Question

Calculate the mass of manganese hydroxide present in \(1300 \mathrm{~mL}\) of a saturated manganese hydroxide solution. For \(\mathrm{Mn}(\mathrm{OH})_{2}, K_{\mathrm{sp}}=\) \(2.0 \times 10^{-13}\) 89\. On a hot day, a \(200.0-\mathrm{mL}\) sample of a saturated solution of \(\mathrm{PbI}_{2}\) was allowed to evaporate until dry. If \(240 \mathrm{mg}\) of solid \(\mathrm{PbI}_{2}\) was collected after evaporation was complete, calculate the \(K_{\mathrm{sp}}\) value for \(\mathrm{PbI}_{2}\) on this hot day.

Short answer

The Ksp value for PbI2 on this hot day is approximately \(7.04 \times 10^{-8}\).

Step-by-step solution

Calculate moles of PbI2

First, we need to convert the mass of PbI2 (240 mg) into moles. We can do this using the molar mass of PbI2. The molar mass of PbI2 is approximately \(461.0 \mathrm{~g/mol}\) (207.2 g/mol for Pb and 2 × 126.9 g/mol for I). So, the moles of PbI2 can be calculated as follows: \[\text{moles of PbI}_{2} = \frac{\text{mass of PbI}_{2}}{\text{molar mass of PbI}_{2}} = \frac{240 \times 10^{-3}\text{ g}}{461.0 \text{ g/mol}} = 5.2 \times 10^{-4}\text{ mol}\]

Determine the molarity of PbI2

Next, we need to calculate the concentration of PbI2 in the original solution. Since the volume of the solution is given as 200 mL, we can calculate the molarity as follows: \[\text{Molarity of PbI}_{2} = \frac{\text{moles of PbI}_{2}}{\text{volume of solution in L}} = \frac{5.2 \times 10^{-4}\text{ mol}}{0.2\text{ L}} = 2.6 \times 10^{-3}\text{ M}\]

Find the concentrations of Pb2+ and I- ions

In a saturated solution of PbI2, the dissociation reaction can be written as: \[\mathrm{PbI}_{2}(s) \rightleftharpoons \mathrm{Pb}^{2+}(aq) + 2\mathrm{I}^-(aq)\] When 1 mole of PbI2 dissociates, it yields 1 mole of Pb2+ and 2 moles of I-. Since the molarity of PbI2 is \(2.6 \times 10^{-3} \text{ M}\), the concentrations of the ions at equilibrium can be determined: \[ [\mathrm{Pb}^{2+}] = 2.6 \times 10^{-3}\text{ M}\] \[ [\mathrm{I}^-] = 2(2.6 \times 10^{-3}\text{ M}) = 5.2 \times 10^{-3}\text{ M}\]

Calculate the Ksp of PbI2

Now we have the concentrations of Pb2+ and I- ions at equilibrium. We can use the Ksp expression to calculate Ksp: \[K_{sp} = [\mathrm{Pb}^{2+}][\mathrm{I}^-]^2\] Plugging the concentrations into the expression: \[K_{sp} = (2.6 \times 10^{-3})(5.2 \times 10^{-3})^2 = 7.04 \times 10^{-8}\] The Ksp value for PbI2 on this hot day is approximately \(7.04 \times 10^{-8}\).

Key concepts

Manganese Hydroxide

Manganese hydroxide, with the chemical formula \(\text{Mn(OH)}_2\), is an interesting compound that forms in certain types of aqueous reactions. This compound often manifests as a result of the interaction between manganese salts and hydroxide ions. In chemistry labs, it's known for its relatively low solubility in water.
One of the key aspects when dealing with manganese hydroxide is its solubility product constant, or \(K_{\text{sp}}\). This constant quantifies the solubility of a compound and is crucial when calculating how much of the material can dissolve in a solution to achieve saturation. For example, in our case, the \(K_{\text{sp}}\) of manganese hydroxide is given as \(2.0 \times 10^{-13}\). Understanding the \(K_{\text{sp}}\) and how it relates to the compound's solubility is essential in predicting and calculating the concentrations of different ions in a saturated solution.

Saturated Solution

A saturated solution is one where a solute is in equilibrium with its dissolved ions in the solution. It is a crucial concept in the field of solubility and solution stoichiometry. When a solution is saturated, it means no additional solute can be dissolved at a given temperature and pressure.
For compounds like manganese hydroxide, reaching saturation is a balance point where any additional solid will remain undissolved in the container. This equilibrium helps to determine the solubility and ultimately the \(K_{\text{sp}}\) value, which is vital for chemists who calculate concentrations in such systems. In this equilibrium state, the solution maintains a constant concentration of dissolved ions, even as some solid continues to dissolve or precipitate simultaneously.

Molarity

Molarity is a common measure of the concentration of a solute in a solution. It's an expression of the number of moles of solute present in one liter of solution, symbolized as \(M\). Calculating molarity is fundamental in chemistry, especially when discussing saturated solutions and calculating their \(K_{\text{sp}}\) values.
To calculate molarity, the amount of solute is divided by the volume of the solution in liters. This measure helps chemists understand how concentrated a solution is. For instance, in a saturated solution of \(\text{PbI}_2\), if we know the number of moles of \(\text{PbI}_2\) and the solution’s volume, we can easily calculate its molarity, which subsequently helps in determining ion concentrations at equilibrium.

PbI2 Dissociation

Dissociation is the process in which an ionic compound separates into its individual ions in solution. \(\text{PbI}_2\) (lead iodide) is an example of such a compound that dissociates into lead ions \((\text{Pb}^{2+})\) and iodide ions \((\text{I}^-)\). The dissociation of \(\text{PbI}_2\) can be expressed as:
\[ \text{PbI}_2(s) \rightleftharpoons \text{Pb}^{2+}(aq) + 2\text{I}^-(aq) \]
When \(\text{PbI}_2\) dissolves in water, it does so to a small extent due to low solubility, but this small amount still establishes an equilibrium between solid \(\text{PbI}_2\) and its ions in solution. Knowing the dissociation stoichiometry helps us calculate the molarity of ions in solution, keeping in mind the balanced equation provides the relationship between the moles of lead and iodide ions produced. This relationship is crucial for using the \(K_{\text{sp}}\) expression to find the solubility product constant and understand the solubility limits of \(\text{PbI}_2\).