Problem 47 - Chemistry An Atoms First Approach

Question

What mass of ZnS \(\left(K_{\mathrm{sp}}=2.5 \times 10^{-22}\right)\) will dissolve in \(300.0 \mathrm{mL}\) of \(0.050 \mathrm{M} \mathrm{Zn}\left(\mathrm{NO}_{3}\right)_{2} ?\) Ignore the basic properties of \(\mathrm{S}^{2-}\).

Step-by-Step Solution

Answer

The mass of ZnS that will dissolve in \(300.0 \mathrm{mL}\) of \(0.050 \mathrm{M} \mathrm{Zn}(\mathrm{NO}_{3})_{2}\) is approximately \(1.5 \times 10^{-19} \mathrm{g}\).

Step 1: Write the Ksp expression and replace the given values

For the reaction ZnS <=> Zn2+ + S2-, the Ksp expression is given by: Ksp = [Zn2+][S^2-] We are given Ksp = 2.5 × 10^-22 and [Zn2+] = 0.050 M, so we have: 2.5 × 10^-22 = (0.050)[S^2-]

Step 2: Solve for the concentration of S^2- ions

Now we need to find the concentration of S^2-, so we simply solve for [S^2-]: [S^2-] = (2.5 × 10^-22) / (0.050) [S^2-] = 5.0 × 10^-21 M

Step 3: Calculate the amount of moles of S^2- ions dissolving in the solution

To find the amount of moles of S^2- ions, we will use the formula: moles(S^2-) = [S^2-] × volume of solution (in L) We are given volume as 300.0 mL, which is equal to 0.300 L. So, moles(S^2-) = (5.0 × 10^-21 M)(0.300 L) moles(S^2-) = 1.5 × 10^-21 mol

Step 4: Calculate the mass of ZnS dissolved in solution

To find the mass of ZnS that will dissolve in the solution, we first need the molar mass of ZnS. Molecular weight (ZnS) = atomic weight (Zn) + atomic weight (S) = 65.38 g/mol + 32.06 g/mol = 97.44 g/mol Now, we multiply the moles of S^2- ions (which will dissolve one mole of ZnS) by the molar mass of ZnS to find the mass: mass(ZnS) = moles(S^2-) × molar mass (ZnS) mass(ZnS) = (1.5 × 10^-21 mol)(97.44 g/mol) mass(ZnS) = approximately 1.5 × 10^-19 g So, the mass of ZnS that will dissolve in 300.0 mL of 0.050 M Zn(NO3)2 is approximately 1.5 × 10^-19 g.

Key Concepts

Dissolution

The concept of dissolution involves the process by which a solute dissolves in a solvent, leading to a formation of a solution. In this example, we focus on zinc sulfide (ZnS) dissolving in a solution to establish a chemical equilibrium. Dissolution starts when the crystalline structure of the solute breaks down and its individual ions are surrounded by solvent molecules. This occurs until a balance is reached between the dissolved ions in solution and the undissolved solid. For ZnS, the equation representing the dissolution process can be written as: ZnS (solid) ⇌ Zn²⁺ (aq) + S²⁻ (aq). The equilibrium state is important because not all the solute dissolves; some amount will always remain as a solid. This equilibrium is quantified using the solubility product constant, represented as Ksp. This constant helps predict how much ZnS can dissolve by considering the concentrations of Zn²⁺ and S²⁻ ions in a saturated solution.

Molar Mass

The term molar mass refers to the mass of one mole of a substance, usually expressed in grams per mole (g/mol). It is a crucial factor when converting between moles and grams. For any compound, the molar mass is computed by summing the atomic masses of all the atoms in its formula. To compute the molar mass of zinc sulfide (ZnS), we add the atomic mass of zinc (Zn) and sulfur (S): - Atomic mass of Zn is 65.38 g/mol. - Atomic mass of S is 32.06 g/mol. Therefore, the molar mass of ZnS is calculated as: Molar mass of ZnS = 65.38 g/mol + 32.06 g/mol = 97.44 g/mol. Knowing the molar mass allows us to convert the number of moles of a substance into grams, which is necessary for practical applications where mass measurements are usually needed.

Concentration Calculation

Concentration calculations are vital in determining the quantities of ions present in a solution. Concentration is typically expressed in molarity (M), which is moles of solute per liter of solution. In this exercise, to find the concentration of sulfide ions ( [S²⁻] ) in the solution, we used the given solubility product constant (Ksp) and the concentration of zinc ions ( [Zn²⁺] ): Ksp = [Zn²⁺][S²⁻] Given Ksp = 2.5 × 10⁻²² and [Zn²⁺] = 0.050 M, the concentration of S²⁻ is found by rearranging the formula: [S²⁻] = Ksp / [Zn²⁺] = (2.5 × 10⁻²²) / 0.050 = 5.0 × 10⁻²¹ M. Once the ion concentration is known, we can calculate the total moles of the ion by multiplying with the volume of the solution in liters: Moles of S²⁻ = [S²⁻] × Volume = (5.0 × 10⁻²¹) M × (0.300 L) = 1.5 × 10⁻²¹ mol. Knowing the moles is essential for determining the mass of solute that dissolves in the solution, closing the loop from dissolution to mass.