CORRESPONDENCE
AN ANALYSIS OF KINETIC POWER FUNCTION MODELS n previous papers, methods were presented for estimat-
I ing reaction order from isothermal, constant volume, batch reactor data (2, 3 ) . These procedures represent a departure from the usual power function analyses which fit data for which error distribution has not been considered. For an order to be estimated from a single concentration-time profile, a method has been presented ( I , 3) which provides a good data fit by a transformation of the dependent variable. This method was later extended (2) to allow a similar estimation of a reaction order for concentration-time data taken at several initial concentrations but a common temperature. The necessity for obtaining a reaction order describing several concentration-time profiles at a given temperature is quite common. T o determine a single reaction order for several such profiles at different temperatures, the nonlinear temperature dependence of the rate constant requires the use of nonlinear least squares (5). We shall here be concerned with the methodology required for examination of several concentration-time profiles at a common temperature. We shall illustrate typical problems that may arise during application of the procedure and indicate how to cope with these problems. As was done earlier, let us choose the variables for the ith data point
Hence y t j is the reciprocal of the fraction of unreacted reactant, and X is related to the reaction order. If we further define
shown ( 2 ) that a maximum likelihood estimate of X can be obtained by minimizing Nj
n
S(X)
=
c c (Z2 - b h J 2
(4)
1=1 z = 1
z,jx = Y W ”
where
(5)
A Z O
(6)
x = 0
g
=
(fi ;
(7)
j = 1 r = l Yt,)l/N
N3
c
=
(j ; = 1I iTI = l c,)l’N
(8)
n
N = C 1 = 1N ,
(9)
I n other words, to estimate a reaction order without the masking effect of error, we should take advantage of the linear nature of the problem of Equation 3 and, 1. Estimate by unweighted linear least squares the parameter minimizing Equation 4-i.e., find
for a given X and calculate the sum of squares S(X)
2.
Plot the minimum sum of squares for several 1
3. Read off the minimum of this plot to obtain the best X,^x
4. Calculate the 99% confidence interval for ^x from 6
In S(X) - In S(X) then the rate equation may be written for any reaction order in the generalized form
If we now assume that a value of X exists which provides goodness-of-fit of the predicted and observed values y z jx , constancy of error variance of y a i X , and normality and independence of error of y a j x , then it has been
5
Xi2 (0.01) -~
N
6.63 N
= __
(11)
For the simplified case in which only one concentration-time profile is available (n = l ) , one needs to check only for an overall lack of fit of the model to the data by an analysis of variance ( 2 ) to prevent gross misapplications of the method: That is, the experimenter should be careful to examine the ability of a power function to represent his data. VOL. 5 9
NO. 1
JANUARY
1967
93
For several concentration-time profiles, it is still important that the power-function model adequately represent the data. Now, however, the data can exhibit an “order with respect to concentration” as well as the “order with respect to time” (4). This would be reflected by a reaction order which changed significantly between profiles when estimated by single-profde methods (3). These considerations may be better discussed and a general method of analyzing multiple profile data considered after their illustration by an example selected to exhibit these characteristics. Example
magnitude. Figures 1 through 5 present the sum of squares of the single profile dependent variables:
+
CHaCHO
+
‘/P
TABLE 1. ESTIMATED VALUES OF m AND THEIR APPROXIMATE CONFIDENCE L I M I T
k
= 1.39
13.48 21.14 25.17
CzHsOH
Data were taken at five different levels of ethyl nitrite initial pressure of 13.48, 21.14, 25.17, 26.14, and 36.57 :m. Hg. The order of reaction was first order, and the rate constant was expressed by
X 1014 e47,7m/nr set.-'
Utilizing this relationship, we estimated the value of k at 209.8’ C. to be 1.19 X lo-“ set.-' We wish to reanalyze these data to determine whether a common reaction order can be obtained, and, if so, its
(12)
i=l
Table I shows the values of m and the 99% confidence interval of the reaction order for five different initial pressures. Notice that the reaction is approximately of first order and initial pressure does not have any effect on reaction rate up to 36.57 cm. Hg of ethyl nitrite.
Steacie and Shaw (6) studied the gaseous homogeneous decomposition of ethyl nitrite at 209.8” C. The reaction is C Z H ~ N O-+ Z NO
c (Z, - btJ2 N1
S(X) =
1.05
0.80
1.05 1 .oo
0.99 0.95
--
1.34 1.11 1.05
1.95 1.93 1.15
See Eguofbn 13.
With the preliminary information thus obtained, we looked for a common reaction order and rate constant. Figure 6 presents the sum of squares of the transformed dependent variables 2,; obtained from Equation 5. An examination of Figure 6 indicates that the reaction order 1.075 f 0.083 may be most appropriate for the system. The 99% confid&c< interval does include the
In the following figures the ordinate is plotted as the sum of squares for the transformed variable 2 (ZA-Z*,J* multiplied by the indicated factor. The inset pressure indicates the pressure of the system in which the decomposition of ethyl nitrite occurred. Figure 6 is the curve for the combined data
‘1
~-
1I
I
value 1.0 which was reported by Steacie and Shaw. The rate constant was also estimated by means of the following relationship:
k=-
6 F' t
where
10. For the reaction order 1.075, the best estimate of the rate constant was 1.992 X lo-* + 0.088 X lo-*. This example shows several advantages of using combined data to estimate the parameters of a rate expression. If the data were analyzed individually, it might be rather difficult to decide which values of constants should be taken. In the method demonstrated, the parameters can be determined explicitly. With the aid of a computer, the entire calculation can be carried out as a routine. A word of caution is that this method assumes the constancy of the two parameters rn and k for different conditions of experimentation. Thus, some preliminary calculation, as shown in this note, is necessary to assure that the system has the constants rn and k.
Allied Chemical Carp. Bufalo, N . Y.
1
L
O00. ' 90 k
4
= geometric mean of all experimental values of the initial concentration = initial concentrations within a particular run j cj . I * = variable defined by Equation 6 k = reaction wnstant m = reaction order n = numbcr of O k v a t i o M Nj = total numba of a k v a t i o n s S(X) = sum of squares of residuals for transformed variable for a given X fi, = reaction time for the itb data point within a particular run j xi, = fractional conversion for the ith data point within a particular run j yi, = dependent variable related to conversion by Equation 1 for the ith data point within a particular run j ytjh = transformed dependent variable given by Equation 2 I/ = geometric mean of all experimental d u e of the untransformed variable y Z1ja = normalized transformed dependent variable given by Equation 5 = parameter related to the reaction order = best estimate of the parameter minimizing the sum of squares of Equation 4 = the absciasa value below which is lOO(1 - .z)yoof the XI' area u n d a the curvt of the chi-squared distribution with one dcgrce of freedom
LITERATURE CITED
W. J. Hill
i
. = transformed rate wnstant of Equation = estimated value of b
(1) Box,G. E. P.,Co.,D.R., J . R q . . % l i r & d S ~ . , s h . B P 6 (2),211 (1964). (2) Knlrcll. J . R.. M a a h R., INO. ENO.Cnru.,in prm. 131 KiWcU, I. R ,Mcmki, R.,Wainon. C . C..lb,d., 58 (51, SI (1966).
Richmond, Calif.
v)
b t
(13)
6, j , and t were defined by Equations 4 through
R. Mezaki Yale uniucrsify New Haven, Conn. J. R. Kittrell Churon Research Co.
NOMENCLATURE
I
