Spectrophotometric Determination of an Equilibrium Constant^1,2

Authors: B. K. Kramer, B. D. Lamp, D. L. McCurdy* and J. M. McCormick

Last Update: December 29, 2006

Introduction

Typically acid-base indicators are themselves weak acids or bases whose acid and base forms have different colors in solution. As the result of the reaction with excess titrant, we convert one form to the other causing a color change that indicated the endpoint of a titration.  If we represent the indicator's acid form as HIn and its basic form as In^-, then the following equilibrium describes the chemical reaction that occurs as the [H⁺] is changed.  If HIn and In^- have different colors, then the solution's color will change as a function of [H⁺] depending on which of the compounds is present in the greater amount.

The acid dissociation equilibrium constant (K_a) for the indicator that describes this reaction is given by Eqn. 1, in terms of the concentrations of the hydrogen ion, In^- and HIn. Because we are working in aqueous solution, it is convenient to rearrange Eqn. 1 to Eqn. 2 by taking the negative logarithm of both sides, and then recognizing the definitions of pK_a and pH, rewriting Eqn. 2 as Eqn. 3, which is simply another version of the Henderson-Hasselbach equation (More Info). Note that Eqn. 3 predicts that the indicator's pK_a corresponds to the pH of an indicator solution when the logarithmic term equals zero (i. e., when [In^-] equals [HIn]).

(2)

(3)

A convenient way to determine the equilibrium constant of a reaction involving colored species and H⁺ is to use absorbance spectroscopy.  If we monitor a wavelength at which either one of the two species strongly absorbs we will see the absorbance as a function of pH change as that species' concentration in solution changes.  From the equilibrium between HIn and In^-, given above, and considering Le Chatelier's principle, we can see that when the [H⁺] is large (low pH ), (Help Me) the equilibrium will shift completely to the left and the indicator will be completely in the HIn form. 

Consider the experiment whose results are shown in Fig. 1.  In this experiment, the absorbance of an indicator solution is measured at two wavelengths as the solution's pH is varied. The acidic form of the indicator, HIn, absorbs strongly at wavelength l₁ and the basic form, In^-, absorbs strongly at l₂.  So when the solution is very acidic, as on the left side of Fig. 1, all of the indicator is in the HIn form, resulting in a large absorbance at l₁ (labeled A_max,l1) but a small absorbance at l₂ (since [In^-] is small).  At high pH, all of the indicator is in the In^- form giving a strong absorbance at l₂ (labeled A_max,l2) and minimal absorbance at l₁.  As the pH changes from acidic to basic, HIn is converted to In^- in accordance with Eqn. 3.  This conversion results in a decrease in [HIn] and a corresponding increase in [In^-].  Since the absorbance at each wavelength is directly proportional to concentration, we observe a decrease in the absorbance at l₁ (because [HIn] decreases), and an increase in the absorbance at l₂ (because [In^-] increases).  From Eqn. 3 is should be obvious that the pH where [HIn] = [In^-] corresponds to the indicator's pK_a.  This occurs when exactly half of the indicator is in the HIn form and half is present as In^-.  In terms of the experiment, this corresponds to the pH where the absorbance for each form is one half of its maximum, as shown by the dotted line in Fig. 1.  Consequently, the pK_a of an indicator corresponds to the pH of the solution at the inflection point in a plot of absorbance as a function of pH (More Info).

Figure 1. Dependence of absorbance HIn and In^- on pH.  The red line shows how the absorbance changes at a wavelength (l₁) where the acidic form of the indicator, HIn, absorbs strongly.  The blue line indicates the behavior at a wavelength (l₂) where In^-, the basic form of the indicator, absorbs strongly.  The pH at which the inflection point in both lines occurs is the indicator's pK_a.

In practice it is difficult to precisely and accurately determine the inflection point in curves of this type.  To get a more precise measure of the pK_a, Eqn. 3 is rearranged to give Eqn. 4. This equation gives a straight line when the solution's pH is graphed as a function of log([In^-]/[HIn]).  The slope of this line should be +1 and the y-intercept,  where log([In^-]/[HIn]) is zero (i. e. [In^-] = [HIn]), is the pK_a, as shown in Fig. 2.

Figure 2. Relationship between pH and log([In^-]/[HIn]) for an indicator.  The pKa of the indicator corresponds to the intersection of the line with the pH axis.

To use Eqn. 4 to determine the pK_a of the indicator, it is necessary to know the pH of solutions that have different ratios of the two indicator species HIn and In^-. Since the pH of the solution will determine the amount of the total indicator that will be in each form, it can be difficult to control exactly how much of the indicator exists as In^- and HIn. It is possible, however, to use absorbance to obtain the ratio [In^-]/[HIn].  We will simply monitor the absorbance at two different wavelengths. The first wavelength (l₁) is chosen where the acidic (HIn) but not the basic (In^-) form of the indicator strongly absorbs radiation. The second wavelength (l₂) is chosen where the basic but not the acidic form strongly absorbs radiation.

If Beer's law is obeyed, the absorbance at l₁ is given by Eqn. 5 where A_l1 is the absorbance at l₁, e_(HIn,l1) is the molar absorptivity of HIn at l₁ and b is the cell pathlength. Since the amount of the indicator that is in the form of HIn depends on the pH, [HIn] can be difficult, or impossible, to determine. What is known, however, is the total concentration (C_T) of the indicator in both forms since a known amount of the indicator was added to the solution at the beginning. At any pH the indicator's C_T is given by Eqn. 6.  In solutions where the pH is sufficiently low, all of the indicator is in the acidic form, and consequently C_T = [HIn].  Substituting C_T = [HIn] into Eqn. 6 gives Eqn. 7.

Likewise, at l₂ we can write an equation analogous to Eqn. 6 (Eqn. 8) where we have simply replaced the molar absorptivity of HIn with that of In^- and the concentration of [HIn] with [In^-].  In basic solution, C_T = [In^-], which leads to Eqn. 9. 

The ratio [In^-]/[HIn] at any pH can be obtained by combining Eqn. 5 and Eqn. 8 to give Eqn. 10. Substituting equations Eqn. 7 and Eqn. 9 into equation Eqn. 10 gives Eqn. 11, which simplifies to Eqn. 12.

So the first step of our experiment will be to determine the value of A_l1(lowest pH) and A_l2(highest pH). A_l1(lowest pH) is the maximum absorbance at l₁ (A_max,l1) in the solution with the lowest pH which should be only due to HIn. A_l2(highest pH) is the maximum absorbance at l₂ (A_max,l2) in the solution with the highest pH and it should be due to only In^-. 

Until this point we have assumed that at l₁ the measured absorbance is due only to any HIn present and at l₂ all absorbance is due to In^-. In fact, the basic form may absorb somewhat at l₁ and the acidic form may absorb at l₂. Because of this, the values of absorbance used in these equations must be corrected to take into account the amount of the absorbance that is due to the other species. To make this correction, we simply subtract the minimum absorbance measured at l₂ from each of the other measurements made at that wavelength. For example, A_l2 = A_l2(measured) - A_l2(mimimum), where the value of A_l2(mimimum) comes from the measurement made on the most acidic solution at l₂, the wavelength at which it should have the least absorbance. The same correction is to be made for l₁ by measuring the absorbance of the most basic solution at l₁.

Experimental

In this experiment, you will determine the pK_a of bromothymol blue (3',3"- dibromo-thymolsulfonephthalein) by the two methods which have been discussed. At a pH which is less than 6, the indicator is yellow and at a pH which is greater than 7.6, the indicator is blue. At an intermediate pH, the blue and yellow combine to yield a green solution. You will use the Ocean Optics spectrometer to measure the absorbance spectrum of the bromothymol blue as a function of pH.  Click here to review the operation of the spectrometer.  You will also be using a pH electrode interfaced to the Logger Pro A/D converter to measure the pH of each solution that you make.  Click here to learn how to make a pH measurement using Logger Pro.

Before coming to the laboratory prepare a table in the Results section of your laboratory notebook like that shown as Table 1.

Table 1. Example of a table to record the results of this experiment.

Pipet 1.00 ml of the bromothymol blue stock solution into each of two 25-mL volumetric flasks. To one of the flasks add 5 mL of distilled or deionized water and 4 drops of concentrated hydrochloric acid; label this flask “Flask 1”. Dilute the solution to the mark with water. The resulting solution should have a pH of approximately 1. To the second flask add 12 drops of 4 M sodium hydroxide solution (CAUTION! the sodium hydroxide solution is very caustic) and fill the flask to the mark with water; label this flask “Flask 9”. The solution should have a pH of about 13.

Prepare the Ocean Optics spectrophotometer to obtain an absorbance spectrum between 365 and 800 nm. Be sure that you use the same cuvette throughout this experiment!  If you do not, then the spectra will be offset from each other and your results will be inaccurate.  The absorbance in the 700 to 800 nm region should be 0 for all the spectra when the spectra have been obtained correctly.  If they are not, correct by adding or subtracting a constant value to all points in the spectrum (do this in Excel^®) before performing the rest of the analysis.

Obtain and save the spectrum of the bromothymol blue solution at pH 1 (Flask 1) and at pH 13 (Flask 9). Remove any bubbles by gently tapping with your finger.  Under absolutely no circumstances are you to tap a cuvette on a table top. 

Use the cursor function in the Ocean Optics software to locate one wavelength (l₁) at which the pH 1 solution absorbs strongly, but the pH 13 solution absorbs weakly, and a second wavelength (l₂) at which the pH 13 solution absorbs strongly but the pH 1 solution absorbs weakly. Be sure to record the absorbance of both solutions at both wavelengths.

Label seven 25-mL volumetric flasks as "Flask 2" through "Flask 8". Use a pipet to deliver 1.00 ml of the bromothymol blue solution to each of the flasks. Add the volumes of the 0.10 M Na₂HPO₄ solution and the 0.10 M KH₂PO₄ solution to each flask that are indicated in Table 2, below, using graduated cylinders. Dilute each solution to the mark with water.

Table 2.   Volumes of 0.10 M KH₂PO₄ solution and 0.10 M Na₂HPO₄ solution to be added to each volumetric flask.

Set up and calibrate the pH electrode using the Logger Pro software.  Measure the pH of the solutions in all nine flasks, and record the pH of each solution in the table in your notebook.  Be sure to copiously rinse the pH electrode with distilled water between each measurement and pat (do not rub) the electrode dry.  When you have finished your measurements, place the electrode in the standard pH 7 buffer solution that you used to calibrate the electrode.

Measure the absorbance of each of the solutions in flasks 2 through 8.  Save all of them.   Use the cursor to determine the absorbance of each solution at the two chosen wavelengths. Write the absorbance values in your table.

Results and Analysis

Import your spectra into Excel^® and prepare a graph of absorbance as a function of wavelength that shows the spectra of all nine flasks.  Set the x-axis range to 365 to 700 nm.  Be sure to clearly differentiate the lines and to label each with its pH.

Input your data from Table 1 into Excel^®. Subtract the minimum absorbance at each wavelength (the minimum absorbance corresponds to the absorbance of the pH 1 solution at l₂ and to that of the pH 13 solution at l₁) from the absorbance of each of the nine solutions at that wavelength. The resulting absorbances are now corrected for background absorbance. Prepare a plot of absorbance as a function of pH for the nine solutions at each of the two wavelengths.  This should look like Fig. 1. Connect the points with a smooth line and determine a value of the pK_a of the indicator from the inflection point in each plot. It is up to you to determine the best way to find the inflection points.

Using Eqn. 12 and the corrected absorbances calculate the ratio [In^-]/[HIn] at each of the nine pHs (also do this in Excel^®).  Graph pH as a function of log ([In-]/[HIn]), as in Fig. 2. From the best-fit line through the data determine the y-intercept and thus the pK_a of bromothymol blue. Share your value with your laboratory section.  Perform a Q-test on the class data, and discard an errant datum, if warranted.  Calculate the estimated standard deviation on the class average pK_a and the confidence limits on the average pK_a at the 95% confidence level.

Conclusions

Use the outline for a measurement laboratory as a guide as your write your conclusions.

Your Summary Table should like Table 3.  Be sure to include the confidence interval for the class pK_a value.

Table 3.   Example of a Summary Table for this exercise.

References

1. Click here to download this file in PDF format. (link not yet active)

 

2. Braun, R. D. Introduction to Chemical Analysis; McGraw-Hill; New York, 1982, pp. 197-199.