PURPOSE

The purpose of this laboratory is to familiarize you with the kinetics of an inorganic reaction and the instrumental and mathematical techniques used to study such reactions. In this experiment you will follow the acid aquation of a cobalt cation [tetraaminecarbonatocobalt(III)]⁺ using spectrophotometry. You will determine the order of the reaction by using both an initial rate study and by using a half life study. Finally you will quantitatively determine the rate constant by finding the best fit of the kinetic data to various integrated rate laws.

INTRODUCTION

The compound [Co(NH₃)₄CO₃]NO₃ dissociates in aqueous solution to yield the tetraaminecarbonatocobalt(III) cation, Co(NH₃)₄CO₃⁺, and the nitrate anion, NO₃^-. The structure of the complex cation is represented as:

Cobalt(III) forms kinetically stable coordination complexes by the bonding of Lewis bases (ligands) to a Lewis acid (the metal) to form six stable bonds. Ligands, like ammonia :NH₃, have a lone pair of electrons that can be donated to the metal to form a coordinate covalent bond. Ammonia is called a monodentate (one bite) ligand because it forms one bond with the metal. Since the carbonate is bound to the cobalt through two separate bonds it is called a bidentate ligand. Most any Lewis base including water can bond with the metal but Cobalt(III) complexes are so kinetically inert that an acid catalyst must be used to speed up the reaction.

In acidic solution Co(NH₃)₄CO₃⁺ undergoes a ligand substitution reaction where carbonate is replaced by water. The net reaction is:¹

The rate of this reaction is defined in terms of the change in concentration with time:

When the time interval Æt is small this average rate approachs the instantaneous rate of the reaction. The empirical rate law, which gives the concentration dependence of the instantaneous reaction rate, is of the form:

where:

k = rate constant (M^-(n+m-1)sec^-1)

n = the order of the reaction with respect to [H⁺]

m = the order of the reaction with respect to [Co(NH₃)₄CO₃⁺]

n+m = the overall order of the reaction

By carrying out the reaction at a constant [H⁺] (buffered solution), the reaction can be studied as a pseudo mth order reaction with a rate law of the form:

where: k_obs = k[H⁺]ⁿ

In a solution of a given pH, the concentration of Co(NH₃)₄CO₃⁺ will be monitored using a spectrophotometer. Measuring the change in concentration with time determines the rate of the reaction. The rate law will then be determined by measuring how the rate of the reaction depends on initial concentrations of reactants.

DATA ANALYSIS

Once the concentration as a function of time is measured there are three common methods used to analyze kinetic data: Initial rate studies where the rate is measured directly, progress curve studies where the concentration is measured as a function of time, and half life measurements where the time needed for half the reaction to be completed is measured. Initial rates and half lives are most useful for determining the order of the reaction and the rate law, while progress curves give the best quantitative measure of the rate constants.

The first method of analysis is usually an initial rate study. Initial rate studies are set up to measure the rate of the reaction quickly before the concentrations have changed by more than 1 percent. Under these conditions all concentrations are constant and essentially equal to the initial concentrations. Hence the name initial rates. By varying the initial concentration of reactants the rate law can be worked out using ratios:

If the ratio of rates is equal to ratio of initial concentrations, we know that the reaction is first order. If the ratio of rates is equal to the square of the ratio of initial concentrations, we know that the reaction is second order. We also can use the properties of logs to numerically determine the order of the reaction:

The order of a reaction should be a whole number or in some special cases it can be a whole multiple of one half. Finally, once the order of the reaction is known the rate constant can be calculated from the rate law.

The second method of analysis is to measure the concentration as a function of time over the whole course of the reaction. This method is called a progress curve analysis. The equation for a progress curve is obtained by integrating a differential rate law. For example, if the reaction was first order in cobalt:

or if the reaction was second order in cobalt:

The type of plot, which is linear, determines the order of the reaction. Linear regression of the line formed by plotting ln[Co]_t vs. t may be used to find the slope and determine a value for the first order rate constant k_obs. Linear regression of the line formed by plotting 1/[Co]_t vs. t may be used to find the slope and determine a value for a second order rate constant k_obs.

The final method of analysis is to measure the progress of the reaction by finding a half life for the reaction. The half life is the time needed to reduce the concentration of the reactant by a factor of two. Formulas for the half life may be determined from the integrated rate equations. For example, if the reaction is first order the time for half the reaction to be completed is:

or if the reaction is second order the time for half the reaction to be completed is:

The time for the concentration to decrease by half is inversely related to the rate constant. Further more, the time needed to go the next half life from Co/2 to Co/4 will determine if the reaction is first or second order. First order reactions have half lives that are independent of the initial concentration while second order half lives depend on the initial concentration. It will take twice as long to reach the second half life if the reaction is second order.

EXPERIMENTAL METHOD

The course of the reaction will be followed using a spectrophotometer, which quantitatively looks at the color of a solution by measuring the absorption of light. Reactions that are well suited to study by this method are those where only one reactant absorbs strongly at a particular wavelength and none of the other reactants or products absorbs appreciably at that wavelength. Since the absorbance, A, of a compound is proportional to its concentration, we can monitor the decrease in absorbance with time and relate this to the decrease in concentration with time:

where:

A = absorbance

e = molar absorptivity (M_-1cm_-1)

l = length of spectrophotometer cell (cm)

c = concentration (M).

The molar absorptivity is constant for a given compound at a given wavelength. The path length, l, will be constant for all the measurements that you will make. In this reaction only Co(NH₃)₄CO₃⁺ and Co(NH₃)₄(H₂O)₂⁺³ absorb light in the visible region of the spectrum. Co(NH₃)₄CO₃⁺ has an absorbance maximum at 525 nm and a molar absorptivity of approximately 100 M_-1cm_-1. Co(NH₃)₄(H₂O)₂⁺³ has a much lower molar absorptivity at this wavelength. For this reason the absorbance will decrease during the course of the reaction but because the both the products and the reactants absorb light the final solution will still be colored. In order to analyze the progress curve it is very important to have a good value for the absorbance (A_oo) at infinite time.

In order to maintain pseudo order reaction conditions, the concentration of the complex must be kept low relative to the buffer (acid) concentration. This requires that we measure very low absorbances on the Spectronic 401 spectrophotometers because the cobalt complex must be kept at very low concentrations.

Kinetic experiments generate lots of data. Some of which may be bad because of experimental errors. Ideally one should begin the analysis of the data before leaving the laboratory so that bad data sets can be found and corrected by repeating the measurements.

GENERAL PROCEDURE

1. Review the instructions in Appendix B for operation of the Spectronic 401.

2. You will be working in groups of three or four. You must turn in your own laboratory report, so make sure that you record all of the group data.

3. Each group must weigh cobalt complex into clean and dry cuvettes. Record all weights precisely. The amounts should be to the nearest ± 0.3 mg. 1 mg samples are needed for the progress curves and 3 and 6 mg samples are needed for the initial rate studies.

4. You will be running two reactions at each pH.

5. Make sure the wavelength of the Spectronic 401 is set to 525 nm. In both cases the concentration is auto-zeroed with a cuvett of distilled water in place. Be sure to always place cuvetts in the spectrophotometer so that the light passes through the clear part of the cuvett.

6. Each group will need a watch to record elapsed time in seconds.

INITIAL RATE STUDY

1. Weigh out 3 mg of cobalt complex into four clean and dry cuvetts.

2. Zero the spectrometer with a distilled water blank.

3. Add 3 mL of pH 3.0 buffer to the sample cuvett, cover the cuvett with parafilm, quickly shake the contents to mix, wipe the outside dry and place in the spectrophotometer. As soon as the reading stabilizes record the reading and the time. Exactly 60 seconds later record the second reading. The difference in these two readings will give the rate.

4. Repeat the run at pH 3.0. Then do two runs at pH 3.3.

5. Weigh out 6 mg (record the precise weights) of cobalt complex into four clean and dry cuvetts. Do two runs at pH 3.0 and two runs at pH 3.3.

6. Record the actual pH from the burette used to deliver the buffer, and use this pH for your calculations.

PROGRESS CURVE AT pH 3

1. Zero the spectrophotometer with a water blank.

2. To a 1 mg sample of cobalt complex in a cuvett add 3 mL of pH 3 buffer, cover with parafilm then quickly shake the contents to mix, remove the parafilm, wipe the outside dry and place in the spectrophotometer.

3. Record the absorbance which corresponds to time, t = 0. The initial absorbance should be at least 0.100 .

4. Start the second sample about 30 seconds after the first sample was started.

5. Record the absorbance and time at 30 second intervals alternating cuvetts (each individual sample will be at 60 second intervals) for 30 minutes. You will need at least two half lives of data, if 30 minutes is not enough extend your measurement time.

6. After 30 minutes place the cuvettes in safe place until the end of the laboratory period. You should also again measure the water blank and record the value to see how much the zero drifted during the course of your measurements.

7. At the end of the laboratory period (at least 1.5 hours, but no more than 2.5 hours), measure the absorbances of both cuvettes. These are A_oo values. A_oo is the absorbance due to the product and unreacted reagent, and will be used to correct the A_t's.

8. Record the actual pH from the burette used to deliver the buffer, and use this pH for your calculations.

PROGRESS CURVE AT pH 2

This pH is optional. Do this run if your pH 3 run failed for some reason. The reaction is much faster at pH 2 and you will be able to finish it quicker. Follow the same procedure as pH 3, but leave the cuvett in the spectrophotometer recording the absorbances at 10 second intervals for a total of 5 minutes. There will not be enough time to switch samples, so run the duplicate sequentially after the first run is done.

CALCULATIONS

INITIAL RATE STUDIES

Fill out the tabulated initial rate table in your report sheet. Calculate the concentration of cobalt complex from your weight measurements [Co]^* and from your initial absorbance [Co] = A/el . Use [Co] for the subsequent calculations but discuss any systematic differences between [Co]^* and [Co]. The hydrogen ion concentrations are calculated from the pH: [H⁺] = 10^-pH . The order of the reaction should be rounded to the nearest whole number.

1. Using the log ratios for experiments which have the same hydrogen ion concentration determine the order of the reaction (m) in terms of cobalt complex then determine kobs.

2. Using log ratios for experiments which have the same cobalt concentration determine the order of the reaction (n) in terms of hydrogen concentration.

3. Given the order of the reaction m and n calculate the rate constant (k) for each set of data.

4. Average your values for k and determine the precision by calculating a standard deviation of the average (the standard devaition of the average is equal to the standard deviation of the sample divided by the square root of N, where N is the number of data points used to form the average) s_av = s/.

HALF LIFE DETERMINATION OF m AND kobs

The half life is the time required for the reaction to be half completed. If A = 0, and we would only need to determine from your progress curve the time needed to give an absorbance half of the initial value but A is not zero. The absorbance at infinite time, A , is due to absorbance by product, by side-reaction products and primarily by unreacted starting material. Unreacted starting material occurs because our reaction actually reaches an equilibrium and does not go to completion.

k_forward

reactants products

k_reverse

The derivation of the integrated rate laws for reversible first or second-order reactions is beyond the scope of this course, but can be found in physical chemistry textbooks._{2 We can use the results of these derivations which allow us to set the quantity At-A proportional to the concentration of reactant at time, t. The half life then will be the time for the reaction to get half the way to completion, or to an absorbance of (Ao -A )/2 . The rate constant determined from progress curves of reversible reactions are actually the sum of the forward and reverse rate constants; in this case the reverse rate constant is much smaller than the forward rate constant.}

1. Fill out the tabulated initial rate table in your report sheet.

2. From the dependence of the half life on initial cobalt concentration determine the order of the reaction in terms of cobalt.

First order half lives are independent of initial concentration. For a first order reaction the time to go from [Co]_{o to [Co]o/2 is exactly the same as the second half life from [Co]o/2 to [Co]o/4 , while for a second order reaction the second half life is twice as long because of the inverse dependence on initial concentration.}

3. From the half life of the reaction determine k_obs.

PROGRESS CURVE DETERMINATION OF m AND kobs

1. Most of the following data analysis can be done on Excel. You can save considerable time if you enter your raw data (absorbances and times) and have the program do all of the calculations. Calculate the value of A_{t-A for each time. Use only values greater than .005 to .01 for the following analysis (you should have at least 10 good points). This is called truncating your data and is a common practice in this sort of analysis. The most reliable points are those collected when the absorbance At-A is large and changing most rapidly. When the absorbance At-A is small and no longer changing, there is a large relative uncertainty in the value of At-A .}

2. Calculate ln(A_{t-A ) for each time.}

3. Calculate 1/(A_{t-A ) for each time.}

4. Since the quantity (A_{t-A ) is directly proportional to concentration and if the reaction is first order in [Co(NH3)4CO3+] (m = 1), a plot of ln(At-A ) versus time should yield a straight line with slope = -kobs:}

5. Conversely a plot of 1/(A_{t-A ) versus time should yield a straight line with slope of kobs if the reaction is second order in [Co(NH3)4CO3+] (m = 2).}

6. Use a linear regression program on your plots of ln(A_{t-A ) versus time and 1/(At-A ) versus time to determine kobs and constant as well as the uncertainties in these fit parameters.}

7. The linear regression program will report the closeness of the fit of the data to a straight line by calculating a correlation coefficient "R2". The closer this value is to 1 or 100%, the better the fit. Look at the "R2" values from your linear regression equation and decide whether a first or a second order rate law is a better fit. Also look at the plots of the residuals (experimental minus fit values), if the fit is good they should be randomly distributed about zero and not trended. Note: a bad A can cause considerable curvature in your plots even when they should be linear.

8. Linear regression is a powerful analysis tool but often one that is miss used by fitting straight lines to data that is in reality curved. The correlation coefficient is markedly insensitive to small amounts of curvature. A better check is to look for consistancy between the fit results and the experimental data. We can check the self consistency of our first or second order model by using the fitted functions to evaluate A :

First order: (A_{t-A ) = e or A = At - e}

Second order: (A_{t-A ) = 1/(kobs t + const) or A = At - 1/(kobs t + const)}

Evaluate A from each time point and average these values to determine A for your fit. Compare this value to your experimentally determined A value.