Kinetics of a Diffusion-Controlled Reaction

Author: Katrina WunderlichPartner: Matthew Kita

12/17/10

Introduction

The purpose of this lab is to determine the rate of diffusion, kdiff, by analyzing the kinetics of a

quenching reaction with excited anthracene and a quenching agent, CBr4. In the calculations of this experiment

it is assumed that there are two reactants A and B that are electrically neutral in random motion in solution.

There is a probability that A and B will come close enough to each other that the radius between them is

roughly equal to the sum of their molecular radii. This arrangement is known as an encounter complex. Unless

acted upon by a chemical reaction they will separate. This separation is inhibited by the neighboring solvent

molecules. This is called the cage effect. Using the chemical equations for these processes and the steady-state

approximation the second-order rate coefficient kobs can be written as

(1)

where k1 is the rate constant of the formation of the encounter complex, k-1 is the rate constant for the

dissociation of the encounter complex and k2 is the rate constant for the unimolecular reaction between A and B

in complex to form products.

If k2<<k-1 the reaction is under chemical control and the reaction of A and B is very slow compared to

the separation from the solvent cage. If k2>>k-1 the reaction is diffusion controlled and the A-B reaction occurs

much faster than the separation from the solvent cage in which case the rate constant can be called kdiff and

calculated using the Stokes-Einstein-Smoluchiowski (SES) Equation,

(2)

where ŋ is the viscosity in Nsdm-2, T is temperature in K, and R is the gas constant 8.314 J/molK.

The reaction between anthracene and CBr4 takes place only when exposed to light and is

intrinsically fast. The CBr4 “extinguishes” the fluorescence of the anthracene. Assuming a diffusion controlled

reaction and using the steady state Stern-Volmer relation where [Q] is the concentration of CBr4, is the

slope of the Stern-Volmer plot, KSV is the Stern-Volmer constant (dm3mol-1), and is the ratio of the

background intensity to the instrument response

(3)

.

kq can be compared with kdiff. The Stern-Volmer plot should be linear following the assumption that there is no

“static” quenching. However, at high concentrations static quenching becomes more significant and deviations

from the linear nature of the S-V plot can be expected. The effects of static quenching can be calculated using

(4)

where is compared to that from Equation 3, [Q] is the concentration of CBr4, is calculated from the slope

of the S-V plot, NA is Avagadro’s number and Y is defined by

(5)

where

(6)

and

(7)

where for Equation 5, 6, and 7, [Q] is in mol dm-3, R 6.7x10-10 m, D=4.5x10-9 m2s-1 and =5.52x10-9 s.

Experimental Procedure

The experiment was completed as written1 with the specifications and modifications that follow. The

stock solutions of anthracene and CBr4 were created by the instructor with the respective concentrations

1.00x10-4 M and 1.50x10-2 M. Ten dilutions were prepared using micropipettes and 10 mL volumetric flasks of

the following concentrations by volume: 0%, 5%, 15%, 25%, 40%, 50%, 65%, 75%, 90%, and

100%. All scans were taken at room temperature. The slit used for all scans was 2.5 nm. One scan of each

sample was taken with the exception of 0% emission of which three scans were taken. Scans were taken from

383 to 450 nm. In between each scan the cuvette used was emptied of the previous sample, then rinsed with

hexane, then rinsed with a small portion of the sample that was to be scanned, the filled with the sample to be

scanned. The emission scans were taken using an excitation wavelength of λ=372.5 nm which was determined

from a sample of 0% CBr4 solution.

Results

An excitation scan was taken of 0% AN/CBr4 was taken in order to determine the maximum intensity

wavelength. This wavelength was found to be 372.5 nm. From this scan the appropriate wavelength range over

which to observe the emission spectra was also determined. The excitation spectrum can be viewed below in

Figure 1.

Figure 1. Excitation Spectrum of 0% AN/CBr4. Maximum intensity wavelength occurs at 372.5 nm.

Using the maximum intensity excitation wavelength of 372.5 emission spectra for the remaining

dilutions were taken over the range of 383 to 450 nm. A compilation of the emission spectra for all dilutions

can be found below in Figure 2.

Figure 2. Emission Spectra for all tested concentrations of AN/CBr4 solutions. Wavelength range: 383-

450 nm.

The maximum wavelength for all of the above spectra was found to be 398 nm. These spectra represent

the raw spectra data. Since the AN/CBr4 solutions were created in n-hexanes a background scan of n-hexane

was taken. The background n-hexane scan was subtracted from the above emission spectra to determine the

corrected intensities. The n-hexane scan and corrected data can be found in Appendix A: Figures 4 and 5.

From these corrected intensities along with the concentrations of CBr4, a Stern-Volmer plot was created,

Figure 3 below. The three emission spectra for 0% CBr4 solution were averaged together. Since there is no

quencher in the 0% solution I0=If. The average was found to be 511.891±4.511, where the error is the standard

deviation associated with the measurements. The Stern-Volmer plot was created by taking the ration of I0/If of

the remaining solutions and subtracting one. The error bars appearing on the Stern-Volmer plot were

determined using the standard deviation associated with the average 0% CBr4 solution and represent the relative

error associated with all intensity measurements. Also appearing in Figure 3 is a residual plot. The residual

data was found using the slope of the regression equation and the concentrations of CBr4 solutions. The

corrected I0/If data was subtracted from the y-values of the regression equation. Since the residual points are

centered on the x-axis there were no obvious error trends in the data.

Figure 3. Stern Volmer Plot with Residual Plot. The y intercept was set to 0 and the regression equation

was found to be -117.99x. Error bar value: 0.0088.

Using Equation 3 it was shown that KSV, the Stern-Volmer constant, equaled the slope of the regression

equation. Using KSV, the slope, and Equation 3 kq was calculated to be 2.14x1010 1/M/s. kdiff was then

calculated using Equation 2, the known values of R, T and ŋ and found to be 2.22x1013 dm3mol-1s-1.

The significance of static quenching involved in this experiment was analyzed. The value of b was

calculated from Equation 6 and found to be 1,928,513 part/s1/2. The value of a was calculated using the value of

b and Equation 7 and found to be 3.42x1011part/s. The value of Y was calculated to be 0.4588 from Equation 5.

I0f/If of static quenching was calculated using these values and Equation 4 and found to be 4119.514. The

percent error between this I0f/If -1 and the value calculated from the data gathered in the lab is 99.96% which

implies that static or non-diffusional quenching played a very minimal part in this experiment.

Discussion

According to the literature by Ware and Novros2, KSV lit was found to be 140 M-1. This value was found

under the similar conditions that this experiment was conducted: 25 °C, with heptanes as the solvent. Following

the same methodology used in this experiment kq lit can be calculated by dividing KSV lit by τ0, the same as used in

this experiment. This results in a value of 2.536x1010 M-1s-1 for kq lit. The kq value found experimentally was

2.14 x1010 M-1s-1. As discussed below, kq can be considered equivalent to kdiff. The percent error between the

literature value of kdiff and the experiment value of kdiff is 15.72%.

As shown by the large percent error of 99.96% when calculating I0f/If -1 through static quenching and the

value of I0f/If -1 found experimentally, it is a sound conclusion that static quenching played a very small role in

this experiment. If in the denominator must be very small to create such a large number in the calculated static

I0f/If which would show that the anthracene is not thoroughly quenched through static quenching. Since our

experimental value was much smaller it means that If was not as small. This implies that the anthracene was not

thoroughly quenched through static quenching but was proportionally more quenched via diffusional

quenching. The relative error associated with the measurements, as determined by the standard deviation of the

0% emission scans is 0.0088.

In order to perform an experiment to determine whether a stable photoproduct, that is a long lasting

stable encounter complex, is formed as a result of flouresnce quenching the cage effect must be maximized. If

the solvent molecules are not preventing the encounter complex from disassociating into its individual parts,

then there is no encounter complex to monitor. If this stable encounter complex is formed, the fluorescent

reactant will not be able to fluoresce as it is loosely bound in the complex. However, if the encounter complex

were able to disassociate the fluorescent reactant would be free to fluoresce since they would not be bound to

the quencher. The emission spectra of this solution could be compared to the emission spectra of a species that

absorbs between the wavelengths 383-450 nm. If a stable long-lasting photoproduct was formed there should

not be a peak for excited anthracene. Low concentration of the quencher should be used in order to avoid the

formation of a ground state complex. Ideally, the ratio between anthracene and CBr4 should be one to one. If

the 1:1 solution were mixed with the other solution, and a true stable long-lasting photoproduct was formed, the

emission spectra should stay the same because the quencher would still be interacting with the anthracene in an

encounter complex.

To fully evaluate and verify that if Y=1, kq kdiff Equation 4 and Equation 8 below must be analyzed.

Equation 8 defines kdiff using the rA and rB, the molecular radii of the reactants, and DA and DB the diffusion

coefficients of the species in the solvent and appears as follows

(8)

where NA is Avagadro’s number, and the units of the diffusion coefficients are dm2s-1. If Y=1 Equation 4

becomes

(9)

By assuming that can be reduced to RD, Equation 8 becomes

Substituting this value into Equation 9 and rearranging the terms it can be seen that

(10)

Comparing Equation 10 to Equation 3 we see that kq kdiff k1.

For a reaction to be diffusion controlled the rate at which the encounter complex chemically bonds

together to form a new product is significantly faster than the rate at which the encounter complex disassociates

into its elemental parts. Since both anthracene and CBr4 are non polar molecules moving in constant random

motion. The solvent, n-hexane is non polar as well, the solvent prevents the encounter complex from

disassociated back its individual components. This is the cage effect discussed in the introduction. As a result

the rate the reaction is roughly equal to the rate at which the encounter complex forms which is comparable to

kdiff. As shown be the results of the experiment kdiff/k1 and kq are similar. This is the case in a diffusion

controlled reaction. Other support is provided by Equation 11

(11)

where kr is the rate constant of the fluorescence of excited anthracene, knr is the rate constant of nonradioactive

decay, and kq[Q] is the reaction rate of quenching. In order to obtain values of I0f/If that were obtained in this

experiment kq[Q] had to be largely more significant than the other rate constants. This is partially due to the

greater concentration of Q than that of excited anthracene. The greater concentration of CBr4 increases the

likelihood that CBr4 will collide with the excited anthracene before it can fluoresce. This reaction is fast because

it has essentially no activation energy, its rate is only limited by the rate of encounter between the excited

anthracene and the quencher CBr4.

Using higher concentrations of CBr4, the quencher, could be used to assess the presence of a ground

state complex. Increasing the concentration of the quencher will increase the likelihood of a collision between

the CBr4 and the unexcited anthracene. In this case there would be a higher ratio of ground state complexes to

excited anthracene which causes fluorescence. This experiment showed that anthracene mixed with low

concentrations of CBr4 fluoresces at a wavelength of 372.5 nm. If the concentration of quencher is increased, in

a different experiment, it will inhibit the absorbance happening at 372.5 nm and there will be a significantly

higher concentration at another wavelength. This absorbance at a different wavelength provides evidence of a

ground-state complex. The rate of this new reaction could not be determined using the exact methodology that

was used in this experiment; high concentrations of quencher are hypothetically used and thus static quenching

would be significant. The equilibrium constant of this reaction is equivalent to the quenching rate at high

concentrations, similar to how the equilibrium constant was determined in this lab using the quenching rate at

low concentrations. Using Equations 4,5,6 and 7 we can calculate I0f/If of a reaction including high

concentrations of the quencher and from there the rate constant could be determined. The Beer-Lambert law

could also be used to calculate the concentration of the quencher and from there the rate constant could be

determined.

If a polar solvent such as acetonitrile were used instead of a non polar solvent like n-hexane, the

reaction would proceed very differently. The polar solvent would inhibit the formation of a ground-state

complex by preventing the collision between anthracene and CBr4. This would accordingly favor the

fluorescence of anthracene; since anthracene is no longer forming the ground-state complex it can be converted

to its excited state and be more likely to fluoresce.

To best design an experiment that maximizes the transient effect in a diffusional process, it would be

ideal to use a fluorescence probe with a longer fluorescence lifetime. The longer the fluorescence lifetime the

longer the period it takes for the excited anthracene (or other chemical under investigation) to decay. This

would maximize the time dependence of the diffusional process. Another way to maximize the diffusion

process is to use a solvent with high viscosity. As viscosity is increased, kdiff decreases as per Equation 2. As

kdiff decreases τ0, the probe’s fluorescence lifetime increases. As seen above, increasing the fluorescence

lifetime maximizes the transient effect in a diffusional process.

References

1 Bunagan, Michelle. Kinetics of a Diffusion –Controlled Reaction. (2010).

2 Ware, William R. Novros, Joel S. Kinetics of Diffusion-Controlled Reactions. An Experimental Test of the

Theory as Applied to Fluorescence Quenching. The Journal of Physical Chemistry. 70:10 (1966). 3250.

Appendix A: Raw Data

Figure 4. Background Spectrum of n-hexane

Figure 5. Fluorescence Intensities of AN/CBr4

solutionsEmission using Excitation λ=372.5 nm

% CBr4λmax emission

(nm)Intensity,

If (Raw)

Intensity, If (corrected)

n-hexane(background) 398.00 9.286  0% 398.00 526.382 517.0960% 398.00 518.408 509.1220% 398.00 518.741 509.455

0% avg 398.00 521.177 511.8915% 398.00 477.582 468.29615% 398.00 407.975 398.68925% 398.00 357.456 348.17040% 398.00 350.320 341.03450% 398.00 278.076 268.79065% 398.00 229.599 220.31375% 398.00 217.624 208.33890% 398.00 214.874 205.588100% 398.00 197.781 188.495

Figure 6. Stern-Volmer Plot DataEmission using Excitation λ=372.5 nm

% CBr4 C(CBr4) Mλmax

emission (nm)

If (raw) If I0f/If -1I0f/If -1

Fit Residual

0 (avg) 0.00000 398 521.177 511.891 0 0 0

5.0 0.00075 398 477.582 468.296 0.0931 0.0885 -0.004615 0.00225 398 407.975 398.689 0.2839 0.2655 -0.018525 0.00375 398 357.456 348.170 0.4702 0.4425 -0.027840 0.00600 398 350.320 341.034 0.5010 0.7079 0.206950 0.00750 398 278.076 268.790 0.9044 0.8849 -0.019565 0.00975 398 229.599 220.313 1.3235 1.1504 -0.173175 0.01125 398 217.624 208.338 1.4570 1.3274 -0.129690 0.01350 398 214.874 205.588 1.4899 1.5929 0.1030100 0.01500 398 197.781 188.495 1.7157 1.7699 0.0542

Figure 7. Results

Value UnitsKSV 117.99 L/molKq 2.14E+10 1/M/sKdiff 2.22E+13 dm3mol-1s-1

b 1928513 part/s.5

a 3.42E+11 part/sY 0.4588  

I0f/If 4119.514  I0f/If-1 4118.514  

Figure 8. Error Analysis% Error Static vs Stern Volmer 99.96%

If avg 511.8910

SD 4.5107

SD/AVG 0.0088

Appendix B: Sample Calculations

Average of 0% CBr4 Emission Intensities (Iof):

Corrected Fluorescence Intensities:

for 5% CBr4 Emission:

for 5% CBr4 Emission from Regression (Fit):

Residual :

kq (empirical):

kdiff (from SES Equation, Equation 2):

Calculating b from Equation 6:

Calculating a from Equation 7:

Calculating Y from Equation 5:

=0.4588

Static Quenching from Equation 4:

Static Quenching :

Appendix C: Error Propogation

Percent Error of using Static and Stern-Volmer Data for 100% CBr4:

Standard Deviation of 0% CBr4 Emission Intensities:

Error Bars for Stern-Volmer Plot (Relative Error):