Problem 88 - Chemistry

Question

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

Step-by-Step Solution

Answer

The equilibrium constant for the reaction is approximately \(K = 3.28 \times 10^{-57}\).

Step 1: Write the given equation and reaction

We are given the dissociation reaction of bismuth subsalicylate in water: \[C_7H_5BiO_4(s) + H_2O(l) \rightleftharpoons C_7H_4O_3^{2-}(aq) + Bi^{3+}(aq) + OH^-(aq)\] Our goal is to calculate the equilibrium constant, K, for this reaction, given the maximum amount of bismuth subsalicylate that can react \(3.2 \times 10^{-19} \ \text{mol/L}\).

Step 2: Set up a table with initial concentrations, change, and equilibrium concentrations

Let's set up a table to help us keep track of the initial concentrations, the change in the concentrations during the reaction, and the equilibrium concentrations: | Compound | Initial (M) | Change (M) | Equilibrium (M) | |-------------------|-------------|------------|-----------------| | C_7H_5BiO_4 | - | - | - | | C_7H_4O_3^{2-} | 0 | +x | x | | Bi^{3+} | 0 | +x | x | | OH^- | 0 | +x | x | The initial concentration of \(C_7H_4O_3^{2-}\), \(Bi^{3+}\), and \(OH^-\) are all zero. The reaction proceeds, and we assume that x moles per liter of the compounds have formed at equilibrium.

Step 3: Calculate x

We are given that the maximum amount of bismuth subsalicylate that reacts by this reaction is \(3.2 \times 10^{-19}\ \text{mol/L}\). This implies that at equilibrium, the concentrations of \(C_7H_4O_3^{2-}\), \(Bi^{3+}\), and \(OH^-\) are: \[x = 3.2 \times 10^{-19} \ \text{mol/L}\] Now, we'll define the equilibrium concentrations of the three ions.

Step 4: Write the expression for equilibrium constant, K

The equilibrium constant K for the given reaction can be written as: \[K = \frac{[C_7H_4O_3^{2-}][Bi^{3+}][OH^-]}{1}\] We don't have the concentration for bismuth subsalicylate and water in the expression, because the former is a solid and the latter a liquid.

Step 5: Substitute the equilibrium concentrations and solve for K

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

Key Concepts

Equilibrium Constant

The equilibrium constant \( K \) is a critical concept in chemistry that helps us understand the extent of a chemical reaction at equilibrium.
It is a number that expresses the ratio of the concentrations of products to reactants, each raised to the power of their respective coefficients in the balanced chemical equation.
In this exercise, we are determining the equilibrium constant for the dissociation reaction of bismuth subsalicylate. To find \( K \), we set up an equation using the concentrations of the ions at equilibrium.
Importantly, solids and liquids do not appear in the equilibrium constant expression because their concentrations do not change. They are considered pure substances with constant activity.
Therefore, only the aqueous ions formed in the reaction, such as \( C_7H_4O_3^{2-} \), \( Bi^{3+} \), and \( OH^- \), are included in the expression for \( K \).

Dissociation Reaction

A dissociation reaction involves the breaking up of a compound into smaller chemical species. In our context, bismuth subsalicylate dissociates in water.
This process involves splitting into negatively charged salicylate ions \( C_7H_4O_3^{2-} \), positively charged bismuth ions \( Bi^{3+} \), and hydroxide ions \( OH^- \).
These ions are the discoed forms of the original compound, and their concentrations can be used to calculate important parameters, such as equilibrium constant \( K \).
By understanding the dissociation reaction, we can predict how the substance behaves in solution, which is vital for applications like medicine or materials science.
Dissociation reactions are fundamental in chemistry because they describe how ionic compounds separate into their respective ions, facilitating reactions and biological processes.

Bismuth Subsalicylate

Bismuth subsalicylate, the active ingredient in medications like Pepto-Bismol, serves multiple purposes, including as an antacid and mild antibiotic.
Upon addition to water, this compound undergoes dissociation, which influences its effectiveness in soothing stomach discomfort.
When bismuth subsalicylate is dissolved, it produces ions that interact with pathogens or excess acidity in the stomach.
Understanding its chemical behavior helps pharmacists and chemists tailor therapies for digestive issues.
This compound's dissociation reaction is particularly interesting due to its low equilibrium constant, indicating a limited tendency to dissociate in water.
The reacted concentration of \(3.2 \times 10^{-19} \ ext{mol/L}\) suggests that only a tiny fraction of bismuth subsalicylate actually dissociates, which might explain its specific therapeutic effects.

Concentration Calculation

In chemistry, calculating concentrations of various species at equilibrium is crucial for understanding reaction dynamics.
Through concentration calculation, we determine how many moles of each product form per liter as the reaction progresses to equilibrium.
In the given exercise, knowing the initial and equilibrium concentrations allows the calculation of the equilibrium constant \( K \).
Since the maximum concentration of the dissociated product is \(3.2 \times 10^{-19} \ ext{mol/L}\), all ions form in equal amounts due to the stoichiometry of the reaction.
This means that, at equilibrium, the concentration of each product ion \( C_7H_4O_3^{2-} \), \( Bi^{3+} \), and \( OH^- \) becomes \( x = 3.2 \times 10^{-19} \ ext{mol/L}\).
Being able to calculate these concentrations accurately is essential for predicting how substances interact in solution, thereby influencing experimental and industrial applications in chemistry.