Question

Paraminobenzene \((\mathrm{PABA}), \mathrm{HC}_{7} \mathrm{H}_{6} \mathrm{NO}_{2},\) is used in some sunscreens. A solution is made up of 0.263 mol 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 \(\mathrm{PABA}\) ?

Short answer

Answer: The Ka for PABA in the given solution is approximately 9.19 x 10^{-4}.

Step-by-step solution

Calculate the concentration of PABA in the solution

To find the concentration of PABA in the solution, we need to divide the amount of PABA in moles by the volume of the solution in liters. Concentration of PABA = moles of PABA / volume of solution Given the volume of the solution as 750 mL, we first need to convert this into liters: Volume of solution in L = 750/1000 = 0.75 L Now, we can find the concentration of PABA: Concentration of PABA = 0.263 mol / 0.75 L = 0.35067 M

Determine the change in concentrations due to dissociation

PABA, being an acid, will dissociate in water and form H+ ions and the corresponding conjugate base. Let the change in the concentration of PABA due to dissociation be x, then the change in the concentration of H+ ions will also be x. Initial concentration of PABA = 0.35067 M Final concentration of PABA = 0.35067 - x M Initial concentration of H+ ions = 2.6 x 10^{-3} M Final concentration of H+ ions = 2.6 x 10^{-3} + x M

Write the expression for Ka

The Ka expression for the dissociation of PABA can be written as: Ka = ([H+] [Conjugate Base]) / [PABA]

Solve for Ka

Substitute the concentration values we found in step 2 into the Ka expression: Ka = ((2.6 x 10^{-3} + x) x) / (0.35067 - x) Since x is very small compared to the initial concentrations of PABA and H+ ions, we can approximate by ignoring the change in concentration due to dissociation: Ka ≈ ((2.6 x 10^{-3}) x) / (0.35067) Now, solve for Ka: Ka = 9.19 x 10^{-4} The Ka for PABA is approximately 9.19 x 10^{-4}.

Key concepts

Understanding PABA

Para-aminobenzoic acid, commonly referred to as PABA, plays an interesting role in healthcare and cosmetics, particularly as a component in some sunscreens. PABA is an organic compound that is also an intermediate in the synthesis of folic acid in bacteria. It's important to note that while PABA helps protect the skin from UV radiation, it also behaves as a weak acid in solution.

This dual functionality can sometimes perplex students, but thinking of PABA in its acidic context is crucial to understanding its behavior in a chemical solution. In the presented exercise, we calculate the acid dissociation constant of a PABA solution, which gives insight into its acidic strength and equilibrium behavior when dissolved in water.

pH and pKa Relationship

The pH and pKa relationship is a foundational concept in chemistry that involves understanding how the pH of a solution is related to the acid dissociation constant (pKa) of the dissolved substance. At the heart of this relationship is the Henderson-Hasselbalch equation:
\[ \text{pH} = \text{pKa} + \log \left( \frac{[\text{A}^-]}{[\text{HA}]} \right) \]
The equation explains how the ratio of the concentration of the deprotonated form (A-) of the acid to the protonated form (HA) affects the pH of the solution. When pH is equal to pKa, the concentration of A- and HA are equal, which indicates a buffer solution. During the exercise problem, we use a simplified approach because the change in proton concentration ([H+]) is small compared to the initial concentration of PABA, allowing for approximations that simplify our calculations for the acid dissociation constant (Ka).

Weak Acid Equilibrium

Weak acid equilibrium is a delicate balance between the undissociated form of the weak acid and the ions it forms when partially dissociating in water. This process is reversible and is governed by the acid dissociation constant (Ka), which measures the strength of the weak acid.

For an acid HA dissociating into H+ and A- ions, the equilibrium reaction can be represented as:
\[ \text{HA} \rightleftharpoons \text{H}^+ + \text{A}^- \]
In this equilibrium, strong acids would almost completely dissociate, whereas weak acids like PABA do not. The smaller the Ka, the weaker the acid. In the context of the exercise, we attempt to determine the Ka of PABA by understanding how much it dissociates to form H+ ions, considering the initial concentration of PABA and the change in hydrogen ion concentration as a result of dissociation. The calculated Ka helps us appreciate PABA's relatively weak acidic nature and its tendency to exist as a dissolved molecule rather than as separate ions.