ChE 551 Lecture 03Analysis of Rate Data

1

General Approach

Initiate reactionmeasure concentration vs timefit data to calculate rates

2

 

0 1 2 3 4 5 6

Time, Hours

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Co

nce

ntr

atio

n

Rate Measurements: An Old Topic

3

 

0 2 4 6 8 10 12

Time, Hours

00.10.20.30.40.50.60.70.80.91

1.1

Co

nce

ntr

atio

n

Figure 3.1 Wilhelmy’s [1850] measurements of the changes in sucrose concentration in grape juice after acid is added.

Many Methods To Do Measurements

Techniques include: conventional, stopped flow, temperature jump…

Differ via time scale of reaction Need to mix reactants and initiate

reaction before reaction is done Different techniques used for fast

reactions than slow ones

4

How Do You Decide What Experimental Method To

Use?

Key Issues:• Direct method or indirect method• Can measurement be done on an

appropriate time scale?

5

Direct vs Indirect Methods

Recall – rate equation is the rate as a function of the concentrations

• Direct method - any method where you actually measure the rate as a function of concentration

• Indirect method - a method where you measure some other property (i.e. concentration vs time) and infer a rate equation.

6

Example: Consider Arsine Doping Of Silicon

7

To Pump

Feed

SiliconWafers

Holder(boat)

3-zone ovenPressure Gauge

Door(Loadlock)

Figure 3.6 A typical arsine decomposition reactor.

2AsH 2As 3H3 2

Direct Measurement

8

P = P eAsH AsH0 k t

3 31

ExhaustFeed

Wafer

Microbalance

HeatLamp

Figure 3.7 A possible apparatus to examine the decomposition of arsine (AsH3) on silicon.

Indirect Measurement

9

0 5 10 15 20 25 30 3510

100

Time, Hours

Pre

ssur

e, x

2, to

rr

Figure 3.8 Typical batch data for reaction(3.7). Data of Tamaru[1955].

A Comparison Of The Advantages And

Disadvantages Of Direct And Indirect Methods

Direct Method Advantages

• Get rate equation directly• Easy to fit data to a rate law• High confidence on final rate

equation

Disadvantages• Difficult experiment• Need many runs• Not suitable for very fast or

very slow reactions

Indirect MethodDisadvantages

• Must infer rate equation• Hard to analyze rate data• Low confidence on final

rate equation

Advantages• Easier experiment• Can do a few runs and

get important information• Suitable for all reactions

including very fast or very slow ones

10

Other Notation

Direct method• differential method• differential reactor

Indirect method• integral method

11

Initial Rate Method

Start with multiple parallel reactors Fill each with a different

concentration Let reaction go & measure

conversion vs time Get rate from slope extrapolated to

zero

12

Next: Start Analysis Of Data From Indirect Reactors:

Which is easier to analyze?• Direct method (rate vs concentration• Indirect method (concentration vs

time)

Direct is easier to analyze.

13

Analysis Of Data From A Differential Reactor

14

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].

Pitfalls Of Direct Measurements

• It is not uncommon for more than one rate equation may fit the measured kinetics within the experimental uncertainties.

• Just because data fits, does not mean rate equation is correct.

• The quality of kinetic data vary with the equipment used and the method of temperature measurement and control.

• Data taken on one apparatus is often not directly comparable to data taken on different apparatus.

15

Pitfalls Continued

• It is not uncommon to observe 10-30% variations in rate taken in the same apparatus on different days.

• Usually, these variations can be traced to variations in the temperature, pressure, or flow rate in the reactor.

• The procedure used to fit the data can have a major effect on the values of the parameters obtained in the data analysis.

• The quality of the regression coefficient (r2) does not tell you how well a model fits your data.

16

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.

17

]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

I prefer the latter, but I wanted to give a picture of the former.

18

Fitting Data Continued

There are two versions of the linear plots: Lineweaver-Burk Plots Eadie-Hofstee Plots   In the Lineweaver-Burk method, one plots

1/rate vs. 1/concentration. Rearranging equation (3.A.1) shows:

19

121p k1

]par[Kk1

r1

(3.A.2)

Fitting Data

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

20

Lineweaver-Burk Plots

Numerical results

From the least squares fit,

21

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?

22

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.

23

Eadie-Hofstee Plot

Avoid the difficulty at low concentrations by instead finding a way to linearize the data without calculating 1rp.

Rearranging equation (3.A.1):rp(1+K2[par])=k1K2[par]

(3.A.4)

Further rearrangement yields:

24

p221p rKKk

]par[

r

(3.A.5)

Eadie-Hofstee Plot

25

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!

26

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:

27

Data

2

2

21p ]par[K1

]par[KkrabsErrorTotal

(3.A.7)

Summary Of Fits

28

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

29

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 that there is no correlation between r2 and goodness of fit.

Pitfalls Of Direct Measurements

• It is not uncommon for more than one rate equation may fit the measured kinetics within the experimental uncertainties.

• Just because data fits, does not mean rate equation is correct.

• The quality of kinetic data vary with the equipment used and the method of temperature measurement and control.

• Data taken on one apparatus is often not directly comparable to data taken on different apparatus.

30

Summary

• Many methods to measure rates • Direct vs indirect• Conventional vs stopped flow vs

temperature jump

• Direct easier to fit• Caution about using r2

31

Class Question

What did you learn new today?

32