Kinetics of the mutarotation reaction of glucose

Introduction

Optical isomers

Most simple chemical compounds can be superimposed on their mirror image. For example, Figure 1 gives a molecule of 2-propanol and the mirror image of the molecule. The mirror image can be superimposed on the original molecule and is therefore not distinguishable, but is in fact the same molecule.

Figure 1. 2-propanol and its mirror image.

There is, however, a class of molecules that cannot be superimposed on their mirror image. A simple example of such a molecule is 2-butanol, shown in Figure 2. The two molecules of 2-butanol in the figure are mirror images of one another, just as was the case in Figure 1. However, unlike the mirror images in Figure 1, the 2-butanol molecule and its mirror image cannot be superimposed on one another, and therefore represent distinct molecules. Molecules that cannot be superimposed on their mirror image are called optically active molecules, for reasons discussed below, with the non-superimposable forms of the molecules called optical isomers.

Figure 2. 2-butanol and its mirror image.

Additional information concerning optically active molecules and optical isomers can be found in reference [1] or most other standard organic chemistry texts.

Rotation of light

The physical property that distinguishes an optically active molecule from an optically inactive molecule is the way in which the molecule interacts with polarized light. Polarized light is light where the electromagnetic oscillations have been oriented in a particular plane perpendicular to the direction in which the light is moving. Polarized light can be produced by reflection of light from a surface or by passing light through a crystal or oriented polymeric film. When polarized light passes through a solution containing only optically inactive molecules, no change in the plane of polarization of the light is observed. However, if an optically active molecule is present in the solution, the plane of polarization of the light will be rotated either to the right (positive, D, or dextrorotatory) or to the left (negative, L, or levorotatory). The angle of rotation of the plane of polarization of the light is given the symbol a. Some of the properties of polarized light and its interaction with optically active molecules are discussed in reference [2].

The amount of rotation that a plane polarized beam of light undergoes when passing through a solution containing optically active molecules depends on many factors, including the concentration of optically active molecules in solution, the wavelength of the polarized light, the pathlength, the temperature, and the solvent. Because of this, a reference condition is needed for reporting optical activity. The specific rotation of an optically active compound, given the symbol [a]D

T, is defined as the observed rotation for a 1 g/mL concentration of the pure optically active compound in a 10 cm polarimeter cell, at a fixed temperature T, using polarized light from the D line of a sodium emission lamp (actually a closely spaced doublet line occurring at 589.3 nm, corresponding to the 2P ® 2S electronic transition). The relationship between aobsd, the rotation observed for a particular concentration and pathlength, and [a]D

T, the specific rotation of the compound is

[a]DT = 10(aobsd)/c (1)

where c is the concentration of the optically active compound (in g/mL) and is the pathlength of the polarimeter cell (in cm).

The mutarotation reaction

Glucose (dextrose) is a monosaccharide, a sugar with the chemical formula C6H12O6. In water, glucose exists in two cyclic forms (which themselves have non-superimposable forms that are their optical isomers), called a-D (+) glucopyranose and -D (+) glucopyranose, as shown in Figure 3. These two forms of glucose are not mirror images of one another because there is more than one optically active site in the molecule. As it happens, both of these molecules are dextrorotatory.

Figure 3. Several representations of the a and isomers of D-glucose.

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The mutarotation reaction occurs by protonation of the ring oxygen atom followed by formation of the linear, or aldehydic, form of the sugar, as indicated in Figure 4. When the cyclic form of the sugar is regenerated either the a or the configuration can occur. Thus one cyclic form of glucose may be converted to the other cyclic form of glucose. Whether we begin with the a or the form of glucose, the aldehydic intermediate is the same. The mutarotation reaction thus generates an equilibrium concentration of a and forms from any initial nonequilibrium concentration of starting material.

Figure 4. The mutarotation reaction of D-glucose. The specific rotations for the two isomersare [a]D

25 = + 112.0 for a-D glucose and [a]D25 = + 18.7 for -D glucose.

Kinetics [3]

The mutarotation reaction is a kinetic system with the form

A « B (2)

whereA = a-D (+) glucose = a-D (+) glucopyranoseB = -D (+) glucose = -D (+) glucopyranosek1 = rate constant for conversion of A into Bk-1 = rate constant for conversion of B into A

The rate law in differential form is then

d[A]/dt = -k1 [A] + k-1 [B] (3)

where d[A]/dt is the rate of change in the concentration of A with time. Note that at equilibrium (with equilibrium quantities labeled eq)

(d[A]/dt)eq = 0 = -k1 [A]eq + k-1 [B]eq (4)

or

[B]eq/[A]eq = Keq = k1/k-1 (5)

which gives a simple relationship between the equilibrium constant for the reaction and the rate constants for the reaction.

Returning to equation (3), we may integrate to obtain the time dependence of the concentration of A. The result [4] is

([A]t - [A]eq)/([A]0 - [A]eq) = exp[ - (k1 + k-1) t] (6)

or

ln{([A]t - [A]eq)/([A]0 - [A]eq)} = - (k1 + k-1)t = - kt (7)

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where we have defined [A]0 as the concentration of A at some convenient starting time t = 0 (which, because the reaction is first order, is arbitrary), [A]eq as the concentration of A at equilibrium, and k as the sum of the forward and reverse rate constants for the reaction (that is, k = k1 + k-1). Note that the rate at which the system approaches equilibrium is determined by the sum of k1 and k-1.

The optical rotation of a solution containing a-D (+) glucose and -D (+) glucose is directly proportional to the concentration of the a-isomer for a fixed path length and total glucose concentration. Equation (7) can therefore be rewritten as

ln{(a0 -aeq)/(at -aeq)} = kt (8)

where at is the observed rotation of the solution at time t, a0 is the observed rotation at t = 0, and aeq is the rotation observed when the system has reached equilibrium. Notice that we do not need to convert the observed rotation of the solution to a specific rotation, nor do we need to know the constant of proportionality that relates at to [A]t.

A plot of the left hand side of equation (8) versus time has a slope equal to k. The equilibrium constant for the mutarotation reaction can be found by measuring the optical rotation of a solution that has achieved equilibrium, given the total concentration of glucose in the polarimeter cell, the path length of the cell, and the specific rotation of each glucose isomer. From k and Keq, the values of k1 and k-1 can be found.

Experimental

A stock solution of perchloric acid (PCA, HClO4) is prepared by carefully diluting 2.0 mL of 70% PCA to a final volume of 100.0 ml with water. This results in a solution that is 0.232 M in PCA. From this stock solution, prepare four additional solutions as follows:

Solution PCA stock solution Water(mL) (mL)

1 5.0 45.0

2 10.0 40.0

3 15.0 35.0

4 20.0 30.0

The hydrogen ion concentration should be calculated for each solution. Remember that PCA is a strong acid, so [H+] = [PCA].

The procedure for carrying out a kinetic run is as follows. Place the 50.0 mL diluted PCA solution and a magnetic stir bar to a 100 mL beaker. Add approximately 5.0 g of glucose to the beaker while stirring the solution vigorously. Be sure to record the exact mass of glucose used as you will need this information in calculating the equilibrium constant for the mutarotation reaction. When the glucose has completely dissolved, fill a 10 cm polarimeter cell with solution so that there is at most only a small air bubble inside the cell (you might want to practice filling the polarimeter cell with distilled water before attempting a kinetic run). Place the polarimeter tube inside the polarimeter, and measure the rotation due to the solution. This initial value of rotation is a0. Continue to measure the rotation of the solution at time intervals over which the rotation changes by about 0.2 , until the initial rotation (which will be about 10) decreases to about 6. Label and save the excess solution. A final measurement of the observed rotation should be made after sufficient time has elapsed so that equilibrium between the two forms of glucose has occurred. The value of the rotation measured at that time is aeq.

On each day that experimental data are taken you should also measure the observed rotation when there is no sample in the pathway. We will call this value for rotation anull. The rotation that is observed should in principle

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be equal to 0.0 . However, it will be slightly different from zero due to drifting in the calibration of the polarimeter. All of your experimental values for rotation should be corrected for this small drift in calibration by subtracting the value of anull from each data point. This must be done before any additional data analysis is carried out, and using the value of anull found on the data that the data were obtained.

You should also record the temperature of the room for each day that data is taken. While the temperature will not be used in your calculations, differences in temperature on different days can affect the rate constant (and, to a lesser extent, the equilibrium constant) for the reaction.

Lab report

Your laboratory report should include the following:

1) The experimental values of k and Keq for each [H+] at which data are taken. You have to derive an equation for finding Keq based on the information you have available to you.

2) The experimental values of k1 and k-1, which can be calculated once k and Keq are known. The details of the calculation of k1 and k-1 from k and Keq should be given. Use your average value for Keq in the calcumtions of k1 and k-1.

3) A discussion of the effect of hydrogen ion concentration on the value of k. How does k depend on [H +], or pH (a plot of k vs [H+] might be useful here)? If possible give a quantitative relationship between hydrogen ion concentration and k. Can you suggest any additional experimental work that could be carried out to test your proposed relationship?

You do not need to find literature values for k or Keq.

References

1. T. W. G. Solomons, Organic Chemistry, Sixth Edition, (Wiley, New York, 1996) pp. 193-199.

2. C. W. Garland, J. W. Nibler, D .P. Shoemaker, Experiments in Physical Chemistry, Seventh Edition, (McGraw-Hill, Philadelphia, 2003), pp. 667-668.

3. A general discussion of first order reversible reactions is given in P. W. Atkins, J. de D .P. Shoemaker, aula, Physical Chemistry, Seventh Edition, (W. H. Freeman Company, New York, 2002), pp. 876-877.

4. S. W. Benson, Foundations of Chemical Kinetics, (McGraw-Hill, New York, 1960), pp. 27-29.

Acknowledgements: Figures 1-4 are taken from T. W. G. Solomons, Organic Chemistry, Sixth Edition, (Wiley, New York, 1996).____________________Revised 1/13

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