Question

The active ingredient of Pepto-Bismol is the compound bismuth subsalicylate, which undergoes the following dissociation when added to water: $$\begin{aligned} \mathrm{C}_{7} \mathrm{H}_{5} \mathrm{BiO}_{4}(s)+\mathrm{H}_{2} \mathrm{O}(l) \rightleftharpoons \mathrm{C}_{7} \mathrm{H}_{4} \mathrm{O}_{3}^{2-}(a q) \\ +\mathrm{Bi}^{3+}(a q)+\mathrm{OH}^{-}(a q) & K=? \end{aligned}$$ 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 (K) for the dissociation reaction of bismuth subsalicylate is approximately \(3.28 \times 10^{-57}\).

Step-by-step solution

1. Write down the balanced chemical equation

The balanced chemical equation for the dissociation reaction is given by: \[\mathrm{C}_{7}\mathrm{H}_{5}\mathrm{BiO}_{4}(s) + \mathrm{H}_{2}\mathrm{O}(l) \rightleftharpoons \mathrm{C}_{7}\mathrm{H}_{4}\mathrm{O}_{3}^{2-}(aq) + \mathrm{Bi}^{3+}(aq) + \mathrm{OH}^{-}(aq)\]

2. Write down the initial concentrations of the reactants and products

Since initially, there is only bismuth subsalicylate solid and water, the initial concentrations can be written as: [\(\mathrm{C}_{7}\mathrm{H}_{4}\mathrm{O}_{3}^{2-}\)] = 0 [\(\mathrm{Bi}^{3+}\)] = 0 [\(\mathrm{OH}^{-}\)] = 0

3. Write down the changes in concentrations at equilibrium

At equilibrium, a maximum of \(3.2 \times 10^{-19}\,\text{mol}\,/\,\text{L}\) of bismuth subsalicylate reacts, so the equilibrium concentrations are: [\(\mathrm{C}_{7}\mathrm{H}_{4}\mathrm{O}_{3}^{2-}\)] = + \(3.2 \times 10^{-19}\,\text{mol}\,/\,\text{L}\) [\(\mathrm{Bi}^{3+}\)] = + \(3.2 \times 10^{-19}\,\text{mol}\,/\,\text{L}\) [\(\mathrm{OH}^{-}\)] = + \(3.2 \times 10^{-19}\,\text{mol}\,/\,\text{L}\)

4. Write the equilibrium constant expression for the reaction

The equilibrium constant expression for the given reaction is: \[K = \frac{[\mathrm{C}_{7}\mathrm{H}_{4}\mathrm{O}_{3}^{2-}][\mathrm{Bi}^{3+}][\mathrm{OH}^{-}]}{[\mathrm{C}_{7}\mathrm{H}_{5}\mathrm{BiO}_{4}]} \] Since the bismuth subsalicylate is a solid and the water is a liquid, their concentrations do not appear in the equilibrium constant expression.

5. Calculate the equilibrium constant K

Now, we can substitute the equilibrium concentrations into the equilibrium constant expression and solve for K: \[K = \frac{(3.2 \times 10^{-19})(3.2 \times 10^{-19})(3.2 \times 10^{-19})}{1}\] \[K = (3.2 \times 10^{-19})^3 = 3.2768 \times 10^{-57}\] The equilibrium constant for the given reaction is approximately \(3.28 \times 10^{-57}\).

Key concepts

Chemical Equilibrium

Understanding chemical equilibrium is essential when studying how chemical reactions reach a state of balance. A reaction is said to be at chemical equilibrium when the rate at which the reactants transform into products equals the rate at which the products transform back into reactants. Even though the individual molecules are still dynamically reacting, there is no net change in the concentrations of the substances involved over time.

In the context of our exercise, the dissociation reaction of bismuth subsalicylate in water is reversible. At equilibrium, the amount of bismuth subsalicylate converted into its ions is equal to the amount of ions recombining to form bismuth subsalicylate, maintaining a constant concentration of each species in solution. Achieving equilibrium does not mean the reactants and products are present in equal amounts, but rather that their respective concentrations are stable over time.

Dissociation Reactions

Dissociation reactions are those in which a compound breaks apart into two or more components. These reactions are particularly common among ionic compounds in aqueous solutions. For instance, bismuth subsalicylate dissociates in water into bismuth ions, salicylate ions, and hydroxide ions.

This process is fundamental to many biological and chemical systems. It's important to note that, in these reactions, the solid compound (like bismouth subsalicylate) often dissociates completely, forming ions that are solvated by the water molecules. However, the extent to which a substance dissociates - and therefore, the concentration of ions in solution - will depend on multiple factors, including the nature of the substance and temperature.

Equilibrium Constant Expression

The equilibrium constant expression, symbolized as K, is a mathematical way to express the extent of a chemical reaction at equilibrium. For the reaction involving bismuth subsalicylate, we consider the concentrations of the ions produced in the reaction, as they are in the aqueous state.

The equilibrium constant expression is given by \[K = \frac{[\mathrm{C}_{7}\mathrm{H}_{4}\mathrm{O}_{3}^{2-}][\mathrm{Bi}^{3+}][\mathrm{OH}^{-}]}{[\mathrm{C}_{7}\mathrm{H}_{5}\mathrm{BiO}_{4}]} \]The concentration of solid bismuth subsalicylate or pure water is not included in the expression because their concentrations are constant and don't change the position of equilibrium. Calculating the equilibrium constant provides a numerical value that indicates whether the reactants or products are favored in a reaction when the system is at equilibrium. In this exercise, the calculation resulted in a K value of approximately \(3.28 \times 10^{-57}\), which is a very small number, suggesting that the dissociation of bismuth subsalicylate in water is not favored under the conditions of the experiment.