Problem 20 - General Chemistry: Principles and Modern Applications

Question

If \(100.0 \mathrm{mL}\) of \(0.0025 \mathrm{M} \mathrm{Na}_{2} \mathrm{SO}_{4}(\mathrm{aq})\) is saturated with \(\mathrm{CaSO}_{4},\) how many grams of \(\mathrm{CaSO}_{4}\) would be present in the solution? [Hint: Does the usual simplifying assumption hold?]

Step-by-Step Solution

Answer

The solution would contain 0.034 grams of \(\mathrm{CaSO}_{4}\).

Step 1: Calculate the molar mass of \(\mathrm{CaSO}_{4}\)

Add the atomic masses of calcium, sulfur, and four oxygen atoms to get the molar mass of \(\mathrm{CaSO}_{4}\): \(\mathrm{Molar \: mass\: of\: CaSO}_{4}=40.08+32.06+4*16.00=136.14 \: \mathrm{g/mol}\)

Step 2: Determine the moles of \(\mathrm{Na}_{2}\mathrm{SO}_{4}\) present in the solution

Use the molarity and volume of the solution to find the moles of \(\mathrm{Na}_{2}\mathrm{SO}_{4}\):\(\mathrm{Moles \: of\: Na}_{2}\mathrm{SO}_{4}=\mathrm{Molarity}\times\mathrm{Volume}=0.0025\: M\times 0.100\: L=0.00025\: \mathrm{moles}\)Given the stoichiometry of the reaction, the moles of \(\mathrm{Na}_{2}\mathrm{SO}_{4}\) equal the moles of \(\mathrm{CaSO}_{4}\). So, you have 0.00025 moles of \(\mathrm{CaSO}_{4}\).

Step 3: Convert moles of \(\mathrm{CaSO}_{4}\) to grams

Use the molar mass of \(\mathrm{CaSO}_{4}\) calculated in Step 1 to convert the moles of \(\mathrm{CaSO}_{4}\) to grams:\(\mathrm{Grams \: of\: CaSO}_{4}=\mathrm{Moles\: of\: CaSO}_{4}\times \mathrm{molar \: mass\: of\: CaSO}_{4}=0.00025\: \mathrm{moles}\times 136.14\: \mathrm{g/mol}=0.034\: \mathrm{g}\)

Key Concepts

Molar Mass Calculation

Understanding how to calculate the molar mass of a compound is crucial in chemistry. The molar mass is the weight of one mole of a substance and usually expressed in g/mol. To calculate the molar mass of calcium sulfate (\(\mathrm{CaSO}_4\)), you need to know the atomic masses of its constituent elements:

• Calcium (Ca): 40.08 g/mol
• Sulfur (S): 32.06 g/mol
• Oxygen (O): 16.00 g/mol (but keep in mind there are four oxygen atoms in the compound)

So, the molar mass of \(\mathrm{CaSO}_4\) is calculated by adding these atomic masses together:\[\mathrm{Molar \: mass \: of \: CaSO}_4 = 40.08 + 32.06 + 4 \times 16.00 = 136.14 \: \mathrm{g/mol}\]This calculation is foundational in converting between moles and grams, essential for various chemistry applications. Always double-check your calculations to ensure accuracy.

Stoichiometry

Stoichiometry is the section of chemistry dealing with the quantitative relationships between reactants and products in a chemical reaction. In our exercise, stoichiometry helps us determine how the substances interact in solution.Given \(100.0 \mathrm{mL} \) of \(0.0025 \mathrm{M} \mathrm{Na}_2 \mathrm{SO}_4\) is saturated with \(\mathrm{CaSO}_4\), the stoichiometry of the reaction dictates that one mole of \(\mathrm{Na}_2 \mathrm{SO}_4\) generates one mole of \(\mathrm{CaSO}_4\). To determine the moles of \(\mathrm{Na}_2 \mathrm{SO}_4\) used, apply the formula:\[\mathrm{Moles \: of \: Na}_2\mathrm{SO}_4 = \text{molarity} \times \text{volume in liters}\]Substitute the given values:\[0.0025 \: \mathrm{M} \times 0.100 \: \mathrm{L} = 0.00025 \: \mathrm{moles}\]Since the moles of \(\mathrm{Na}_2 \mathrm{SO}_4\) equal the moles of \(\mathrm{CaSO}_4\), equate them directly. This implies you have 0.00025 moles of \(\mathrm{CaSO}_4\), demonstrating the stoichiometric relationship and enabling conversion to grams. Stoichiometry ensures you use quantities effectively by applying direct mole relationships.

Molarity and Solutions

Molarity is a measure of the concentration of a solute in a solution. It’s defined as moles of solute per liter of solution and indicated as "M" (molar). Getting this concept right is key to solving the given exercise.Consider our problem setup with \(0.0025 \mathrm{M} \mathrm{Na}_2 \mathrm{SO}_4\). Molarity becomes useful as you determine the number of moles present. Molarity can also predict changes in concentrations during chemical reactions, providing a ratio for reaction stoichiometry.For example, in the exercise:

• The volume of solution: 100.0 mL or 0.100 L
• Molarity: 0.0025 M

Multiplying the molarity by the volume lets you find the moles:\[0.0025 \: \mathrm{M} \times 0.100 \: \mathrm{L} = 0.00025 \: \mathrm{moles \: of \: Na}_2 \mathrm{SO}_4\]This information is crucial in converting to other units, like grams, using molar mass. Understanding molarity not only helps in calculations but also paints a clear picture of the solution’s chemical strength, helping you anticipate its behavior in reactions.