Problem 46 - Chemistry: Principles and Reactions

Question

Para-aminobenzoic acid (PABA), \(\mathrm{HC}_{7} \mathrm{H}_{6} \mathrm{NO}_{2}\), is used in some sunscreen agents. A solution is made by dissolving \(0.263\) mol of PABA in enough water to make \(750.0 \mathrm{~mL}\) of solution. The solution has \(\left[\mathrm{H}^{+}\right]=2.6 \times 10^{-3} \mathrm{M}\). What is \(K_{\mathrm{a}}\) for PABA?

Step-by-Step Solution

Answer

Answer: The acid dissociation constant (\(K_a\)) for para-aminobenzoic acid (PABA) is approximately \(1.94 \times 10^{-5}\).

Step 1: Write the chemical reaction for PABA ionization

First, we need to write the chemical equation for the ionization of PABA in water. PABA is a weak acid, so it will donate a proton (H+) to water, forming its conjugate base and a hydronium ion: \(\mathrm{HC}_{7} \mathrm{H}_{6} \mathrm{NO}_{2}(aq) + \mathrm{H}_{2}\mathrm{O}(l) \rightleftharpoons \mathrm{C}_{7} \mathrm{H}_{6} \mathrm{NO}_{2}^{-}(aq) + \mathrm{H}_{3}\mathrm{O}^{+}(aq)\)

Step 2: Write the expression for the acid dissociation constant, \(K_a\)

The \(K_a\) is the equilibrium constant for the ionization of an acid in water. It can be written as the product of the concentrations of the products, divided by the concentration of the reactant. \(K_a = \frac{[\mathrm{C}_{7} \mathrm{H}_{6} \mathrm{NO}_{2}^{-}][\mathrm{H}_{3}\mathrm{O}^{+}]}{[\mathrm{HC}_{7} \mathrm{H}_{6} \mathrm{NO}_{2}]}\) Since the concentration of hydronium ions is given, we can represent it as \([\mathrm{H}^{+}]\) instead of \([\mathrm{H}_{3}\mathrm{O}^{+}]\)

Step 3: Calculate the initial concentration of PABA

We are given that there are \(0.263\) moles of PABA dissolved in \(750\) mL of solution. To calculate the initial concentration of PABA, we divide the moles by the volume of the solution in liters: Initial concentration of PABA = \(\frac{0.263 \,\text{mol}}{0.750 \, \text{L}} = 0.350 \,\text{M}\)

Step 4: Find the concentrations at equilibrium

Since we know the concentration of hydrogen ions in the solution, we can assume the same concentration of para-aminobenzoate ions, as one hydrogen ion is released for each PABA ion: \([\mathrm{C}_{7} \mathrm{H}_{6} \mathrm{NO}_{2}^{-}] = [\mathrm{H}^{+}] = 2.6 \times 10^{-3} \,\text{M}\) Next, we need to find the concentration of PABA at equilibrium. As each PABA molecule ionizes, one hydrogen ion is released. Therefore, the change in concentration of PABA is equal to the concentration of hydrogen ions: Change in concentration of PABA = \(2.6 \times 10^{-3} \,\text{M}\) Now, we subtract this change from the initial concentration of PABA to find the equilibrium concentration of PABA: \([\mathrm{HC}_{7} \mathrm{H}_{6} \mathrm{NO}_{2}] = 0.350 \,\text{M} - 2.6 \times 10^{-3} \,\text{M} = 0.347 \,\text{M}\)

Step 5: Calculate the \(K_a\) for PABA

Now that we have all the equilibrium concentrations, we can plug them into the expression for \(K_a\): \(K_a = \frac{[\mathrm{C}_{7} \mathrm{H}_{6} \mathrm{NO}_{2}^{-}][\mathrm{H}^{+}]}{[\mathrm{HC}_{7} \mathrm{H}_{6} \mathrm{NO}_{2}]}\) \(K_a = \frac{(2.6 \times 10^{-3} \,\text{M})(2.6 \times 10^{-3} \,\text{M})}{0.347 \,\text{M}} = 1.94 \times 10^{-5}\) So, the acid dissociation constant (\(K_a\)) for para-aminobenzoic acid (PABA) is approximately \(1.94 \times 10^{-5}\).

Key Concepts

PABA Ionization

Para-aminobenzoic acid (PABA), a compound commonly found in sunscreen, ionizes in water by donating a hydrogen ion (H+) to form a conjugate base, p-aminobenzoate ion, and a hydronium ion (H₃O⁺ or simply H⁺). Understanding PABA ionization is crucial for figuring out its behavior in solutions, particularly, its strength as an acid. Weak acids like PABA do not dissociate completely in water, which means they establish an equilibrium between the non-ionized acid and the products of their ionization.

The chemical equation for PABA ionization could be visually represented, aiding students to understand how each molecule of PABA contributes a single H+ to the solution, leading to a corresponding amount of p-aminobenzoate ions.

Equilibrium Constant

The equilibrium constant, designated as K_a for acids, quantifies the extent of ionization in a solution and is a measure of an acid's strength. It is determined by the ratio of the product of the concentrations of the ionized species to the concentration of the original acid, at equilibrium. In the case of PABA, by expressing the K_a formula, students can visualize the dynamic nature of the chemical equilibrium. It demonstrates that the direction of the reaction depends on the relative concentrations of the reactants and products.

As an educator, I would stress the importance of the equilibrium constant by using examples that relate to everyday life, like the role of PABA in sunscreen, thereby making the concept of equilibrium constants more relatable and easier to grasp.

Chemical Concentration Calculation

Chemical concentration calculation involves determining the amount of solute in a given volume of solution. This exercise provides an excellent opportunity to apply these calculations, taking the amount of PABA and dissolving it in a specific volume of water to find the concentration. Recognizing the importance of this skill, one could elucidate how concentration impacts various real-life situations such as the efficacy and safety of sunscreen.

For students to fully understand concentration calculations, I would recommend converting between moles and liters frequently. This helps solidify the concept that concentration is a ratio and can help students with future calculations in various contexts, beyond just acid-base chemistry.