We will now use nonlinear regression to determine the rate law parameters from concentration–time data obtained in batch experiments. We recall that the combined rate law-stoichiometry-mole balance for a constant-volume batch reactor is
We now integrate Equation (7-6) to give
Rearranging to obtain the concentration as a function of time, we obtain
Now we could use Polymath or MATLAB to find the values of α and k that would minimize the sum of squares of the differences between the measured and calculated concentrations. That is, for N data points,
we want the values of α and k that will make s2 a minimum.
If Polymath is used, one should use the absolute value for the term in brackets in Equation (7-16), that is,
Another way to solve for the parameter values is to use time rather than concentrations:
That is, we find the values of k and α that minimize
Finally, a discussion of weighted least squares as applied to a first-order reaction is provided in the Professional Reference Shelf R7.4 on the DVD-ROM.
Example 7-3. Use of Regression to Find the Rate Law Parameters
We shall use the reaction and data in Examples E7-1 and E7-2 to illustrate how to use regression to find α and k′.
The Polymath regression program is included on the DVD-ROM. Recalling Equation (E5-1.5)
and integrating with the initial condition when t = 0 and CA = CA0 for α ≠ 1.0
We can proceed two ways from this point, both of which will give the same result. We can search for the combination α and k that minimizes [σ2 = Σ(tim – tic)2], or we could solve Equation (E7-4.3) for CA and find α and k that minimize [σ2 = Σ(CSim – CAic)2]. We shall choose the former.
Substituting for the initial concentration CA0 = 0.05 mol/dm3
The Polymath tutorial on the DVD-ROM shows screen shots of how to enter the raw data in Table E7-2.1 and to carry out a nonlinear regression on Equation (E7-3.2). For CA0 = 0.05 mol/dm3, that is, Equation (E7-3.1) becomes
We want to minimize s2 to give α and k′.
The result of the first and second Polymath regressions are shown in Tables E7-3.1 and E7-3.2.
The first regression gives α = 2.04, as shown in Table E7-3.1. We shall round off α to make the reaction second order, (i.e., α = 2.00). Now having fixed α at 2.0, we must do another regression (cf. Table E7-3.2) on k′ because the k′ given in Table E.7-3.1 is for α = 2.04. We now regress the equation
The second regression gives k′ = 0.125 dm3/mol · min. We now calculate k
Analysis: In this example we showed how to use non-linear regression to find k′ and α. The first regression gave α = 2.04 which we rounded to 2.00 and then regressed again for the best value of k′ for α = 2.0 which was k′ = 0.125 (dm3/mol)/min giving a value of the true specific reaction rate of k = 0.25 (mol/dm3)2/min. We note that the reaction order is the same as that in Examples 7-1 and 7-2; however, the value of k is about 8% larger. The r2 and other statistics are in Polymath’s output.
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