Problem 46 - Chemistry The Central Science

Question

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

Step-by-Step Solution

Answer

(a) To calculate the molality of the KCl solution, use the formula: \(M_\text{solution}\) = moles of solute(KCl) / mass of solvent(H₂O) in kg. Given moles of KCl = 1.12 mol, and moles of water = 16 moles, first, find the mass of 16 moles of water in grams and then convert it to kg. Then, plug the values into the formula: \(M_\text{solution}\) = 1.12 mol / mass of water in kg to find the molality. (b) To find the amount of sulfur (S_8) needed to make a 0.12 m solution in 100.0 g of naphthalene (C_10H_8), rearrange the molality formula to find the moles of sulfur: Moles of S_8 = molality * mass of solvent(C_10H_8) in kg. Convert the mass of naphthalene to kg and plug the values into the rearranged formula to find the moles of sulfur. Finally, use the molar mass of S_8 to find the required mass of sulfur: Mass of S_8 = moles of S_8 * molar mass of S_8.

Step 1: Calculate molality (a)

To calculate the molality of the KCl solution, we will use the formula: \(M_\text{solution}\) = moles of solute(KCl) / mass of solvent(H₂O) in kg Given: moles of KCl = 1.12 mol, moles of water = 16 moles Mass of 1 mol of water = 18.015 g/mol To find mass of 16 moles of water: Mass of 16 moles of water = 16 * Mass of 1 mol of water

Step 2: Convert mass of water to kg

We need the mass of water in kg for the molality formula. To convert the mass of water in g to kg, simply divide by 1000. Mass of water in kg = Mass of water in g / 1000

Step 3: Calculate molality (a)

Now we can calculate the molality of the KCl solution: \(M_\text{solution}\) = moles of KCl / mass of water in kg #Phase 2#

Step 4: Calculate molality (b)

To find the amount of sulfur (S_8) needed to make a 0.12 m solution in 100.0 g of naphthalene (C_10H_8), use the molality formula: \(M_\text{solution}\) = moles of solute(S_8) / mass of solvent(C_10H_8) in kg We know the mass of naphthalene and the desired molality of the solution so we can rearrange the formula to find the moles of sulfur. Moles of S_8 = molality * mass of solvent(C_10H_8) in kg Given: mass_of_naphthalene = 100 g

Step 5: Convert mass of naphthalene to kg

To convert the mass of naphthalene in g to kg, divide by 1000. Mass of naphthalene in kg = mass_of_naphthalene / 1000

Step 6: Calculate moles of S_8

Plug the given values into the rearranged molality formula to find the moles of sulfur: Moles of S_8 = molality * mass of solvent(naphthalene) in kg

Step 7: Calculate mass of S_8

Using the molar mass of S_8, we can find the mass of S_8 required to make the 0.12 m solution. The molar mass of S_8: 32.06 g/mol * 8 Mass of S_8 = moles of S_8 * molar mass of S_8 This will give us the required mass of sulfur to make a 0.12 m solution in 100 g of naphthalene.

Key Concepts

Molar Mass

Understanding molar mass is critical for mastering molality calculations. Molar mass is the weight of one mole of a substance and is expressed in grams per mole (g/mol). It serves as a bridge between the microscopic world of atoms and molecules and the macroscopic world of grams and kilograms that we can measure. Each element's molar mass is found on the periodic table, but for compounds like potassium chloride (KCl) or sulfur (S₈), you sum the molar masses of the constituent elements. For example, the molar mass of KCl can be calculated by adding the molar masses of potassium (K) and chlorine (Cl). Likewise, for S₈, multiply the molar mass of a single sulfur atom by 8, as there are eight sulfur atoms in each molecule of S₈.

Molar mass is vital in stoichiometry and mole-to-mass conversions, allowing us to calculate the grams of a substance when given the number of moles or vice versa, which is exactly what we need when solving for the mass of S₈ in part (b) of our exercise.

Solution Concentration

The concentration of a solution reflects the amount of solute present in a given quantity of solvent. There are various ways to express this concentration, and molality is one of them. Molality (\(m\)) is defined as the number of moles of solute per kilogram of solvent. Unlike molarity, which depends on the volume of the solution, molality depends only on the mass, making it independent of temperature because mass doesn't change with temperature. This makes molality particularly useful in scenarios where temperature can fluctuate.

In part (a) of our exercise, the molality calculation requires knowing the amount of KCl in moles and the mass of water in kilograms. For part (b), the goal is to find out how much sulfur is needed to achieve a specific molality, again focusing on the mass of the naphthalene solvent and the moles of sulfur solute.

Stoichiometry

The stoichiometry of a chemical reaction involves the quantitative relationship between reactants and products. In the context of solution preparation, stoichiometry is the art of measuring and calculating the exact amounts of substances required. It's all about using the balanced chemical equations to understand the proportions in which the substances react or combine.

While our exercise doesn't involve a chemical reaction per se, the foundational concept of stoichiometry still applies: quantities of substances must relate through conversion factors. Here, we're interested in the stoichiometric relationship between moles and mass based on the molar mass, ensuring that we add the precise amount of solute to the solvent to achieve the desired concentration.

Mole-to-Mass Conversion

Mole-to-mass conversion is a fundamental concept in chemistry that relates the moles of a substance to its mass in grams, employing the molar mass as a conversion factor. It's a two-way street—sometimes you start with moles and want to find mass, other times you have mass and need to find moles. The molar mass tells you how many grams one mole of a substance weighs, so by multiplying the number of moles by the molar mass, you get the mass in grams.

In the steps provided for solving part (b) of our exercise, after calculating the moles of sulfur (S₈), we perform a mole-to-mass conversion using the molar mass of S₈. This step is crucial to determine the exact mass of S₈ required to create a solution with a particular molality. The correct use of mole-to-mass conversion guarantees precise and accurate formulation of solutions, a skill essential in many fields, including chemistry, biology, pharmaceuticals, and more.