Question

Find the minimum volume of \(\mathrm{H}_{2} \mathrm{O}\) needed to dissolve \(1 \mathrm{~g} \mathrm{CaSO}_{4}\) at \(248 \mathrm{~K}\left[\mathrm{~K}_{\mathrm{SP}}\right.\) of \(\left.\mathrm{CaSo}_{4}^{2}=9.1 \times 10^{-6}\right]\) : (a) \(2.43 \mathrm{~L}\) (b) \(1.215 \mathrm{~L}\) (c) \(4.86 \mathrm{~L}\) (d) \(24.3 \mathrm{~L}\)

Short answer

The minimum volume is 2.43 L, so the answer is (a) 2.43 L.

Step-by-step solution

Calculate Molar Mass of \(\mathrm{CaSO}_{4}\)

To find out how much water is needed to dissolve the compound, first calculate the molar mass of \(\mathrm{CaSO}_{4}\). This can be done by adding the atomic masses of calcium (40.08 g/mol), sulfur (32.07 g/mol), and four oxygens (4 x 16.00 g/mol). Thus, molar mass = 40.08 + 32.07 + 64.00 = 136.15 g/mol.

Calculate Moles of \(\mathrm{CaSO}_{4}\)

Next, calculate the number of moles in 1 g of \(\mathrm{CaSO}_{4}\) using its molar mass. The formula to find moles from mass is \( \text{moles} = \frac{\text{mass}}{\text{molar mass}} \). Thus, \( \frac{1 \text{ g}}{136.15 \text{ g/mol}} \approx 0.00734 \text{ moles} \).

Determine Solubility from \(K_{\text{sp}}\)

The solubility product \(K_{\text{sp}}\) for \(\mathrm{CaSO}_{4}\) is given as \(9.1 \times 10^{-6}\). Since CaSO4 dissociates into Ca and SO4 ions in a 1:1 ratio, \(s^2 = 9.1 \times 10^{-6}\), where \(s\) is the solubility in mol/L. Solving for \(s\), \(s = \sqrt{9.1 \times 10^{-6}} \approx 0.003015 \text{ mol/L} \).

Calculate Volume of Water Needed

To find the minimum volume of water needed to dissolve the \(\mathrm{CaSO}_{4}\), use the formula \(\text{volume} = \frac{\text{moles}}{\text{solubility}}\). Therefore, \( \frac{0.00734 \text{ moles}}{0.003015 \text{ mol/L}} \approx 2.43 \text{ L}\).

Identify the Correct Answer

From the calculated volume, the minimum volume of water needed is approximately \(2.43 \text{ L}\). Thus, the correct choice is (a) \(2.43 \text{ L}\).

Key concepts

Calcium Sulfate

Calcium sulfate (7, 9) is a chemical compound comprising calcium, sulfur, and oxygen. It is commonly found in nature in its hydrated forms as gypsum (7323) and as the anhydrous form used in this exercise. Calcium sulfate is often used in industries for making plaster, as a drying agent, and in educational experiments related to solubility and chemical equilibria. It is known for its relatively low solubility in water compared to other sulfate salts, which makes it an interesting subject of study when exploring dissolution processes. In its purest form, calcium sulfate crystallizes with a regular lattice structure that involves tight packing of calcium and sulfate ions. This crystalline arrangement contributes to its low solubility because the energy required to break these bonds exceeds the typical interactions it forms in water.

Molar Mass Calculation

Molar mass is a critical concept in chemistry, especially when dealing with reactions and dissolutions. It is the mass of one mole of substance and is calculated by summing the atomic masses of its constituent elements, each multiplied by their respective subscript in the chemical formula. For calcium sulfate, 79, the calculation is:

• Calcium (7) = 40.08 g/mol
• Sulfur (9) = 32.07 g/mol
• Oxygen9 (3 4) = 64.00 g/mol (16.00 g/mol each)

Thus, the total molar mass is 136.15 g/mol. Understanding molar mass is crucial for connecting mass measurements with moles, which is essential for quantifying substances in chemical reactions and solutions.

Chemical Solutions

Chemical solutions are homogeneous mixtures composed of solutes dissolved in solvents. In the context of this exercise, water acts as the solvent, and calcium sulfate as the solute. When creating a solution, solutes are dispersed at the molecular or ionic level throughout the solvent, resulting in a uniform phase.
Solubility, a key property of chemical solutions, describes the maximum concentration of solute that can be dissolved in the solvent at a specified temperature. Calcium sulfate's solubility in water is determined by its 1, providing insight into whether a given amount of the compound will dissolve under specific conditions. Complete solubility results in a "saturated" solution, where no additional solute can dissolve without altering the system conditions, such as temperature or pressure.

Dissolution Process

The dissolution process involves breaking the interactions within a solute to allow solute particles to interact with the solvent. For 79 in water, this involves the dissociation of calcium ions (73) and sulfate ions (93). These ions are surrounded by water molecules, forming hydration shells that stabilize them in solution.
When considering the solubility product K2, this is a measure of the extent to which 79 dissociates in water. The equation is given as 2 = 0 0 where 0 is the solubility in moles per liter. By calculating K2, we find the amount of 79 that will dissolve before it reaches equilibrium, where the rate of dissolution equals the rate of precipitation.
This knowledge helps in determining how much of a solvent is needed to dissolve a specified amount of solute, as seen in this task.