Problem 68 - Chemistry: The Central Science in SI Units, Global Edition

Question

Using the value of \(K_{s p}\) for \(\mathrm{Ag}_{2} \mathrm{~S}, K_{a 1}\) and \(\mathrm{K}_{a 2}\) for \(\mathrm{H}_{2} \mathrm{~S},\) and \(K_{f}=1.1 \times 10^{5}\) for \(\mathrm{AgCl}_{2}^{-},\) calculate the equilibrium constant for the following reaction: $$ \mathrm{Ag}_{2} \mathrm{~S}(s)+4 \mathrm{Cl}^{-}(a q)+2 \mathrm{H}^{+}(a q) \rightleftharpoons 2 \mathrm{AgCl}_{2}^{-}(a q)+\mathrm{H}_{2} \mathrm{~S}(a q) $$

Step-by-Step Solution

Answer

The overall equilibrium constant for the given reaction is \(K = K_{sp}\cdot K_{a1}\cdot K_{a2}\cdot K_{f}^2\). Plugging the given values for \(K_{sp}\), \(K_{a1}\), \(K_{a2}\), and \(K_f\), we can calculate the value of \(K\).

Step 1: Write down the reactions with their equilibrium constants

The three separate reactions with their given equilibrium constants are: 1. Dissolution of silver sulfide (\(Ag_2S\)): \[Ag_2S(s) \rightleftharpoons 2Ag^+(aq) + S^{2-}(aq)\] Given: \(K_{sp} = [Ag^+]^2[S^{2-}]\) 2. The first dissociation of hydrogen sulfide (\(H_2S\)): \[H_2S(aq) \rightleftharpoons H^+(aq) + HS^-(aq)\] Given: \(K_{a1} = [H^+][HS^-]/[H_2S]\) 3. The second dissociation of hydrogen sulfide: \[HS^-(aq) \rightleftharpoons H^+(aq) + S^{2-}(aq)\] Given: \(K_{a2} = [H^+][S^{2-}]/[HS^-]\) 4. Formation of the silver chloride complex (\(AgCl_2^-\)): \[Ag^+(aq) + 2Cl^-(aq) \rightleftharpoons AgCl_2^-(aq)\] Given: \(K_f = [AgCl_2^-]/([Ag^+][Cl^-]^2)\)

Step 2: Add the equations and find the overall equation

To obtain the overall balanced equation, we will sum up the individual reactions and cancel out the species that appear on both sides of the overall equation: \[Ag_2S(s) + 4Cl^-(aq) + 2H^+(aq) \rightleftharpoons 2AgCl_2^-(aq) + H_2S(aq)\]

Step 3: Calculate the overall equilibrium constant

Since we have the individual equilibrium constants, we can now find the overall equilibrium constant for the reaction. When adding the individual reactions, we multiply their corresponding equilibrium constants. Overall equilibrium constant (\(K\)): \[K = K_{sp}\cdot K_{a1}\cdot K_{a2}\cdot K_{f}\cdot K_f\] (Note that we multiply \(K_f\) twice, as there are two moles of \(AgCl_2^-\) in the overall equation) Now, you can plug in the given values of \(K_{sp}\), \(K_{a1}\), \(K_{a2}\), and \(K_f\) into the equation above and solve for the overall equilibrium constant, \(K\).

Key Concepts

Chemical Reactions

In chemistry, understanding chemical reactions is like unraveling a complex puzzle. A chemical reaction involves the transformation of one or more substances into a different set of substances. These transformations occur when chemical bonds are broken and new ones are formed. It's the backbone of everything from baking a cake to rust forming on iron. The essence of chemical reactions involves the reactants and products, along with the rearrangement of atoms.
In a balanced chemical equation, we ensure that the number of each type of atom is the same on both sides of the equation. This conformity aligns with the Law of Conservation of Mass, asserting that matter is neither created nor destroyed in a chemical reaction.

• Reactants: The starting substances.
• Products: The substances that are formed as the reaction progresses.

Equilibrium is a state in chemical reactions where the rate of the forward reaction equals the rate of the reverse reaction. The equilibrium constant, denoted as \( K \), describes the ratio of concentrations of products to reactants when the system is at equilibrium. Each reaction has its own specific equilibrium constant, except for reactions where no change occurs.

Silver Sulfide

Silver sulfide \( (Ag_2S) \) is a compound that forms when silver reacts with sulfur. It's responsible for the black tarnish often seen on silverware. In chemistry, we often need to dissolve silver sulfide in reactions, which isn't always straightforward.
The process of dissolving silver sulfide involves breaking it down into its ions in an aqueous solution. The reaction for this process is:

• \(Ag_2S(s) \rightleftharpoons 2Ag^+(aq) + S^{2-}(aq)\)

This reaction has its equilibrium constant, \( K_{sp} \), which represents the solubility product of the compound. It is particularly crucial in quantifying how much of the compound will dissolve in a given amount of solvent. Interestingly, \( K_{sp} \) helps predict whether a precipitate will form during a reaction or if the compound will remain in solution.
Silver sulfide's low solubility is why some silver objects tend to corrode when exposed to air, as they react slowly but continuously with sulfur compounds in the environment.

Hydrogen Sulfide Dissociation

Hydrogen sulfide \( (H_2S) \) is a gas renowned for its rotten egg smell, which arises from its typical occurrence in sewers and volcanic gases. In an aqueous environment, hydrogen sulfide can dissociate into ions in two main steps. The dissociation occurs as follows:

• First dissociation: \(H_2S(aq) \rightleftharpoons H^+(aq) + HS^-(aq)\)
• Second dissociation: \(HS^-(aq) \rightleftharpoons H^+(aq) + S^{2-}(aq)\)

Each step has its equilibrium constant, \( K_{a1} \) and \( K_{a2} \) respectively, which measure the extent of the dissociation in water. The first dissociation is usually more significant than the second, as \( H_2S \) doesn't fully dissociate in most environments.This dual-step dissociation is pivotal when calculating the concentration of sulfide ions in solutions, which is critical in areas such as wastewater treatment and understanding the biochemical pathways where \( H_2S \) is involved.

Silver Chloride Complex Formation

The formation of silver chloride complexes is an interesting chemical phenomenon. In solutions containing chloride ions, \( Ag^+ \) ions can form a complex ion known as \( AgCl_2^- \). This process is characterized by the reaction:

• \(Ag^+(aq) + 2Cl^-(aq) \rightleftharpoons AgCl_2^-(aq)\)

The equilibrium constant for this reaction is denoted as \( K_f \). It indicates the stability of the complex formed and tells us how the silver and chloride ion interactions are favored.This complex formation is crucial in photographic processing and analytical chemistry where the solubility of silver compounds in various conditions needs precision control. By manipulating \( K_f \), chemists are able to predict and control the precipitation and dissolution of such compounds in diverse chemical processes.