Howard K. Zimmerman
University of the Pacific Stockton, California
Method for Determining Order of a Reaction
Frequently the properties of a kinetic system are such that the degree of advancement of the reactiou can be determined only once for any given sample; i.e., neither periodic determinations of a characteristic property nor periodic removal of aliquots for subsequent aualysis is feasible. When this sitnation is encountered in reactions with moderate rates, one usually nms parallel samples on which determinations are made a t different times following inception of the process. For very fast reactions, however, this approach becomes complicated and is often impossible, while for very slow ones (e.g., geochemical processes) it is also often impossible. In the study of very fast processes, numerous ingeuious stratagems' (e.g., the various flow reactors or the application of relaxation methods) have beeu devised to permit the iuterpretation of rate data. However, most of these methods suffer either from the disadvantage of extreme waste in the consumption of materials during the experiments or from the high cost of the requisite apparatus. Since a general method is ueeded for the treatment of practical data dealing with both extremes of rate phenomena, the present paper will attempt t o lay a groundwork for that purpose. The basic procedure used is a substitution process which expresses the various rate laws in terms of one initial concentration only. Remaining concentratiou factors are reduced to dimensionless parameters expressing the ratios of initial concentrations and a reaction index which is unity a t the beginniug of a process and decreases toward zero as the reaction progresses.? Notation kr specific rate constant C,: instantaneous concentration of dominant reagent, A Ca, C,: instantaneous concentrations of minor reagents (O,' , Cbo,Cco: concentrations of above reagents at zero time 1: time measured from inception of reaction t i : time corresponding to the reaction index, i 1 = C,/Cao: index of reaction n: ratio of initial concentrations, Ca@/Cao . ratio of initial concentrations, C.O/CCn z: advancement of reaction, CCo- C. r: reaction order This work was supported by the National Science Foundation. EIGEN, M., A N D JOHNSON, J. S., Ann. Revs. Phys. Chem., 11, 3 0 7 3 4 (1960). 3 Although this is commonly done with zeroth order reactions and has been done occasionally with first order ones (e.g., BROWN, H. C., AND FLETCHER, E. A., J. Am. Chem. Soc., 7 3 , 2 8 0 8 ( 1 9 5 1 ) ) as well as in several problems conducted in our laboratory (vie. ZIMMBRMAN, H. K., ET AL., Ann., 628, 37 (10.51)); 655, 48, 54 ( 1 9 6 2 ) ) , its general utility appears not to have been i d l y appreciated and applied.
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Zeroth Order Reactions
The integrated rate law for a zeroth order reaction, the rate of mhich is independent of concentration of reagent, may he witten as k =
(C.0
- C.)/t = C$(1 - i ) / t i
(1)
The specific rate may be determined directly from equation (1). Alternatively, a plot of CaOagainst t, is linear with a slope such that k = (1-i) (slope). (The reactiou index most commonly used is i = in vhich case 1c = '/? (slope).) Yet another method of data treatmeut xhich may be d r a w from equation (1) is a plot of i against t , to yield a straight line originating at i = 1 and having a slope of -k 'Can. First Order Reactions
The integrated rate law for a first order reaction may be readily written in terms of the reaction index asfollows:
equation (Z), which implies that t t is iudependent of the initial concentration of reagent, permits direct, calculation of the specific rate; and for the so-called "half-life" it reduces to the familiar relation, k = 0.693/f,12. When t, values are k n o ~ ufor a variety of different reaction indexes, as in the method of parallel rate nms, a plot of log i against ti is liuear with its int,ercept at the origin and a slope of -A!2.303. Second Order Reactions
Among second order reactions, in which the rate of reaction is proportional to t x o concentrations, two experimental cases are usually distinguished according to whether the initial concentrations of reagents are equal or different from each other. Case I, Ca = Cn. When the initial concentrations are equal, the integrated rate lav yields
The specific rate may be computed direct,ly from equation (3) through a kuowledge of the ti corresponding to any reactiou index. When the half-time is used, equation (3) reducesto the familiar k = 1/Ca0tliy. When ti values are known for a particular reactiou index employiug a series of different initial coucentrations, it follows from equation (3) that a plot of l,'CaO against ti will be linear with the iutercept at the origin and a slope of ki/(l - i). On the other hand, vhen one plots the function (1 - i)/i against t i for different
reaction indexes obtained from a fixed initial coucentration, a straight line from the origin is ohtained with a slope equal to /,.Ca0. Case I I , C. Z Cb. In the general case of a second order rate process, the concentrations of the two reageuts are uneqnal. One may then write the rate law in terms of the dominant reactant and the advancement of the reaction in the following manner:
Case IZa Co = Ca # C.
Case IIb, C,
-- C, f Ca
Case ZIZ, C. #
Cb
f
Cc
I t should he noted that the notation for the species A and B is selected so that n > 1. Integration of equation (4) betweeo limits leads to
Introduction of the reaction index, in terms of which
yields
from vhich the specific rate may be directly obtained. I+om this relation it may also he shown that, a t a constant reactioo index and ratio of initial concentrations, a plot of ti against 1/C2 is linear from the origin with a slope given by the expression,
Similarly, if both of the initial concentrations are held constant, the plot of t , against the function,
In each of these special cases, the specific rate may, of course, he ascertained directly from computation based on one of the equations (7) through (10). Graphically, a plot of t c against either 1/C.0)2 (at constant index) or the function of i shown in brackets in each case (at constant initial composition) gives a linear relation in which the slope is inversely proportional to the specific rate. When i = equation (7) reduces to the familiar "half-life" expression for a third order process. In equation (8) through (10) the same principle is valid suhiect to the limitation that i 2 1 - I/n, since these three expressions possess the same type of singularity already noted for second order reactions. Moreover, in equation (8) the numerator of the logarithmic argument shows that indexes smaller than '/*are meaningless even when n lies between 1 and 2. In equation (10) there are two formal singularities, namely a t i = 1 - l/n and a t i = 1 - l/m. However, since m > n, the latter controls. The General Case
is linear from the origin with a slope given by
In considering the application of equation (6), it must he remembered that the limitation that n > 1 imposes certain restrictions. Specifically, examination of the logarithmic argument in equation (6) reveals a singularity when i = 1 - l/n, which follows from the definition of n. Thus equation (6) becomes physically meaningless for i I 1 - l/n, since for such values the advancement of reaction would exceed Cao. Within this limitation, however, equation (6) shows that the half-life concept (i.e., correspondence of ti to a given index) is just as valid for unequal concentrations as it is for equal ones. Third Order Reactions
Normally three types of composition circumstances may he envisioned with third order reactions. When the reaction index and the dimensionless composition parameters are employed in a manner analogous to that already indicated, the resulting integrated rate laws are found in the form summarized below: Case I, C. = Cs = C.
One may visualize a rate law of the rth order, in which it may be shown that the integrated rate law is
The same methods of using the equation to dete~miue specific rates that have already been mentioned also apply here, and when the reaction index is the usual "half-life" relation emerges. As a result of the treatment developed here, it is seen that the integrated rate laws may be expressed in a general sense in terms of dimensionless concentration terms except for the concentration of the major reagent. Kot only does this procedure clarify the dimensional relationships involved, hut it also emphasizes the generality of the reaction index as a concept commou to all simple rate laws. In consequence of this generalization, these conclusions follow: (1) The applicability of the "half-life" concept to cases of reaction orders greater than unity is seen to be valid within the limits of the condition, 1 > i > 1 - 1/n1 (where n' is the ratio of initial concerrtration of the most plentiful reagent to that of the least plentiful one), even for the general situation wherein the initial concentrations of the reagents are unequal. (2) From the first conclusion, it follows that it is possible to determine the specific rate of any reaction Volume 40, Number 7, luly 1963
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of simple order from a knowledge of reaction index at a single point in time together with a knowledge of the reagent concentrations a t the beginning of the reacti~n.~ (3) Even in the most general situation, the order of a simple reaction conforms t o the conditiona that,
for any given permitted reaction index and initial ratio of reactants, the product, ti(Cao)'-', is a constant. From this fact, it may be shown that the order is given by 7
=
K
+ log(C.O/ti) log
C2
(12)
This condition is well known in testing reaction orders by the K follows from the appropriate ~ i ~ ~ ~ where i ~ ~the , constant, n method, "&, LAIDLER, K, J., plot of t , against 1/(Ca0)'-'. McGraw-Hill, New York, 1950, p. 14. a
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