Page 76 - Introduction to chemical reaction engineering and kinetics

58 Chapter 3: Experimental Methods in Kinetics: Measurement of Rate of Reaction

Generally, the primary objective of parameter estimation is to generate estimates
of rate parameters that accurately predict the experimental data. Therefore, once es-
timates of the parameters are obtained, it is essential that these parameters be used to
predict (recalculate) the experimental data. Comparison of the predicted and experi-
mental data (whether in graphical or tabular form) allows the “goodness of fit” to be
assessed. Furthermore, it is a general premise that differences between predicted and
experimental concentrations be randomly distributed. If the differences do not appear
to be random, it suggests that the assumed rate law is incorrect, or that some other
feature of the system has been overlooked.
At this stage, we consider a reaction of the form of (A) in section 3.1.2:

lvAIA + /vulB + IV& + products (A)

and that the rate law is of the form of equations 3.1-2 and 3.1-8 combined:
.
(-rA) = kAcic[. . = Aexp(-E,lRT)czcg . . .
(In subsequent chapters, we may have to consider forms other than this straightfor-
ward power-law form; the effects of T and composition may not be separable, and, for
complex systems, two or more rate laws are simultaneously involved. Nevertheless, the
same general approaches described here apply.)
Equation 3.4-17 includes three (or more) rate parameters in the first part: kA, a, j?,
. ..) and four (or more) in the second part: A, EA, (Y, p, . . . . The former applies to data
obtained at one T, and the latter to data obtained at more than one T. We assume that
none of these parameters is known a priori.
In general, parameter estimation by statistical methods from experimental data in
which the number of measurements exceeds the number of parameters falls into one of
two categories, depending on whether the function to be fitted to the data is linear or
nonlinear with respect to the parameters. A function is linear with respect to the param-
eters, if for, say, a doubling of the values of all the parameters, the value of the function
doubles; otherwise, it is nonlinear. The right side of equation 3.4-17 is nonlinear. We
can put it into linear form by taking logarithms of both sides, as in equation 3.4-4:

ln(-rA) = lnA-(E,/RT)+aclnc,+Plncu+... (3.4-18)

The function is now ln(-rA), and the parameters are In A, EA, a, p, . . . .
Statistical methods can be applied to obtain values of parameters in both linear and
nonlinear forms (i.e., by linear and nonlinear regression, respectively). Linearity with
respect to the parameters should be distinguished from, and need not necessarily be
associated with, linearity with respect to the variables:

(1) In equation 3.4-17, the right side is nonlinear with respect to both the parameters
(A, EA, (Y, p, . . .) and the variables (T, CA, cB, . . .).
(2) In equation 3.4-18, the right side is linear with respect to both the parameters and
the variables, if the variables are interpreted as l/T, ln CA, ln cn, . . . . However,
the transformation of the function from a nonlinear to a linear form may result
in a poorer fit. For example, in the Arrhenius equation, it is usually better to esti-
mate A and EA by nonlinear regression applied to k = A exp( -E,/RT), equation
3.1-8, than by linear regression applied to Ink = In A - E,IRT, equation 3.1-7.
This is because the linearization is statistically valid only if the experimental data
are subject to constant relative errors (i.e., measurements are subject to fixed
percentage errors); if, as is more often the case, constant absolute errors are ob-
served, linearization misrepresents the error distribution, and leads to incorrect
parameter estimates.