Saccharin, a sugar substitute, is a weak acid with pKa = 2.32 at 25 °C. It ionizes in aqueous solution as follows: HNC7H4SO31(aq) ⇌ H+(aq) + NC7H4SO3-(aq). What is the pH of a 0.10 M solution of this substance?

Verified step by step guidance

1. Identify the given information: The concentration of saccharin is 0.10 M, and the pKa is 2.32.
2. Use the relationship between pKa and Ka: \( \text{pKa} = -\log_{10}(K_a) \). Calculate \( K_a \) by rearranging the formula to \( K_a = 10^{-\text{pKa}} \).
3. Set up the expression for the ionization of saccharin: \( K_a = \frac{[H^+][NC_7H_4SO_3^-]}{[HNC_7H_4SO_3]} \). Assume \([H^+] = [NC_7H_4SO_3^-] = x\) and \([HNC_7H_4SO_3] = 0.10 - x\).
4. Substitute the expressions into the \( K_a \) equation: \( K_a = \frac{x^2}{0.10 - x} \). Since \( K_a \) is small, assume \( x \ll 0.10 \), simplifying to \( K_a \approx \frac{x^2}{0.10} \).
5. Solve for \( x \) (which is \([H^+]\)) using the simplified equation: \( x = \sqrt{K_a \times 0.10} \). Finally, calculate the pH using \( \text{pH} = -\log_{10}(x) \).

Key Concepts

pKa and pH Relationship

The pKa value of an acid indicates its strength; a lower pKa means a stronger acid. The pH of a solution is a measure of the concentration of hydrogen ions (H+) in that solution. The relationship between pKa and pH is crucial for understanding how much of the acid will ionize in solution, which directly affects the pH.

Weak Acid Ionization

Weak acids do not completely dissociate in solution, establishing an equilibrium between the undissociated acid and its ions. The ionization of saccharin can be represented by the equilibrium expression, which allows us to calculate the concentration of H+ ions produced. This is essential for determining the pH of the solution.

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation relates the pH of a solution to the pKa of the acid and the ratio of the concentrations of the deprotonated and protonated forms. It is expressed as pH = pKa + log([A-]/[HA]). This equation is particularly useful for calculating the pH of weak acid solutions, such as saccharin, by incorporating the concentrations of the ionized and non-ionized forms.