Question

Calcium sulfate, \(\mathrm{CaSO}_{4}\), is only soluble in water to the extent of approximately 2.05 \(\mathrm{g} / \mathrm{L}\) at \(25 \quad \mathrm{C} .\) Calculate \(K_{\mathrm{sp}}\) for calcium sulfate at \(25 \mathrm{C}\)

Short answer

The solubility product constant (Ksp) for calcium sulfate at 25°C is approximately \(2.27 × 10^{-4}\).

Step-by-step solution

Calculate the molar mass of calcium sulfate

First, we need to calculate the molar mass of CaSO4: Calcium (Ca): 40.08 g/mol Sulfur (S): 32.07 g/mol Oxygen (O): 16.00 g/mol (but there are 4 oxygen atoms, so multiplied by 4) Molar mass of CaSO4 = 40.08 g/mol (Ca) + 32.07 g/mol (S) + 4 × 16.00 g/mol (O) = 136.14 g/mol

Calculate the number of moles of CaSO4 dissolved

Next, we need to find the number of moles of CaSO4 dissolved in 1 L of water. To do this, we will use the mass of CaSO4 (2.05 g) and divide it by the molar mass of CaSO4 (136.14 g/mol). Moles of CaSO4 = (2.05 g CaSO4) / (136.14 g/mol CaSO4) = 0.01506 mol CaSO4

Calculate the concentration of ions

Since the reaction of CaSO4 dissociating in water is 1:1 for Ca2+ and SO42- ions, the concentration of ions in the solution can be found using the number of moles of CaSO4. Since we are given 1 L of water as the volume, we can directly find the concentration of ions. Concentration of Ca2+ ions = 0.01506 mol / 1 L = 0.01506 M Concentration of SO42- ions = 0.01506 mol / 1 L = 0.01506 M

Calculate Ksp

Now we can plug in the values of Ca2+ ions and SO42- ions into the Ksp expression to find the solubility product constant for calcium sulfate. Ksp = [0.01506 M Ca2+][0.01506 M SO42-] = (0.01506)^2 = 2.27 × 10^(-4) The value of Ksp for calcium sulfate at 25°C is approximately 2.27 × 10^(-4).

Key concepts

Solubility Product

In chemistry, the solubility product constant, often symbolized as \(K_{sp}\), is a crucial value used to describe how much of a compound can dissolve in water. It represents the extent to which a salt can dissolve to form its constituent ions in a saturated solution. For any ionic compound, the expression for \(K_{sp}\) is derived from its dissociation equilibrium.Consider the dissolution of calcium sulfate, \(\text{CaSO}_4\), disassociating into ions in water:\[ \text{CaSO}_4 (s) \rightleftharpoons \text{Ca}^{2+} (aq) + \text{SO}_4^{2-} (aq) \]The equilibrium expression for the solubility product is given by:\[ K_{sp} = [\text{Ca}^{2+}] [\text{SO}_4^{2-}] \]This shows that the \(K_{sp}\) is dependent on the product of the molar concentrations of the ions. The smaller the \(K_{sp}\), the less soluble the compound is in water. It's vital in predicting whether a precipitate will form under a given set of conditions in a solution.

Calcium Sulfate Solubility

Calcium sulfate is a compound characterized by limited solubility in water. At 25°C, its solubility reads as approximately 2.05 g/L. This solubility can be used to calculate the molar solubility, which can further help in determining the \(K_{sp}\).The process begins with the equation that describes the dissociation of calcium sulfate in water:\[ \text{CaSO}_4 \rightarrow \text{Ca}^{2+} + \text{SO}_4^{2-} \]Given its solubility is 2.05 g/L, it indicates that this much quantity of calcium sulfate can dissolve in one liter of water. Knowing this helps us to work out the concentration of ions in the solution, contributing directly to the \(K_{sp}\) calculation.

Molar Mass Calculation

To find out how much of a substance like calcium sulfate is in a solution, we first need to calculate its molar mass. The molar mass is the mass of one mole of a compound. Follow these steps:

• Identify the elements in the compound and their respective atomic masses. In calcium sulfate (\(\text{CaSO}_4\)), they are calcium (Ca), sulfur (S), and oxygen (O).
• The atomic masses are approximately 40.08 g/mol for Ca, 32.07 g/mol for S, and 16.00 g/mol for O.
• Multiply the atomic mass of oxygen by 4 since there are four oxygen atoms in \(\text{CaSO}_4\).
• Sum these values: 40.08 + 32.07 + (4 \times 16.00) = 136.14 g/mol.

With the molar mass calculated, it becomes straightforward to relate the solubility (in g/L) to the number of moles (in mol/L). This step is pivotal for further calculations of ionic concentrations.

Ionic Concentration Calculation

Calculating ionic concentration is essential to finding out how much each ion contributes to the overall solubility product. For calcium sulfate, upon dissolving, there's a one-to-one stoichiometric relationship between \(\text{Ca}^{2+}\) ions and \(\text{SO}_4^{2-}\) ions. Here's how it works:

• Use the molar solubility, found by dividing the solubility in grams per liter by the molar mass, to calculate the concentration of \(\text{Ca}^{2+}\) and \(\text{SO}_4^{2-}\).
• From the example: with a solubility of 2.05 g/L, converting this to moles using the molar mass of 136.14 g/mol gives approximately 0.01506 moles per liter.
• This value represents both \([\text{Ca}^{2+}]\) and \([\text{SO}_4^{2-}]\) in the solution.

Calculating each ion's concentration allows us to further compute the solubility product \(K_{sp}\), vital for understanding the precipitation potential of calcium sulfate in different environmental conditions.