Question

Challenge The solubility of silver chloride (AgCl) is \(1.86 \times 10^{-4} \mathrm{g} / 100 \mathrm{g}\) of \(\mathrm{H}_{2} \mathrm{O}\) at 298 \(\mathrm{K}\) . Calculate the \(K_{\mathrm{sp}}\) for AgCl.

Short answer

The \(K_{sp}\) value for silver chloride (AgCl) at 298 K is \(1.69 \times 10^{-10}\).

Step-by-step solution

Calculate molar solubility of AgCl

Since the solubility of AgCl is given in grams per 100g of water, let's convert it into molarity. We know that: Solubility in g/100g of H2O = \(1.86 \times 10^{-4}\) g/100g of H2O To find the molar solubility, we will divide the solubility in grams by the molar mass of AgCl (which is the sum of molar mass of Ag and Cl). Molar mass of Ag = 107.87 g/mol Molar mass of Cl = 35.45 g/mol Molar mass of AgCl = 107.87 g/mol + 35.45 g/mol = 143.32 g/mol Therefore, molar solubility of AgCl = \(\frac{1.86 \times 10^{-4}\, \mathrm{g}}{143.32\, \mathrm{g/mol}}\).

Write the balanced dissolution equation for AgCl

The balanced dissolution equation for AgCl in water is: AgCl (s) <-> Ag+ (aq) + Cl- (aq) From the balanced equation, we see that for every one mole of AgCl, one mole of Ag+ and Cl- ion is formed. So the molar solubility of Ag+ and Cl- ions would be the same as that of AgCl.

Write the expression for Ksp of AgCl

The expression for \(K_{sp}\) for AgCl can be written as: \(K_{sp} = [\mathrm{Ag}^+] [\mathrm{Cl}^-]\) Now we will substitute the molar solubility value we calculated in step 1: Molar solubility of AgCl = \(\frac{1.86 \times 10^{-4}\, \mathrm{g}}{143.32\, \mathrm{g/mol}} = S\) mol/L So, \(K_{sp} = [S] [S] = S^2\)

Calculate Ksp

Now, we can plug in the value of 'S' and find the \(K_{sp}\): \(K_{sp} = \left(\frac{1.86 \times 10^{-4}\, \mathrm{g}}{143.32\, \mathrm{g/mol}}\right)^2\) Solve for Ksp: Ksp = \(1.69 \times 10^{-10}\) Therefore, the \(K_{sp}\) value for AgCl at 298 K is \(1.69 \times 10^{-10}\).

Key concepts

Solubility Product Constant

The Solubility Product Constant, denoted as K_sp, is a fundamental principle in chemistry that measures the maximum product of the molar concentrations of the ions in a saturated solution of a sparingly soluble salt. In simple terms, it tells us how much of a solid substance can dissolve in a solvent at a given temperature to reach equilibrium, without additional solid precipitating out. Each salt has its unique K_sp value.

For example, in the case of silver chloride (AgCl), the K_sp represents the concentration of silver ions (Ag⁺) times the concentration of chloride ions (Cl^-) at the point where the solution is saturated. This K_sp value is crucial for predicting the solubility of AgCl under various conditions and is particularly important in fields like analytical chemistry, where precipitation reactions are common.

Molar Solubility

Molar Solubility refers to the number of moles of a solute that can dissolve in a liter of solution before the solution becomes saturated. It is often expressed in mol/L and is directly linked to the solubility product constant. When a slightly soluble ionic compound dissolves, it dissociates into its ions, and the molar solubility can then be used to determine the concentration of these ions in a saturated solution.

To calculate the molar solubility, one must know the amount of the substance that can dissolve in a given volume of solvent. In practical terms, like with silver chloride (AgCl), finding the molar solubility involves converting the solubility from grams per liter to moles per liter by dividing by the molar mass, as shown in the provided exercise. This measure is extremely useful in predicting how much of the substance will dissolve under certain conditions and is key to tasks such as designing a proper chemical reaction setup.

Dissolution Equation

The Dissolution Equation describes how a solid solute dissolves in a solvent to form its constituent ions. It represents the process in a balanced chemical equation format. For ionic compounds like AgCl, the equation would be AgCl (s) <-> Ag⁺ (aq) + Cl^- (aq). This indicates that solid silver chloride separates into silver ions and chloride ions when it dissolves in water.

Understanding and writing the correct dissolution equation is crucial because it allows us to determine the stoichiometry of the dissolved species. This stoichiometric relationship is necessary for calculating the solubility product constant K_sp. Moreover, it sets the stage for applying the principles of chemical equilibrium to the dissolution process, thus linking these concepts together for a comprehensive analysis of solubility behavior.

Chemical Equilibrium

Chemical Equilibrium is the state in which the rate of the forward reaction equals the rate of the reverse reaction, meaning that the concentrations of reactants and products remain constant over time. It is a dynamic balance rather than a static one, as the reactions continue to occur, but with no net change in concentration. In the context of solubility, equilibrium is reached when the dissolution of a solute and the precipitation of a solute occur at the same rate.

For sparingly soluble salts like AgCl, it takes a specific combination of ion concentrations to achieve this balance, which is described by the solubility product constant, K_sp. Understanding chemical equilibrium helps us not only in predicting whether a precipitate will form in a reaction but also in manipulating reaction conditions through Le Châtelier's principle. For instance, changing the concentration of one ion may shift the equilibrium, thereby affecting the solubility of the compound.