ChE 452 Lecture 06

Analysis of Direct Rate Data

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Objective

How do you fit data Least squares vs lowest variance

Strengths, weaknesses Problem with r2

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Analysis Of Direct Rate Data

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0.020

0.100

10 100

Oxygen Pressure (Torr)

Etc

h R

ate

(mic

rons

/min

)

Metallic Color

Oxide Color

Slope = 0.5

0.030

0.040

0.050

0.060

0.080

0.010

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 r2

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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.

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]par[K1

]par[Kkr

2

21p

(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.

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Methodology

There are two versions of the linear plots:• Lineweaver-Burk Plots• Eadie-Hofstee Plots 

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(3.A.2)

Methodology

 In the Lineweaver-Burk method, one plots 1/rate

vs. 1/concentration.

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12121

2

21p k

1

]par[Kk

1

]par[Kk

]par[K

]par[Kk

1

r

1

(3.A.2)

Methodology

]par[K1

]par[Kkr

2

21p

Rearranging

A B C D E F

01 k_1 139.400 0.194 2.000

02 K_2 0.037 0.007 9642.8

03 r2z 0.900 Calculated

04 conc rate 1/conc 1/rate rate error

05 0.000 0.000 0.000 0.000

06 2.000 10.400 0.500 0.096 9.602 0.636

07 3.600 12.800 0.278 0.078 16.382 12.829

08 4.000 23.200 0.250 0.043 17.967 27.379

09 5.200 17.600 0.192 0.057 22.488 23.897

10 7.800 46.400 0.128 0.022 31.216 230.566

11 8.000 23.200 0.125 0.043 31.833 74.534

12 8.000 46.400 0.125 0.022 31.833 212.189

13 11.000 32.000 0.091 0.031 40.319 69.210

Table 3.A.3 The numerical values in the spreadsheet for the Lineweaver Burke plot

Table continues

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Lineweaver-Burk Plots

Numerical results

From the least squares fit,

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121p k

1

]par[Kk

100717.0

]par[

194.0

r

1

(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

How Well Does It Fit?

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0 0.1 0.2 0.3 0.4 0.5 0.60

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

1/R

ate

1/Concentration0 10 20 30 40 50 60 70 80 90

0

50

100

150

Rat

e

Concentration

Data

Lineweaver-Burk

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/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.

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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:

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p221p rKKk

]par[

r

(3.A.5)

Eadie-Hofstee Plot

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0 50 100 1501

2

3

4

5

6

7

Rat

e/co

ncen

trat

ion

Rate0 10 20 30 40 50 60 70 80 90

0

50

100

150

Rat

e

Concentration

Data

Eadie-Hofstee

r2=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

r2 Does Not Indicate Goodness Of Fit

Eadie-Hofstee gives much lower r2 but better fit to data!

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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:

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Data

2

2

21p ]par[K1

]par[KkrabsErrorTotal

(3.A.7)

Summary Of Fits

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0 10 20 30 40 50 60 70 80 900

50

100

150

Rat

e

Concentration

Data

Non-linear least squares

Non-linear least squares

0 10 20 30 40 50 60 70 80 900

50

100

150

Rat

eConcentration

Eadie-Hofstee

Lineweaver-Burk

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

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Method k1 K2 Total error r-squared

Lineweaver-Burk 139 0.037 9643 0.910 (linear plot)

Eadie-Hofstee 267 0.0156 6809 0.344 (linear plot)

non-linear least squares 204 0.0221 4919 0.905 (non-linear)

Table 3.A.5 A comparison of the various fits to the data in Table 3.A.1

Note: 1)Results change according to fitting method2)there is no correlation between r2 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 r2

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Class Question

What did you learn new today?

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