Problem 46 - Chemistry: The Central Science in SI Units, Global Edition

Question

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

Step-by-Step Solution

Answer

The molality of the KCl solution is \(3.88 \mathrm{~m}\). To make a \(0.12 \mathrm{~m}\) solution of sulfur in naphthalene, 4.684 g of sulfur must be dissolved in 100.0 g of naphthalene.

Step 1: Understand the definition of molality

Molality (m) is the number of moles of solute per kilogram of solvent. It is given by the formula: \(m = \frac{\text{moles of solute}}{\text{kilograms of solvent}}\)

Step 2: Identify the given values

We are given: Moles of KCl = 1.12 mol Moles of water = 16.0 mol First, we need to convert moles of water to kilograms. We know that the molar mass of water is 18.015 g/mol.

Step 3: Convert moles of water to kilograms

To convert moles of water to kilograms, we'll use the following equation: Kilograms of solvent = moles of solvent * molar mass of solvent * (1 kg/1000 g) Kilograms of water = 16.0 mol * 18.015 g/mol * (1 kg/1000 g) = 0.28824 kg

Step 4: Calculate the molality of KCl solution

Using the molality formula from Step 1 and the values from Steps 2 and 3, we can now calculate the molality of KCl solution: \(m = \frac{1.12 \ \text{mol}}{0.28824 \ \text{kg}} = 3.88 \ \text{mol/kg}\) So, the molality of the KCl solution is 3.88 m. #b. Determine the mass of sulfur in a naphthalene solution#

Step 1: Convert given molality to a mass-based formula

The given molality is 0.12 m. In this case, the molality represents the number of moles of sulfur \(\mathrm{(S_8)}\) per kilogram of naphthalene \(\mathrm{(C_{10}H_8)}\). We need to find the mass of sulfur required for a 100.0 g sample of naphthalene. Let's set up the mass-based equation: \(\text{molality} = \frac{\text{mass of sulfur} \cdot \text{conversion factor} }{\text{mass of naphthalene}}\) Here, the conversion factor is the molar mass of \(\mathrm{S_8}\).

Step 2: Determine the molar mass of sulfur and naphthalene

The molar mass of sulfur \(\mathrm{(S_8)}\) is 256.48 g/mol. The molar mass of naphthalene \(\mathrm{(C_{10}H_8)}\) is 128.16 g/mol.

Step 3: Calculate the mass of sulfur required

Using the equation from Step 1 and the values from Step 2, we can find the mass of sulfur required as follows: \(0.12 \ \text{m} = \frac{\text{mass of sulfur} \cdot 256.48 \ \text{g/mol} }{100.0 \ \text{g}}\) Now, we need to solve for the mass of sulfur: Mass of sulfur = \((0.12 \ \text{m} * 100.0 \ \text{g}) / 256.48 \ \text{g/mol} = 4.684 \ \text{g}\) So, 4.684 g of sulfur must be dissolved in 100.0 g of naphthalene to make a 0.12 m solution.

Key Concepts

Moles of solute

Moles of solute refer to the amount of solute molecules present in a solution. It is a fundamental unit in chemistry that represents a specific quantity, known as Avogadro's number, which is approximately \(6.022 \times 10^{23}\) molecules. Calculating moles is essential in determining other properties of a solution, such as molality.

To calculate the moles of a solute, you need its mass and molar mass.

• The formula to find moles is: \( \text{moles of solute} = \frac{\text{mass of solute}}{\text{molar mass}} \).
• The molar mass is usually given in grams per mole (g/mol).

In exercises, the moles might be directly provided, like the 1.12 moles of KCl in the example problem. It's important to note that understanding the concept of moles helps in rearranging and utilizing different chemical formulas effectively.

Kilograms of solvent

The solvent in a solution is the substance in which the solute is dissolved. The weight of the solvent is crucial when calculating molality because it influences the number of particles solvated per unit weight. Molality is defined as moles of solute per kilogram of solvent and thus requires this parameter.

Here's how to convert moles to kilograms for a solvent:

• Find the molar mass of the solvent, which is the weight of one mole of solvent molecules.
• Multiply the moles of solvent by its molar mass to find grams, then convert grams to kilograms by dividing by 1000.

For example, converting 16.0 moles of water to kilograms involves using its molar mass 18.015 g/mol, leading to \( 16.0 \times 18.015 \div 1000 = 0.28824 \text{ kg} \). Conversions like these are necessary for any precise scientific calculation involving solutions.

Molar mass

Molar mass is the weight of a compound's molecules and is a vital aspect of calculating both moles and solution characteristics like molality. It is expressed in grams per mole (g/mol) and reflects the sum of the atomic masses of all atoms in a molecule.

To find the molar mass of a substance:

• Identify and add the atomic masses of each element present in the molecule, found on the periodic table.
• Consider the number of each type of atom in the molecular formula.

For example, sulfur \(\mathrm{(S_8)}\) has a molar mass of 256.48 g/mol, calculated by multiplying the atomic mass of sulfur by 8, its atom count in \(\mathrm{S_8}\).

Molar mass is crucial because it allows the conversion from grams to moles, enabling the calculations of properties like molality. In any problem involving solutions, knowing how to accurately determine and apply molar mass is indispensable.