ChE 452 Lecture 06 Analysis of Direct Rate Data 1

Objective How do you fit data Least squares vs lowest variance Strengths, weaknesses Problem with r 2 2

Analysis Of Direct Rate Data 3 General method – least squares with rate vs time data Figure 3.10 The rate of copper etching as a function of the oxygen concentration. Data of Steger and Masel [1998].

Usually Not So Easy Results vary with how fitting is done Cannot tell how well it works by looking at r 2 4

Example: Fitting Data To Monod’s Law Table 3.A.1 shows some data for the growth rate of paramecium as a function of the paramecium concentration. Fit the data to Monod’s Law: where [par] is the paramecium concentration, and k 1 and K 2 are constants. 5 (3.A.1)

There are two methods that people use to solve problems like this: Rearranging the equations to get a linear fit and using least squares Doing non-linear least squares to minimize variance I prefer the latter, but I wanted to give a picture of the former. 6 Methodology

There are two versions of the linear plots: Lineweaver-Burk Plots Eadie-Hofstee Plots 7 (3.A.2) Methodology

In the Lineweaver-Burk method, one plots 1/rate vs. 1/concentration. 8 (3.A.2) Methodology Rearranging

9 Lineweaver-Burk Plots

Numerical results From the least squares fit, 10 (3.A.3) Comparison of equations (3.A.2) and (3.A.3) shows: k 1 = 1/.00717=139.4, K 2 =1/(0.194*k 1 )=0.037, r 2 =0.900

How Well Does It Fit? 11 Figure 3.A.1 A Lineweaver- Burk plot of the data in Table 3.A.1 Figure 3.A.2 The Lineweaver- Burk fit of the data in Table 3.A.1

Why Systematic Error? We got the systematic error because we fit to 1/r p. A plot of 1/r p gives greater weight to data taken at small concentrations, and that is usually where the data is the least accurate. 12

Eadie-Hofstee Plot Avoid the difficulty at low concentrations by instead finding a way to linearize the data without calculating 1/r p. Rearranging equation (3.A.1): r p (1+K 2 [par])=k 1 K 2 [par] (3.A.4) Further rearrangement yields: 13 (3.A.5)

Eadie-Hofstee Plot 14 r 2 =0.34 Figure 3.A.3 An Eadie- Hofstee plot of the data in Table 3.A.1 Figure 3.A.4 The Eadie- Hofstee fit of the data in Table 3.A.1

r 2 Does Not Indicate Goodness Of Fit Eadie-Hofstee gives much lower r 2 but better fit to data! 15

Non-linear Least Squares Use the solver function of a spreadsheet to calculate the best values of the coefficients by minimizing the total error, where the total error is defined by: 16 (3.A.7)

Summary Of Fits 17 Figure 3.A.5 A nonlinear least squares fit to the data in Table 3.A.1 Figure 3.A.6 A comparison of the three fits to the data

Comparison Of Fits 18 Note: 1)Results change according to fitting method 2)there is no correlation between r 2 and goodness of fit.

Summary Fit data using some version of least squares Results change drastically according to How You fit data Caution about using r 2 19

Class Question What did you learn new today? 20