ChemLab.Truman
Spectrophotometric Determination of an Equilibrium Constant^{1,2}
Authors: B. K. Kramer, B. D. Lamp, D. L. McCurdy* and J. M. McCormick
Last Update: April 21, 2011
Introduction
Typically acidbase 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 indicates 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 baseten logarithm of both sides (More Info). By convention, we use the prefix "p" to denote the negative baseten logarithm, and so logK_{a} becomes pK_{a} and the log[H^{+}] becomes pH. This leads to a very convenient way of writing the very small [H^{+}] that occur in aqueous solution as numbers that generally fall between 0 and 14. We can then rewrite Eqn. 2 as Eqn. 3 (which some of you may recognize as simply another version of the HendersonHasselbach equation). 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]).

(1) 
(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 a solution of the indicator bromthymol blue is measured as the solution's pH is varied. As the solution pH changes from acidic to basic, we observe an evolution from spectra where the peak centered around 430 nm (denoted by λ_{1}) is largest, to spectra where the peak at about 630 nm (denoted by λ_{2}) dominates. Further examination of the spectra reveals one wavelength, noted by a star, where the absorbance is essentially independent of pH. This is called an isosbestic point and results from the both forms of the indicator (HIn and In^{}) having the same molar absorptivity at this wavelength. It is not uncommon to see one or more isosbestic points in a set of spectra.
Figure 1. Spectra of the indicator bromothymol blue as a function of pH.
A closer examination of the trends in the spectra is shown in Fig. 2. When the solution is very acidic, as on the left side of Fig. 2, all of the indicator is in the HIn form, resulting in a large absorbance at λ_{1} (labeled A_{max,}_{λ1}) but a small absorbance at λ_{2} (since [In^{}] is small). At high pH, all of the indicator is in the In^{} form giving a strong absorbance at λ_{2} (labeled A_{max,}_{λ2}) and minimal absorbance at λ_{1}. As the pH changes from acidic to basic, the position of the equilibrium changes such that 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 λ_{1} (because [HIn] decreases), and an increase in the absorbance at λ_{2} (because [In^{}] increases). It is important to realize that even though we may only be collecting absorbance values at two wavelengths in our experiment, the entire spectrum is undergoing a transformation as we cause the relative amount of HIn to decrease and In to increase by increasing the pH of the system. In principle, we could choose any wavelength where HIn absorbs for λ_{1}, and any wavelength where In^{} absorbs for λ_{2} in our analysis. In practice, however, it is beneficial to choose the wavelength of maximum absorbance for each species. Doing so provides the greatest sensitivity (ability to distinguish small changes in absorbance) for our measurement.
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. 2. 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 2. Dependence of absorbance HIn and In^{} on pH. The red line shows how the absorbance changes at a wavelength (λ_{1}) where the acidic form of the indicator, HIn, absorbs strongly. The blue line indicates the behavior at a wavelength (λ_{2}) 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}.


(4) 
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 yintercept, where log([In^{}]/[HIn]) is zero (i. e. [In^{}] = [HIn]), is the pK_{a}, as shown in Fig. 3.
Figure 3. 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 (λ_{1}) is chosen where the acidic (HIn) but not the basic (In^{}) form of the indicator strongly absorbs radiation. The second wavelength (λ_{2}) is chosen where the basic but not the acidic form strongly absorbs radiation.


(5) 
If Beer's law is obeyed, the absorbance at λ_{1} is given by Eqn. 5 where A_{λ}_{1} is the absorbance at ε_{1}, λ_{HIn,λ}_{1)} is the molar absorptivity of HIn at λ_{1} and b is the cell path length. 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.


(6) 


(7) 
Likewise, at λ_{2} 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.


(8) 


(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_{λ}_{1}(lowest pH) and A_{λ}_{2}(highest pH). A_{λ}_{1}(lowest pH) is the maximum absorbance at λ_{1} (A_{max}_{,λ}_{1}) in the solution with the lowest pH which should be only due to HIn. A_{λ}_{2}(highest pH) is the maximum absorbance at λ_{2} (A_{max,}λ_{2}) in the solution with the highest pH and it should be due to only In^{}.


(10) 


(11) 


(12) 
Until this point we have assumed that at λ_{1} the measured absorbance is due only to any HIn present and at λ_{2} all absorbance is due to In^{}. In fact, the basic form may absorb somewhat at λ_{1} and the acidic form may absorb at λ_{2}. 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 λ_{2} from each of the other measurements made at that wavelength. For example, A_{λ}_{2} = A_{λ}_{2}(measured)  A_{λ}_{2}(mimimum), where the value of A_{λ}_{2}(mimimum) comes from the measurement made on the most acidic solution at λ_{2}, the wavelength at which it should have the least absorbance. The same correction is to be made for λ_{1} by measuring the absorbance of the most basic solution at λ_{1}.
Experimental
In this experiment, you will determine the pK_{a} of bromothymol blue (3',3" dibromothymolsulfonephthalein) 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 Spectronic 20 Genesys spectrometer to measure the absorbance of a bromothymol blue solution at the two specified wavelengths as a function of pH. 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. Be sure that you have read and understand the operation of both instruments before coming to the lab. Note that there will be two or three spectrometers in the lab for this exercise and you may be sharing with up to three other tables.
Before coming to the laboratory prepare a table in the Results section of your laboratory notebook like that shown as Table 1.
Flask Number 
Absorbance at λ_{1}  Absorbance at λ_{2}  pH 
Solution Color 
1  
2  
3  
4  
5  
6  
7  
8  
9 
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 25mL 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 Spectronic 20 Genesys to obtain at 430 nm or at 630 nm according to the instrument's operating instructions. You should discuss with the other groups sharing a particular spectrometer on how best make your measurements so as to use your time effective. Obtain and record the absorbance of the bromothymol blue solution at pH 1 (Flask 1) and at pH 13 (Flask 9) at both wavelengths. Be sure that you use the same cuvette throughout this experiment! Remove any bubbles from the cuvette by gently tapping with your finger. Under absolutely no circumstances are you to tap a cuvette on a table top.
Label seven 25mL 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 Na_{2}HPO_{4} solution and the 0.10 M KH_{2}PO_{4} solution to each flask that are indicated in Table 2, below, using graduated cylinders. Dilute each solution to the mark with water.
Flask Number 
Volume of KH_{2}PO_{4} Solution (mL) 
Volume of Na_{2}HPO_{4} Solution (mL) 
2  5.0  0.0 
3  5.0  1.0 
4  10.0  5.0 
5  5.0  10.0 
6  1.0  5.0 
7  1.0  10.0 
8  0.0  5.0 
Table 2. Volumes of 0.10 M KH_{2}PO_{4} solution and 0.10 M Na_{2}HPO_{4} 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 at each wavelength. Record the absorbance values in your table.
Results and Analysis
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 λ_{2} and to that of the pH 13 solution at λ_{1})_{ }from the absorbance of each of the nine solutions at that wavelength. The resulting absorbance values are now corrected for background absorbance. Prepare a graph of absorbance as a function of pH for the nine solutions at each of the two wavelengths. This should look like Fig. 2. 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. 3. From the bestfit line through the data determine the yintercept and thus the pK_{a} of bromothymol blue. Share your value with your laboratory section. Perform a Qtest 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.
Our pK_{a} 
Number of Class Values 
Class pK_{a} 
Table 3. Example of a Summary Table for this exercise.
References
 1. Click here to download this file in PDF format. Note that hyperlinks are not active in the pdf version.
 2. Braun, R. D. Introduction to Chemical Analysis; McGrawHill; New York, 1982, pp. 197199.