 = - (n, 1.*5), then the points in Fig. 1 are obtained. Therefore, the reaction appears to be of an order less than two ( p = 1.5) and is inhibited by a product (Y = - I , (Y < 0). Subsequent data from the determination of initial rates indicate an order with respect t o initial concentration of approximately 1.G9 and a n inhibition by C1-, a product of the reaction. (3) H. F. Launer, W. K. Wilson and J. H. Flynn, J . Research Natl. Bur. Standards, 61, 237 (1953). (4) J. H. Flynn and W. L. Morrow, unpublished work.
c
