Question

(a) What is the molality of a solution formed by dissolving 1.12 mol of \(\mathrm{KCl}\) in 16.0 \(\mathrm{mol}\) of water? (b) How many grams of sulfur \(\left(S_{8}\right)\) must be dissolved in 100.0 g of naphthalene \(\left(\mathrm{C}_{10} \mathrm{H}_{8}\right)\) to make a 0.12 \(\mathrm{m}\) solution?

Short answer

a) To find the molality of the KCl solution, first, convert the moles of water to mass (in kg) using the molar mass of water (18.02 g/mol): Mass of water (kg) = (16.0 mol × 18.02 g/mol) ÷ 1000 = 0.288 kg Then, calculate the molality of the KCl solution using the formula: Molality (m) = 1.12 mol ÷ 0.288 kg = 3.89 m b) To find the mass of S₈ needed, first, convert the mass of naphthalene to kg: Mass of naphthalene (kg) = 100.0 g ÷ 1000 = 0.1 kg Then, find the moles of S₈ using the given molality: Moles of S₈ = 0.12 m × 0.1 kg = 0.012 mol Finally, convert the moles of S₈ to mass using the molar mass of S₈ (256.52 g/mol): Mass of S₈ = 0.012 mol × 256.52 g/mol = 3.08 g

Step-by-step solution

Identify the given values

In this problem, we are given: - Moles of KCl = 1.12 mol - Moles of water = 16.0 mol

Convert moles of water to mass of water in kg

We know that the molar mass of water (H₂O) is 18.02 g/mol. Therefore, to find the mass of water, we can use the following equation: Mass of water = Moles of water × Molar mass of water Now, we will convert this mass of water to kg: Mass of water (kg) = Mass of water (g) ÷ 1000

Calculate the molality of the KCl solution

Now we can use the molality formula: Molality (m) = moles of KCl ÷ mass of water (kg) Plug in the values and find the molality of the KCl solution. #b) Find the mass of S₈ needed to make a 0.12 m solution in 100.0 g of naphthalene#

Identify the given values

In this problem, we are given: - Molality (m) = 0.12 m - Mass of naphthalene (C₁₀H₈) = 100.0 g

Convert mass of naphthalene to mass of naphthalene in kg and find moles of solute

To find the mass of naphthalene in kg, divide the given mass by 1000: Mass of naphthalene (kg) = Mass of naphthalene (g) ÷ 1000 Next, use the molality formula to find the moles of S₈ (solute): Molality (m) = moles of S₈ ÷ mass of naphthalene (kg) Rearrange the formula to find the moles of S₈: Moles of S₈ = Molality (m) × mass of naphthalene (kg) Plug in the values to find the moles of S₈.

Convert moles of S₈ to mass of S₈

We know that the molar mass of S₈ (sulfur) is 256.52 g/mol. Therefore, to find the mass of S₈, we can use the following equation: Mass of S₈ = Moles of S₈ × Molar mass of S₈ Calculate the mass of S₈ required to make a 0.12 m solution in 100.0 g of naphthalene.

Key concepts

Solution Concentration

Solution concentration refers to the amount of a solute that is dissolved in a given quantity of solvent. Understanding solution concentration is essential in chemistry, as it helps determine how concentrated or dilute a solution is. In this context, molality (\( m \)) is a useful measure of concentration, especially when temperature changes are involved since it is independent of temperature. Molality is defined as the number of moles of solute per kilogram of solvent.

To calculate the molality, first, the amount of solute in moles is determined, then the mass of the solvent is measured in kilograms. The formula is simple: \[ m = \frac{{\text{{moles of solute}}}}{{\text{{mass of solvent (kg)}}}} \] For instance, if you dissolve 1.12 moles of potassium chloride (\( \text{{KCl}} \) in 16.0 moles of water (\( \text{{H}}_2\text{{O}} \) the molality of the solution can be calculated using the steps given in the textbook solution.

Molar Mass

Molar mass is the mass of one mole of a substance and it is expressed in grams per mole (g/mol). It is a fundamental concept for converting between the mass of a substance and the number of moles. Each element has a unique molar mass, which is equivalent to its atomic weight listed on the periodic table. For compounds, the molar mass is the sum of the molar masses of all the atoms in the molecule.

For example, the molar mass of water (H₂O) is calculated by adding twice the molar mass of hydrogen (about 1.01 g/mol) and once the molar mass of oxygen (about 16.00 g/mol), giving us approximately 18.02 g/mol. Similarly, to find the mass of sulfur octamer (\( S_8 \) which is needed to create a solution, knowing that its molar mass is 256.52 g/mol allows the direct conversion from moles to grams and vice versa.

Moles to Mass Conversion

The process of converting moles to mass and vice versa is critical in preparing solutions of precise concentrations. This fundamental calculation requires the molar mass of the solute. To convert the number of moles of a substance to mass, you multiply the number of moles by the molar mass of the substance. The formula is: \[ \text{{Mass (g)}} = \text{{Moles}} \times \text{{Molar mass (g/mol)}} \] Conversely, to find the number of moles when you have the mass, you divide the mass by the molar mass: \[ \text{{Moles}} = \frac{{\text{{Mass (g)}}}}{{\text{{Molar mass (g/mol)}}}} \]

Using sulfur octamer (\( S_8 \) as an example from the original exercise, once its molar mass is known (256.52 g/mol), it is straightforward to convert the calculated moles of \( S_8 \) needed for the solution into mass by simply multiplying the moles by the molar mass. This is how, after calculating the molality of the solution, we find how many grams of \( S_8 \) are required to obtain the desired solution concentration.