In the case of very weak acids, 3H3O+ 4 from the dissociation of water is significant compared with 3H3O+ 4 from the dissociation of the weak acid. The sugar substitute saccharin 1C7H5NO3S2, for example, is a very weak acid having Ka = 2.1 * 10-12 and a solubility in water of 348 mg/100 mL. Calculate 3H3O+ 4 in a saturated solution of saccharin. (Hint: Equilibrium equations for the dissociation of saccharin and water must be solved simultaneously.)

Verified step by step guidance

1. Convert the solubility of saccharin from mg/100 mL to molarity (mol/L) by using its molar mass.
2. Write the equilibrium expression for the dissociation of saccharin in water: \( \text{C}_7\text{H}_5\text{NO}_3\text{S}_2 \rightleftharpoons \text{H}^+ + \text{C}_7\text{H}_4\text{NO}_3\text{S}_2^- \).
3. Write the equilibrium expression for the dissociation of water: \( \text{H}_2\text{O} \rightleftharpoons \text{H}^+ + \text{OH}^- \).
4. Set up the equilibrium expressions for both reactions: \( K_a = \frac{[\text{H}^+][\text{C}_7\text{H}_4\text{NO}_3\text{S}_2^-]}{[\text{C}_7\text{H}_5\text{NO}_3\text{S}_2]} \) and \( K_w = [\text{H}^+][\text{OH}^-] \).
5. Solve the system of equations simultaneously to find the concentration of \([\text{H}^+]\) in the saturated solution.

Key Concepts

Weak Acids and Their Dissociation

Weak acids are substances that partially dissociate in solution, establishing an equilibrium between the undissociated acid and its ions. The strength of a weak acid is quantified by its acid dissociation constant (Ka), which indicates the extent of dissociation. For saccharin, with a Ka of 2.1 x 10^-12, this means that only a small fraction of the acid molecules donate protons (H+) to the solution, making the calculation of hydronium ion concentration (H3O+) crucial.

Equilibrium and the ICE Table

In chemical equilibria, the concentrations of reactants and products remain constant over time. The ICE (Initial, Change, Equilibrium) table is a useful tool for organizing the concentrations of species involved in a reaction at different stages. For the dissociation of saccharin and water, setting up an ICE table allows for the systematic calculation of H3O+ concentrations by accounting for the contributions from both the weak acid and the autoionization of water.

Simultaneous Equations in Equilibrium Calculations

When dealing with multiple equilibria, such as the dissociation of a weak acid and the autoionization of water, simultaneous equations are often necessary to find the concentrations of all species involved. In this case, the concentration of H3O+ from both the dissociation of saccharin and the water must be considered together. Solving these equations allows for a comprehensive understanding of the system's behavior and the accurate calculation of the hydronium ion concentration in the saturated solution.