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Determining the Rate Law of a Reaction

Using the TI-83Plus Graphing Calculator

n-butyl chloride reacts with water as shown in the following reaction equations:

C4H9Cl + H2O→ C4H9OH + HCl

An experiment is done to determine the rate law of this reaction by monitoring theconcentration of n-butyl chloride (C4H9Cl) as a function of time. The data collected from

the experiment is collected in the table below:

Rate Data for the reaction of C4H9Cl with Water

Time, t (sec) [C4H9Cl]

0.0 0.100

50.0 0.0905

100.0 0.0820

150.0 0.0741

200.0 0.0671

300.0 0.0549

400.0 0.0448

500.0 0.0368

800.0 0.0200

1. Enter the data into the L1 and L2 data vectors in the calculator.

a. Press STAT, EDIT

 b. Use the Arrow keys below and to the right of the display screen to move around

the data matrix.

c. Enter the times into the L1 data vector. This data vector will become the X-axis

in all graphs generated in this exercise.d. Enter the concentrations into the L2 data vector. This data vector will become the

Y-axis in at least one of the graphs generated in this exercise.

e. Press 2nd

 Quit after all the data has been entered into the matrix and checked.

2. Set the calculator to graph the data in the L1 and L2 data vectors.

a. Press 2nd

 Stat Plot

 b. Plot1 should be turned On and Plot2 and Plot3 should be turned Off.

c. Plot1 should be set to plot L1 and L2. The settings for Plot2 and Plot3 can be left

as they are as long as those plots are turned off.

d. Use the Enter key to access the screen for Plot1.

e. Use the Arrow keys to move around the screen and place the cursory to define

any new settings.f. Use the Enter key to confirm that new setting.

g. Arrow up to the line with Plot1, Plot2, and Plot3 to move between the screens for

the individual plots.

h. Press 2nd

 Quit after all settings have been set as required.

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3. Set the range of data to plotted along the X and Y axes to match that the data in the

vectors. In this example: the minimum value along the X-axis is 0, the maximum

value along the X-axis is 800, the minimum value along the Y-axis is 0, and the

maximum value along the Y-axis is 0.10

a. Press Window b. Enter Xmin = 0

c. Enter Xmax = 2000

d. Leave Xscl = 1

e. Enter Ymin = 0

f. Enter Ymax = 0.1

g. Leave Yscl = 1

h. Leave Xres = 1

i. Press 2nd

 Quit after all settings have been set as required.

4. Display the graph (after a moment of “I hope this works!”).

a. Press Graph.

 b. Press 2nd

 Quit to exit the graph.

This particular plot of time (X-axis) versus concentration (Y-axis) is also the graphical

test data that reflects a zero order rate law. If this plot had been linear, which it clearly

isn’t, we would have concluded that this reaction followed zero order kinetics. What

would we have done if the plot looked linear? Is there a mathematical test for linearity?

The answer is obviously “yes”, and one method we can use is linear regression. Linear

regression is used to calculate the best  straight line that can be fitted to the data. The

algorithm available in the TI-83Plus returns the slope and intercept of that line, and the

correlation coefficient. The correlation coefficient is a measured of the goodness of fit  of

the calculated straight line to the experimental data. If all the data points fell perfectly on

the fitted line, the correlation coefficient would be 1.00, if the relationship between X andY is direct, or -1.00, if that relationship is inverse. As the data strays from the fitted line,

the correlation coefficient degrades toward zero. The closer the correlation coefficient is

to one or minus one the better the fit; the closer to zero the poorer the fit.

1. Make sure that the calculator is set to display all the available diagnostics Press 2nd

 

Catalog and scroll down the list with the Down Arrow key to DiagnosticOn. Press

the Enter key, then press the Enter key a second time. The calculator should respond

with the message “Done”.

2. Press the STAT key.

3. Select CALC by pressing the Right arrow key.

4. Scroll down the screen the selection number 4 (LinReg (ax+b)) and press the Enterkey.

5. You know must tell the calculator what data is to be fit to a straight line using linear

regression. The X-data is stored in the L1 data vector while the Y-data is stored in

the L2 data vector. Press the 2nd

 key and then the 1 key to specify L1. Remember the

X data is specified first! Then, press comma key. Press the 2nd

 key and then the 2

key to specify L2. Remember the Y data is specified second! Press the Enter key to

execute the calculation.

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The calculator should respond with numerical values for a, b, r 2 and r. As indicated on

the display, “a” is the slope of the best straight line fit to the data while “b” is the

intercept. The correlation coefficient is “r” while “r 2” is the probability that the fit is

“good”. The correlation coefficient is surprisingly high given the obvious curve in the

display. A value of 0.947 for “r 

2

” is actually low; we’re looking for values much closer.98-.99 to be confident that the data is really linear.

If the data had been linear, the slope (a) would be equal to the negative of the rate

constant (-k) and the intercept (b) would be equal to the initial concentration of the

species being monitored.

The graphical test for data that reflects a first order reaction is time on the X-axis versus

the logarithm of concentration on the Y-axis. If this plot is linear, we would conclude

that this reaction follows first order kinetics. The time data is already stored in the

calculator in the L1 data vector. We need to take logarithm of the concentration data

which is in L2, and store it in L3. With the main screen displayed press the LN button, press 2

nd L2, press the right parentheses), press STO, and press 2

nd L3. Now repeat the

steps described above to generate a graph of the data in L1 and L3, and repeat the linear

regression calculation. Be sure to use STAT EDIT to view the data and note the

minimum and maximum values for the Y-axis. You will also need to use STAT PLOT to

turn off Plot1, and turn on Plot2 using the data in L1 as the X-axis and L3 as the Y-axis.

You should infer from the results of linear regression that this data is indeed linear. The

value for “r 2” is essentially 1.0 (0.99998). That means this particular reaction follows

first order kinetics. The numerical value for the slope (a) is equal to the negative of the

rate constant (-k) and the intercept (b) is equal to the logarithm of the initial

concentration of the species being monitored. The value of the initial concentration can

 be calculated by taking the inverse ln of “b”.

Even though it is clear that this reaction does follow first order kinetics, it would be a

good idea to practice testing for second order kinetics data. The graphical test for data

that reflects a second order reaction is time on the X-axis versus the inverse of the

concentration (1/conc) on the Y-axis. If this plot is linear, we would conclude that this

reaction follows second order kinetics. The time data is already stored in the calculator

in the L1 data vector. We need to take inverse of the concentration data which is in L2,

and store it in L4. With the main screen displayed press 2nd

 L2, the press the X-1

, press

STO, and press 2nd

 L4. Now repeat the steps described above to generate a graph of the

data in L1 and L4, and repeat the linear regression calculation. Be sure to use STAT

EDIT to view the data and note the minimum and maximum values for the Y-axis. Youwill also need to use STAT PLOT to turn off Plot2, and turn on Plot3 using the data in L1

as the X-axis and L4 as the Y-axis. Not surprisingly, you should infer from the results of

linear regression that this data is indeed NOT linear. The value for “r 2” is again low

(0.941). That means this particular reaction does NOT follow second order kinetics. If

this data had been linear, then the numerical value for the slope (a) would have been

equal to the rate constant (k) and the intercept (b) would have bee equal to the inverse of

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the initial concentration of the species being monitored. The value of the initial

concentration could have been calculated by taking the inverse of “b”.

Note: You will need to be able to do these calculations for lecture quizzes and exams. If

you are having any questions, do not hesitate to ask your lecture instructor!