Acid-Base Background

Henderson-Hasselbach Equation. Many organic acids (HA), including carboxylic acids (RCO₂H), phenols (ArOH), and ammonium ions (RNH₃⁺), are so strong that they can protonate water.

Equilibrium between the acid (HA), its conjugate base (A^-), and water, is achieved so rapidly (on the same time scale as mixing of the ingredients) that the acid-base reaction may appear to be instantaneous.

The tendency of an acid to protonate water is expressed by K_a, the acid dissociation constant. When the solution is dilute, K_a is related to the equilibrium concentrations of the acid and its conjugate base by:

Note on [...] symbols. All of the concentrations that appear in these formulas refer to equilibrium concentrations.

Because K_a values range over many powers of 10, it is can be more convenient to define and work with pK_a values.

The last term in this equation, -log[H⁺], equals pH for dilute solutions.

Rearrangement of the pK_a equation gives the famous Henderson-Hasselbach (H-H) equation:

An important consequence of the H-H equation is that the percentages of acid, HA, and conjugate base, A^-, don't change much with pH except when pH is in the immediate vicinity of the acid's pK_a:

Spectrophotometric measurement of pKa. The goal of this experiment is to measure an acid's pK_a and the H-H equation tells you which variables you need for this. First, you need a solution that contains the acid and its conjugate base at equilibrium and in measurable quantities. Second, you need to measure the solution's pH. Third, you need to measure the ratio of conjugate base to acid. Inserting these quantities in the H-H equation will let you solve for pK_a.

It is quite a simple matter to bring an acid and its conjugate base into equilibrium with water (this happens almost instantaneously after the acid is dissolved in water). It is even a simple matter to measure pH. This can be done with a pH meter. Unfortunately, measuring the concentrations, or even the relative concentrations, of conjugate base and acid is a bit of a problem. Some clever methods for measuring these concentrations have been devised, but none are more clever (at least in my opinion) than measuring how much light is absorbed by a mixture of the acid and its conjugate base (this is called a spectrophotometric measurement).

Let's see how this might work. First, you need to recall that the absorbance of a solution containing one light-absorbing species is summarized by the Beer-Lambert (B-L) law:

A = abc

where A represents absorbance, a represents the compound's molar absorptivity (also called the "extinction coefficient"), and b and c represent the sample thickness (in cm) and concentration (in moles/liter), respectively. Clearly, if we can measure absorption, we might be able to obtain the concentration of the light-absorbing compound.

But now some problems enter the picture. First, molar absorptivity a varies with wavelength. If we measure absorption over a range of wavelengths, and a varies across this range, we cannot determine c. Therefore, we need to make sure that our apparatus can make absorbance measurements over a very restricted wavelength range.

Second, there are four separate parameters in the B-L law. We need to know three of them in order to get the fourth. How do we get around that? It turns out that A and b are both easily measured, so we mainly need to worry about a and c. If we make up a sample of known concentration c, we can calculate the molar absorptivity, a. Then, we can use this value of a to determine c for other samples.

Our experiment will actually take a more devious route. Recall that we want the concentrations of acid and conjugate base. Two separate species. One absorption measurement cannot give us two concentrations, but three measurements can.

To see how this works, suppose we have two species, HA (acid) and A^- (conjugate base). If these species absorb light independently, we can write a separate B-L equation for each species like the one shown above. The absorbance we measure, A, will be the sum of these absorbances. In other words:

A = A_HA + A_A = a_HAbc_HA + a_Abc_A

Because A depends on two molar absorptivities and two concentrations, it is impossible to convert A into either concentration directly. This is where some cleverness is needed. Suppose you begin with a sample solution of initial acid concentration C_o and allow it to equilibrate with water. Once the acid and its conjugate base reach equilibrium:

C_o = c_HA + c_A

Now, if you add just one drop of very concentrated strong acid to a suitable quantity of this solution, all of the A^- will be protonated. Numerically speaking, c_A goes to zero and C_o = c_HA. If you measure the absorbance of this acidic solution (call it A_a(cid)) you can write:

A_a = a_HAbC_o

Now suppose you start over with another portion of the original solution, but this time add just one drop of very concentrated strong base. All of the HA will be deprotonated making c_HA go to zero and C_o = c_A. Calling the absorbance of this basic solution A_b(ase), you can write:

A_b = a_AbC_o

Still making sense? It turns out that if you combine these "extreme" absorbance values with the absorbance A of the original sample, the one that contains some HA and some A^-, you can obtain the ratio of conjugate base concentration to acid concentration. Here's the magic formula:

A note on "magical" math: It's hard (for me) to see where this formula came from so don't worry about that. Instead, please verify for yourself that the formula works. To do this, replace each absorbance term on the right-hand side with the appropriate "abc" combination from the B-L law. It will look very complicated at first, but everything should eventually fall away except the ratio of the two concentrations. Stumped? Here's how I did it.

This is a good point to look around and see how things stand. 1) We want pK_a. 2) The H-H equation tells us we need a pH measurement and the equilibrium ratio [A^-]/[HA], 3) pH can be measured with a pH meter and 4) the ratio of equilibrium concentrations can be obtained by combining three absorbance measurements.

If you actually went into the lab to do this, you might try the following: 1) make up a solution of your acid with a known initial concentration (C_o), 2) measure its absorbance (A) and pH, 3) take a small portion of this solution and add a drop of very strong acid; measure its absorbance (A_a), 4) take a separate small portion and add a drop of very strong base; measure its absorbance (A_b). Combine your measured absorbances to get [A^-]/[HA], insert this ratio and pH into the H-H equation, and voila: pK_a.

Unfortunately, this probably won't work. If you simply make up a solution of acid in water, there is a good chance that the solution will end up being nearly all HA (if the acid is rather weak) or all A^- (if the acid is fairly strong). In either case, the ratio [A^-]/[HA] will so fantastically small, or so overwhelmingly huge, that it will be impossible to measure it reliably. The problem is simple: absorbances don't change much once a solution is dominated by one species so either the numerator or the denominator in our "magic" formula will be about zero (and whatever we think it is, that value will be unreliable).

If we want reliable data, we must have both compounds, HA and A^-, present in significant quantities. This is the only way to make all three absorbance measurements different from each other.

What conditions produce this type of mixture? As was shown in the previous section, this kind of mixture occurs when the solution's pH falls within one unit of the acid's pK_a. We don't know this pK_a, but we can set up solutions with different pH values and hope that one or more of them will provide the necessary mixture and data. A technique for creating solutions of well-defined pH is described in the next section.

Buffer solutions. In this experiment you will prepare a series of acetic acid-sodium acetate buffer solutions in which the pH methodically increases from a low value to a high value. A little bromocresol green (BCG) will be added to each solution, and BCG is an acid, but this will have no substantial effect on the pH because of the behavior of the buffering agent. In any case, you will measure the pH after you have added BCG and use your measured value to estimate BCG's pK_a.

A longer description of buffer solutions can be found at Wikipedia, but some key points are mentioned below.

Buffer solutions are usually made by combining an acid, HBuff, and a salt that contains the acid's conjugate base, Buff^-. In our experiment, acetic acid (CH₃COOH) and sodium acetate (CH₃COONa) are the acid and salt, respectively.

Buffering agents maintain a constant pH only when the percentages of acid and conjugate base are both substantial. From our previous discussion of the H-H equation, this means a buffering agent only stabilizes pH near the pK_a of HBuff.

When a weak acid and its conjugate base (salt) are combined in similar amounts, the amount of acid and conjugate base in solution are almost equivalent to the amount used to make the solution. Therefore, if the acid's pK_a is known, we can estimate the solution's pH using a modified H-H equation:

The pK_a of acetic acid is well-known, but you will be asked to estimate it for yourself in this experiment and test the validity of the modified H-H equation given above. To do this, you will plot your pH data against log([moles salt used]/[moles acid used]) and decide whether the expected relationship is observed (what kind of curve does this equation predict?). You will then use curve-fitting software to determine the pK_a of acetic acid.

Spectroscopic background

Bromocresol green. Bromocresol green is a neutral solid with the following structure:

image from Wikipedia and used without permission

In water, BCG ionizes by losing a proton and breaking the extremely polar C-O bond. The electrical charge on BCG- enhances its water-solubility. This is the "acidic" or yellow form of the indicator.

image from Wikipedia and used without permission

At higher pH, BCG loses a second proton and becomes a dianion. This is the "basic" or blue form of the indicator.

image from Wikipedia and used without permission

Some interesting questions can be asked about these compounds: Why are they colored? Why does changing the ionization state also change the color? Can we anticipate or explain why the color changes the way it does?

Padias p. 94 explains the fundamental principles of light absorption and concludes that conjugated systems tend to absorb lower energy (longer wavelength) light.

BCG anion (acidic form) and BCG dianion (basic form) are both conjugated compounds. In fact, we say they contain extended conjugation because the conjugation goes beyond a single benzene ring. It is not surprising, therefore, that they both absorb low energy visible light.

As it happens, we can draw many more resonance structures for BCG dianion (only two are shown above) than for BCG anion. This indicates that electron delocalization is more important in the dianion, which suggests in turn that the dianion will absorb lower energy light. Comparing the UV-vis spectra of the "acid" anion (yellow) and "basic" dianion (blue) confirms this:

image from Wikipedia and used without permission

The region of maximum light absorption shifts from shorter wavelength (higher energy) light for the anion (450 nm) to longer wavelength (lower energy) light for the dianion (>620 nm). This may seem confusing because the anion looks yellow and the dianion looks blue. Remember, however, the complementary relationship that exists between absorbed light and perceived color. You don't see the light that the compound absorbs. A compound that absorbs "blue" (short wavelength) light will appear yellow, while a compound that absorbs "yellow" (long wavelength) light will appear blue.