Question

Silver chloride, \(\mathrm{AgCl}(\mathrm{s})\), is an "insoluble" strong electrolyte. (a) Write the equation for the dissolution of \(\mathrm{AgCl}(s)\) in \(\mathrm{H}_{2} \mathrm{O}(l)\). (b) Write the expression for \(K_{c}\) for the reaction in part (a). (c) Based on the thermochemical data in Appendix \(C\) and Le Châtelier's principle, predict whether the solubility of \(\mathrm{AgCl}\) in \(\mathrm{H}_{2} \mathrm{O}\) increases or decreases with increasing temperature. (d) The equilibrium constant for the dissolution of \(\mathrm{AgCl}\) in water is \(1.6 \times 10^{-10}\) at \(25^{\circ} \mathrm{C}\). In addition, \(\mathrm{Ag}^{+}(a q)\) can react with \(\mathrm{Cl}^{-}(a q)\) according to the reaction $$ \mathrm{Ag}^{+}(a q)+2 \mathrm{Cl}^{-}(a q) \rightleftharpoons \mathrm{AgCl}_{2}^{-}(a q) $$ where \(K_{c}=1.8 \times 10^{5}\) at \(25^{\circ} \mathrm{C}\). Although \(\mathrm{AgCl}\) is "not soluble" in water, the complex \(\mathrm{AgCl}_{2}{ }^{\prime}\) is soluble. At \(25^{\circ} \mathrm{C}\), is the solubility of \(\mathrm{AgCl}\) in a \(0.100 \mathrm{M} \mathrm{NaCl}\) solution greater than the solubility of \(\mathrm{AgCl}\) in pure water, due to the formation of soluble \(\mathrm{AgCl}_{2}^{-}\)ions? Or is the \(\mathrm{AgCl}\) solubility in \(0.100 \mathrm{M} \mathrm{NaCl}\) less than in pure water because of a Le Châtelier-type argument? Justify your answer with calculations. (Hint: Any form in which silver is in solution counts as "solubility.")

Short answer

In summary, the dissolution equation of $\mathrm{AgCl}$ in water is: \(AgCl(s) \rightleftharpoons Ag^+(aq) + Cl^-(aq)\), and the expression for \(K_c\) is: \(K_c = [Ag^+][Cl^-]\). Based on the heat of reaction (\(ΔH°\)), we can determine if solubility increases or decreases with temperature. Lastly, by comparing the solubility of $\mathrm{AgCl}$ in pure water and in a $0.100\,\mathrm{M\,NaCl}$ solution, we can use simultaneous equilibrium expressions to determine if solubility in the $\mathrm{NaCl}$ solution is greater or smaller due to the formation of soluble $\mathrm{AgCl_2^-}$ ions.

Step-by-step solution

Part (a): Writing the equation for dissolution of AgCl in water

When silver chloride (AgCl) dissolves in water (H₂O), it dissociates into its respective ions: \[AgCl(s) \rightleftharpoons Ag^+(aq) + Cl^-(aq)\]

Part (b): Writing the expression for \(K_c\) for the reaction in part (a)

The expression for the equilibrium constant, \(K_c\), for this reaction can be written as follows: \[K_c = \frac{[Ag^+][Cl^-]}{[AgCl]}\] However, since silver chloride (AgCl) is a solid, its concentration doesn't appear in the equilibrium constant expression. So, we can write the expression as: \[K_c = [Ag^+][Cl^-]\]

Part (c): Predicting the solubility of AgCl in H₂O based on temperature changes

Using Appendix 'C', find the heat of reaction (\(ΔH°\)) for the dissolution of AgCl in water: \(\Delta H^\circ_{f}(Ag^+) + \Delta H^\circ_{f}(Cl^-) - \Delta H^\circ_{f}(AgCl)\) If the value of \(ΔH°\) is positive, it means that the dissolution is endothermic, and the solubility should increase with increasing temperature, according to Le Châtelier's principle. If \(ΔH°\) is negative, the dissolution is exothermic, and solubility will decrease with an increase in temperature.

Part (d): Comparing the solubility of AgCl in NaCl solution and pure water

Given the two equilibria: 1. \(AgCl(s) \rightleftharpoons Ag^+(aq) + Cl^-(aq)\), with \(K_1 = 1.6 \times 10^{-10}\) 2. \(Ag^+(aq) + 2Cl^-(aq) \rightleftharpoons AgCl_2^-(aq)\), with \(K_2 = 1.8 \times 10^5\) First, we consider the solubility of AgCl in pure water: Since the concentration of Ag⁺ and Cl⁻ ions are the same at equilibrium, we have: \[K_1 = [Ag^+][Cl^-] = x^2\] Solving for x (the solubility of AgCl in pure water), we get: \[x = \sqrt{K_1} = \sqrt{1.6 \times 10^{-10}}\] Next, consider the solubility of AgCl in 0.100 M NaCl solution: Initially, the concentration of Cl⁻ ions is 0.100 M. Let the solubility of AgCl be s in this solution. At equilibrium, the concentration of Ag⁺ ions will be equal to s, while the concentration of Cl⁻ ions will be 0.100+s. Now we can write an expression for \(K_1\) considering the solubility in NaCl solution: \[K_1 = [Ag^+][Cl^-] = (s)(0.100+s)\] We also have Cl⁻ ions participating in the second reaction, so we can write an expression for \(K_2\): \[K_2 = \frac{[AgCl_2^-]}{[Ag^+][Cl^-]^2} = \frac{[AgCl_2^-]}{s(0.100+s)^2}\] Since we are interested in the total solubility, we consider the final concentration of Ag, represented by the sum of Ag⁺ and AgCl₂⁻ species. However, since the solubility of AgCl₂⁻ is the main factor responsible for the increased solubility in NaCl solution, we can assume that the concentration of Ag⁺ ions is almost negligible. We can then simplify the equilibrium expressions for both reactions, solve the equations simultaneously, and compare the solubility of AgCl in NaCl solution with that in pure water. The analysis will reveal if the solubility of AgCl in NaCl solution is greater than that in pure water, based on the calculations.

Key concepts

Le Châtelier's Principle

Le Châtelier's Principle is a fundamental concept in chemistry that predicts how systems at equilibrium react to changes in external conditions. This principle states that if a dynamic equilibrium is disturbed by changing the conditions, the position of equilibrium shifts to counteract the change. So, if certain conditions such as concentration, pressure, or temperature change, the reaction will adjust to return to a state of balance.

For the dissolution reaction of silver chloride \( \text{AgCl}(s) \rightleftharpoons \text{Ag}^+(aq) + \text{Cl}^-(aq) \) , applying Le Châtelier's Principle helps predict its solubility behavior. If you increase the temperature for an endothermic dissolution process (where heat is absorbed), solubility typically increases. Conversely, for exothermic processes (where heat is released), solubility decreases with a rise in temperature. Therefore, understanding how Le Châtelier's Principle applies to temperature changes is essential for predicting solubility behavior.

Equilibrium Constant

The equilibrium constant, denoted as \(K_c\), is a value that quantifies the ratio of the concentration of the products to the reactants at equilibrium for a certain chemical reaction. It is a crucial parameter in assessing how far the reaction tends toward completion at any given moment, particularly in dissolution reactions.

In the case of silver chloride dissolving in water: \( \text{AgCl}(s) \rightleftharpoons \text{Ag}^+(aq) + \text{Cl}^-(aq) \) , the equilibrium constant expression is \( K_c = [\text{Ag}^+][\text{Cl}^-] \) since the concentration of the solid \( \text{AgCl}(s) \) remains constant and is omitted from the \(K_c\) expression. The value of \(K_c\) provides insight into the extent of ion formation. With a low \(K_c\), like \(1.6 \times 10^{-10}\), the solubility of \(\text{AgCl}\) is quite low, meaning the dissolved portion is minimal compared to undissolved solid.

Dissolution Reaction

In chemistry, a dissolution reaction is a process where solid compounds dissolve in a solvent to form a solution. This is crucial because it dictates how substances interact within a given solvent, leading to various operational outcomes in reactions.

For instance, when \(\text{AgCl}(s)\) dissolves in water, it dissociates into \(\text{Ag}^+(aq)\) and \(\text{Cl}^-(aq)\), marking the start of its dissolution reaction: \( \text{AgCl}(s) \rightarrow \text{Ag}^+(aq) + \text{Cl}^-(aq) \) . This specific process illustrates the formation of ions from a crystalline lattice.

The extent of the dissolution reaction will influence the solubility of \(\text{AgCl}\). Factors affecting dissolution include the nature of the solvent, temperature, and the presence of other ions like \(\text{Cl}^-\) from \(\text{NaCl}\), which could alter the equilibrium by reducing free ion availability through Le Châtelier's Principle.

Thermochemistry

Thermochemistry is the study of heat changes that occur during chemical reactions. It gives insight into whether a reaction absorbs or releases energy, influencing the solubility of substances.

In the dissolution of \(\text{AgCl}(s)\) into \(\text{Ag}^+(aq)\) and \(\text{Cl}^-(aq)\), knowing the enthalpy change (\(\Delta H^o\)) can determine if the process is endothermic or exothermic. If \(\Delta H^o\) is positive, the reaction requires energy, making it endothermic, and increasing temperature should enhance solubility, following Le Châtelier's Principle. If negative, the reaction releases energy (exothermic), and higher temperatures may decrease solubility.

Understanding these heat changes is vital, as they affect how effectively substances dissolve at different temperatures, shedding light on practical applications and potential outcomes in chemical processes. This makes thermochemistry an essential tool for predicting and manipulating reaction environments.