Correlations activation parameter errors

The precision of the rate constants as a function of temperature determines the standard deviations of the activation parameters. The absolute error, not the percentage error in the activation parameters, represents the agreement to the model, because of the exponential functions. If, for example, one wished to examine the values of AS for two reactions that were reported as -4 3 and 26 3 J mol 1K 1, then it should be concluded that the two are known to the same accuracy. Since AS and A// are correlated parameters, the uncertainty in AS will be about 1/Tav times that in A//. At ambient temperature this amounts to an approximate factor of three (that is, 1000/T, converting from joules for AS to kilojoules for A// ). Thus, the uncertainty in A//, 0 of 2.50 kJ mol 1 is consistent with the uncertainty in ASn of 7.21 J mol1 K-1 at Tav - 350 K. 

Only true rate constants (i.e., those with no unresolved concentration dependences) can properly be treated by the Arrhenius or transition state models. Meaningful values are not obtained if pseudo-order rate constants or the rates themselves are correlated by Eq. (7-1) or Eq. (7-2). This error is found not uncommonly in the literature. The activation parameters from such calculations, A and AS in particular, are meaningless. 

Error correlation and co-variance. Parameters may affect one another, as do the pre-exponential factor A and the activation energy Ei in the Arrhenius equation. If so, the errors are correlated, and a co-variance must be included in the error-propagation formula. For a quantity y that is a function of two parameters xl and x2, the co-variance is denoted oXlX2, and the variance in y is... 

A plot of In (k/T) versus 1/T is linear, Fig. 2.4. Least squares analysis of the data gives AH = 50.5 kJ tnof and AS 29s -135 JK mof with a correlation coefficient of 0.9980. Using these activation parameters gives the calculated values of k,q (calc) quoted in Table 2.1. There are frequently many errors in the activation parameters quoted in the literature and it is often advisable to check the values using the quoted rate constants. 

If the standard error is small on AI ( 0.02 eV), it is certainly high on A because the value of A on surface or at the steady state cannot be ascertained. Moreover, several solids are tried for ODH of alkanes, as already mentioned, with little success as far as selectivity is concerned. For sure, however, the correlations show that a catalyst for ODH must be less acidic than for MOX of alkanes. It must be emphasized that these correlations are given for a theoretical selectivity but do not apply to activity parameters, as far as different reactions are compared. Up to now, and to give an example, our attempts failed in correlating optical basicity of oxides with the turnover frequency measured for various oxidation reactions by using the very thorough work presented by Wachs et al. [92]. 

The parameters in Eq. (2.59) are usually determined from the condition that some function mean-square deviations between the experimental and calculated curves (the error function). The search for the minimum of the function Nelder-Mead algorithm.103 As an example, Table 2.2 contains results of the calculation of the constants in a self-accelerating kinetic equation used to describe experimental data from anionic-activated e-caprolactam polymerization for different catalyst concentrations. There is good correlation between the results obtained by different methods,as can be seen from Table 2.2. In order to increase the value of the experimental results, measurements have been made at different non-isothermal regimes, in which both the initial temperature and the temperature changes with time were varied. 

Statistical analysis of the results was performed using the software Statistica 5.5 (Stat Soft). Maximum lipase activities and time to reach the maximum were calculated through fitting of kinetic curves. The maximum was estimated by derivation of the fits. Empirical models were built to fit maximum lipase activity in the function of incubation temperature (T), moisture of the cake (%M), and supplementation (%00). The experimental error estimated from the duplicates was considered in the parameter estimation. The choice of the best model to describe the influence of the variables on lipase activity was based on the correlation coefficient (r2) and on the x2 test. The model that best fits the experimental data is presented in Table 2. 

The biological activity satisfies the second and third requirements, since each measurement is done independently (see, however, below). Upon repeated measurements the values will be normally distributed, and will have an equal variance. The independent variables in the Hansch approach are obtained from experimental measurements and contain therefore an experimental error. They cannot be considered as fixed variates. The experimental error in these parameters, however, is usually much smaller than the error in the dependent variable they can be treated therefore as fixed variates. The complications arising from correlations between observations that include experimental errors have been thoroughly analyzed (224). 

---