ChE 551 Lecture 03 Analysis of Rate Data 1

General Approach Initiate reaction measure concentration vs time fit data to calculate rates 2

Rate Measurements: An Old Topic 3 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 Figure 3.6 A typical arsine decomposition reactor.

Direct Measurement 8 Figure 3.7 A possible apparatus to examine the decomposition of arsine (AsH 3 ) on silicon.

Indirect Measurement 9 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 Method Disadvantages 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 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 (r 2 ) 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 k 1 and K 2 are constants. 17 (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 (3.A.2) Fitting Data

20 Lineweaver-Burk Plots

Numerical results From the least squares fit, 21 (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? 22 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. 23

Eadie-Hofstee Plot Avoid the difficulty at low concentrations by instead finding a way to linearize the data without calculating 1r p. Rearranging equation (3.A.1): r p (1+K 2 [par])=k 1 K 2 [par] (3.A.4) Further rearrangement yields: 24 (3.A.5)

Eadie-Hofstee Plot 25 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! 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 (3.A.7)

Summary Of Fits 28 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 Note that there is no correlation between r 2 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 r 2 31

Class Question What did you learn new today? 32