ChE 452 Lecture 06

1. ChE 452 Lecture 06 Analysis of Direct Rate Data

2. Objective • How do you fit data • Least squares vs lowest variance • Strengths, weaknesses • Problem with r2

3. Analysis Of Direct Rate Data 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].

4. Usually Not So Easy • Results vary with how fitting is done • Cannot tell how well it works by looking at r2

5. 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 k1 and K2 are constants. (3.A.1)

6. Methodology • 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.

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

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

9. Lineweaver-Burk Plots

10. Numerical results From the least squares fit, (3.A.3) Comparison of equations (3.A.2) and (3.A.3) shows: k1 = 1/.00717=139.4, K2=1/(0.194*k1)=0.037, r2=0.900

11. How Well Does It Fit? 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

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

13. Eadie-Hofstee Plot Avoid the difficulty at low concentrations by instead finding a way to linearize the data without calculating 1/rp. Rearranging equation (3.A.1): rp(1+K2[par])=k1K2[par] (3.A.4) Further rearrangement yields: (3.A.5)

14. Eadie-Hofstee Plot r2=0.34 Figure 3.A.4 The Eadie-Hofstee fit of the data in Table 3.A.1 Figure 3.A.3 An Eadie-Hofstee plot of the data in Table 3.A.1

15. r2 Does Not Indicate Goodness Of Fit Eadie-Hofstee gives much lower r2 but better fit to data!

16. 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: (3.A.7)

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

18. Comparison Of Fits • Note: • Results change according to fitting method • there is no correlation between r2 and goodness of fit.

19. Summary • Fit data using some version of least squares • Results change drastically according to How You fit data • Caution about using r2

20. Class Question • What did you learn new today?