Question

The active ingredient of Pepto-Bismol is the compound bismuth subsalicylate, which undergoes the following dissociation when added to water: \(\begin{aligned} \mathrm{C}_{7} \mathrm{H}_{5} \mathrm{BiO}_{4}(s)+\mathrm{H}_{2} \mathrm{O}(l) \rightleftharpoons \mathrm{C}_{7} \mathrm{H}_{4} \mathrm{O}_{3}{ }^{2-}(a q) \\\ &+\mathrm{Bi}^{3+}(a q)+\mathrm{OH}^{-}(a q) \quad K=? \end{aligned}\) If the maximum amount of bismuth subsalicylate that reacts by this reaction is \(3.2 \times 10^{-19} \mathrm{~mol} / \mathrm{L}\), calculate the equilibrium constant for the preceding reaction.

Short answer

The equilibrium constant for the dissociation of bismuth subsalicylate in water is approximately \(3.3 \times 10^{-57}\).

Step-by-step solution

Write down the dissociation equation and the respective equilibrium equation

The dissociation equation is given as: \[\mathrm{C}_{7} \mathrm{H}_{5} \mathrm{BiO}_{4}(s) + \mathrm{H}_{2} \mathrm{O}(l) \rightleftharpoons \mathrm{C}_{7} \mathrm{H}_{4} \mathrm{O}_{3}{ }^{2-}(a q) + \mathrm{Bi}^{3+}(a q) + \mathrm{OH}^{-}(a q)\] We can represent the equilibrium equation using the equilibrium constant K: \[K = \frac{[\mathrm{C}_{7} \mathrm{H}_{4} \mathrm{O}_{3}{ }^{2-}][\mathrm{Bi}^{3+}][\mathrm{OH}^{-}]}{[\mathrm{H}_{2}\mathrm{O}]}\] However, since the concentration of water ([H2O]) remains relatively constant, we can express the equilibrium constant as: \[K' = [\mathrm{C}_{7} \mathrm{H}_{4} \mathrm{O}_{3}{ }^{2-}][\mathrm{Bi}^{3+}][\mathrm{OH}^{-}]\] In this form, K’ will be the equilibrium constant that we need to find.

Use the given information and stoichiometry to find the concentrations at equilibrium

The maximum amount of bismuth subsalicylate reacted is \(3.2 \times 10^{-19}\; mol/L\). According to the stoichiometry of the reaction, for every 1 molecule of bismuth subsalicylate that dissociates, 1 molecule of \(\mathrm{C}_7\mathrm{H}_4\mathrm{O}_3^{2-}\), 1 molecule of \(\mathrm{Bi}^{3+}\), and 1 molecule of \(\mathrm{OH}^-\) are produced. So, at equilibrium, the concentration of these ions is the same as the mols of bismuth subsalicylate reacted: \[[\mathrm{C}_{7} \mathrm{H}_{4} \mathrm{O}_{3}{ }^{2-}] = [\mathrm{Bi}^{3+}] = [\mathrm{OH}^{-}] = 3.2 \times 10^{-19}\;mol/L\]

Calculate K' using the equilibrium concentrations

Now substitute the equilibrium concentrations found in step 2 into the equation for K’: \[K' = (3.2 \times 10^{-19})(3.2 \times 10^{-19})(3.2 \times 10^{-19})\] Solve for K': \[K' \approx 3.3 \times 10^{-57}\] Thus, the equilibrium constant for the dissociation of bismuth subsalicylate in water is approximately \(3.3 \times 10^{-57}\).

Key concepts

Dissociation Reaction

A dissociation reaction occurs when a compound breaks apart into its component ions. This process is common in solutions and is usually reversible, meaning the ions can recombine to form the original compound. When bismuth subsalicylate, the active ingredient in Pepto-Bismol, is added to water, it dissociates into three types of ions:

• Bi³⁺
• C₇H₄O₃^2-
• OH^-

The dissociation of this compound in water is represented by the equation: \[\mathrm{C}_{7} \mathrm{H}_{5} \mathrm{BiO}_{4}(s) + \mathrm{H}_{2} \mathrm{O}(l) \rightleftharpoons \mathrm{C}_{7} \mathrm{H}_{4} \mathrm{O}_{3}{ }^{2-}(aq) + \mathrm{Bi}^{3+}(aq) + \mathrm{OH}^{-}(aq)\] The double arrow in the equation indicates that the reaction can proceed in both directions, meaning it can both dissociate and recombine. The extent to which the compound dissociates is a measure of the equilibrium. Understanding this balance is vital in chemistry, especially when calculating the equilibrium constant.

Stoichiometry

Stoichiometry involves the calculation of reactants and products in chemical reactions. It is based on the conservation of mass where the mass of the reactants equals the mass of the products. In the dissociation of bismuth subsalicylate, stoichiometry tells us how many moles of each product ion are produced per mole of reactant. For this reaction, the stoichiometric relationship can be understood as each mole of bismuth subsalicylate dissociating to form:

• One mole of C₇H₄O₃^2-
• One mole of Bi³⁺
• One mole of OH^-

Applying these ratios helps us determine that when 3.2 x 10^-19 mol/L of bismuth subsalicylate reacts, it produces an equivalent amount of each dissociated ion in the solution.

Bismuth Subsalicylate

Bismuth subsalicylate is widely known as the active ingredient in over-the-counter medications like Pepto-Bismol, which is used to treat temporary discomforts of the stomach and gastrointestinal tract. Chemically, it is a compound that, when dissolved in water, undergoes a dissociation reaction. This reaction is particularly interesting because it creates ions that contribute to the medication's function. The release of

• C₇H₄O₃^2-
• Bi³⁺
• OH^-

ions suggests how the medication may neutralize stomach acid and settle the stomach. Understanding this dissociation at a molecular level provides insight into how such medications perform their role and underscores the importance of equilibrium constants in pharmaceutical applications.

Equilibrium Concentrations

Equilibrium concentrations refer to the concentrations of the reactants and products at a point where the forward and reverse reactions occur at the same rate. In our given reaction, once equilibrium is reached, the concentrations of the product ions remain constant, meaning the process of dissociation and recombination continues without changing the concentrations further. For the dissociation of bismuth subsalicylate, the equilibrium concentrations of:

• C₇H₄O₃^2-
• Bi³⁺
• OH^-

are measured to be 3.2 x 10^-19 mol/L. This calculation, achieved using stoichiometry and the law of mass action, allows us to find the equilibrium constant \(K'\). It represents the balance point of the reaction according to the equation: \[K' = [\mathrm{C}_{7} \mathrm{H}_{4} \mathrm{O}_{3}{ }^{2-}][\mathrm{Bi}^{3+}][\mathrm{OH}^{-}]\]This value helps chemists predict how the reaction will behave under different conditions, offering insights into chemical processes and their efficiencies.