Page 279 - Elements of Chemical Reaction Engineering 3rd Edition

See. 5.5 Least-Squares Analysis 251

We have three linear equations and three unknowns which we can solve for:
a,, a,, and a,. A detailed example delineating the lunetics of the reaction

using linear least-squares analysis can be found in Example 10-2. If we set
a, = 0 and consider only two variables, Y and X, Equations (5-31) and (5-32)
reduce to the familiar least-squares equations for two unknowns.
/ Example 5-5 Using Least-Squares Analysis to Determine Rate Luw Paramteters

The etching of semiconductors in the manufacture of computer chips is another
important solid-liquid dissolution reaction ,(see Problem P5-12 and Section 12.10).
The dissolution of the semiconductor MnO, was studied using a number of diflerent
acids and salts. The rate of dissolution was found to be a function of the reacting
liquid solution redox pptential relative to the energy-level conduction band olf the
semiconductor I: was found that the reaction rate could be increased by a factor of
A lo5 fold increase lo5 simply by changing the anion of the acid9!! From the data below, determine the
in reaction rate! ! ! reaction order and specific reaction rate for the dissolution of MnO, in HBr.
C, (mol HBrMm3) 0.1 0.5 1.0 2.0 4.0

-ri, (mol HBr/m2. h) X lo2 0.073 0.70 f.84 4.86 12.84

Solution
We assume a rate law of the form
- rkB = (E5 -5.1)
Letting A = HBr, taking the In of both sides of (E5-5.1), and using the initial rate
and concentration gives

In(-ri,) = lnk+a lnCAo (E5-5.2)
Let .Y = In(-ri,), a = Ink, b = a, and X = lnCAo. Then
Y=a+bX (E5-5.3)
The least-squares equations to be solved for the best values of a and b are for N runs

N N N
%X,Y, =az~~+bx~,~ (E5-5.5)
z=I 1=1 z=1
where i = run number. Substituting the appropriate values from Table E5-5.1 into
Equations (E5-5.4) and (E5-5.5) gives

S. E. Le Blanc and H. S. Fogler, AIChE J., 32, 1702 (1986).