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Determining Rate Constants
First we will examine how to determine the rate constant of the reaction from last week's data. For pseudo zeroth order conditions, with large and constant concentrations of H+ and cyclohexanone, the rate law,

Rate = -d([I2]+[I3-])/dt = k [H+] [cyclohexanone] = constant

results in a constant reaction rate and a linear change in iodine plus triiodide concentration with time. Thus, plots of absorbance versus time for each run should give a straight line. The reaction rate is related to the slope of this line, ΔA/Δt, since Beer's Law relates the change in concentration to the change in absorbance.

A plot of sample data from run v is shown in Figure 1, below.

Figure 1. Data for run v

The slope of the line,
ΔA/Δt, can be determined by subtracting the values of absorbance at t = and t = 0, where t = is the end of the reaction, when all I2 + I3- has been consumed, and t = 0 is the start of the reaction. Because we want the rate in units of ΔC/Δt, we will determine the change in concentration over the course of the reaction directly, by subtracting the initial concentration from the final concentration of I2 + I3-, which is zero. The plot must be used to determine the value of t = , either by observing the absorbance reach zero, when all the I2 + I3- has been used up, or by extrapolation. A convenient and accurate way to extrapolate to t = is to use the Linear Least Squares applet available on the ChemLab website.

To determine the rate of reaction, and from that the rate constant, you will use an expression which relates the reaction rate to the concentration and time changes:

The initial concentration of I2 + I3- for run n is labeled Cn , shown as Cv in Figure 1. The rate of reaction can be calculated as Cn/tn, where tn is the t = value for run n, shown as tv in Figure 1. The initial I2 + I3- concentration Cn and absorbance An (init) should be the same for each run, provided your pipetting is accurate and reproducible. Cn is the "value of ([I2] + [I3-])init for run n". To calculate this, multiply the dilution factor for your reaction mixtures (2.0 mL/25.0 mL) by the stock iodine concentration (approximately 0.04
M).

Example: The data of Figure 1 were obtained from run (v). Assume [iodine stock] = 0.0364
M. An aliquot of 2.00 mL of the iodine was placed in a reaction solution with a final volume of 25.00 mL. Therefore, the initial concentration of iodine in run v is 0.00291 M.

The time axis intercept is found at 11.2 minutes and the rate is readily calculated. Rates and rate constants are usually reported with time measured in seconds, so we include a conversion from minutes to seconds:

Rate = 0.00291 M/(60 s/min)(11.2 min)= 0.00000433 M/s

Recall that the experimental rate law you observed last week for the halogenation of ketones has the form

For our pseudo zeroth order conditions, this reaction rate is a constant, since the concentrations of cyclohexanone and H+ are constant. Thus, the rate of reaction is equal to k', the effective rate constant. After we determine the value of k' from the changes in concentration and time for each run, we can determine the value of k, knowing the concentrations of cyclohexanone and H+. Note that the units of k' and k are not the same and units must be taken into account when calculating one rate constant from the other. The value of the overall rate constant k is related to the rate constants and equilibrium constants appearing in mechanism steps (1)-(4). The validation of this rate law can be checked by calculating k for each of the runs and determining how closely it remains constant.

Example: The same run (v) discussed above will be used to illustrate the calculation of k. First, it is necessary to obtain values for [cyclohexanone] and [H+]. Since these values change slightly during the reaction, they should be calculated for both t = 0 and t = .

t = 0:

t = :   Recall the net reaction stoichiometry:

For every mole of I2 (or I3-) that is consumed, one mole of cyclohexanone disappears, and one mole of hydrogen ions is formed. For the present example, ([I2] +[ I3-])init is 2.9 x 10-3
M, and all of this iodine is consumed. The stoichiometry tells us that this same amount of cyclohexanone will be consumed and the same amount of H+ will be produced in the reaction.

[cyclohexanone]t = = 9.2 x 10-2 M - 2.9 x 10-3 M = 8.9 x 10-2 M

[H+]t = = 0.14 M + 2.9 x 10-3 M = 0.14 M

These results show that the assumption that the cyclohexanone and hydrogen ion concentrations remain constant throughout the run is good to within 3% or better. For the concentration of H+, accurate to within ±0.01 M, the concentration difference is negligible. For the calculation of k, average values will be used; this refinement borders on being unnecessary, however.

[cyclohexanone]ave = (9.2 x 10-2 M + 8.9 x 10-2 M )/2 = 9.05 x 10-2 M

[H+]ave = (0.140 M + 0.143 M )/2 = 0.1415 M

To calculate the rate constant, we rearrage the rate law to obtain:

The Temperature Dependence of Reaction Rate Constants

The temperature dependence of rate constants is described by the Arrhenius equation,

Here k is the rate constant, T is the absolute temperature, Ea is the activation energy of the reaction, and A is a constant called the pre-exponential factor. We can take the natural log of both sides of this equation to obtain

In this form, the Arrhenius equation shows a linear relationship between ln k and 1/T. The activation energy can be calculated from rate constants at different temperatures by plotting ln k versus 1/T. The slope of this plot will be -Ea/R, and the y intercept will be ln A. Note that the temperature must be in units of Kelvin and the gas constant R must be in units of J mol-1 K-1 for this relationship to provide an activation energy in the units of J/mol.

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