Question

Calculate the molality of the following (a) A solution containing \(174 \mathrm{~g}\) of \(\mathrm{HCl}\) in \(757 \mathrm{~g}\) of \(\mathrm{H}_{2} \mathrm{O}\) (b) A solution containing \(16.5 \mathrm{~g}\) of naphthalene \(\left(\mathrm{C}_{10} \mathrm{H}_{8}\right)\) in \(53.3 \mathrm{~g}\) of benzene \(\left(\mathrm{C}_{6} \mathrm{H}_{6}\right)\)

Short answer

(a) 6.3 mol/kg, (b) 2.42 mol/kg

Step-by-step solution

Determine the molar mass of the solute

For HCl, the molar mass is calculated as follows: \( \text{Molar mass of HCl} = 1 + 35.5 = 36.5 \text{ g/mol} \). For naphthalene (\( \text{C}_{10} \text{H}_{8} \)), the molar mass is: \( \text{Molar mass of} \text{ C}_{10} \text{H}_{8} = 10 \times 12 + 8 \times 1 = 128 \text{ g/mol} \).

Calculate the number of moles of the solute

For HCl: \( \text{Moles of HCl} = \frac{174 \text{ g}}{36.5 \text{ g/mol}} = 4.77 \text{ moles} \).For naphthalene: \( \text{Moles of C}_{10} \text{H}_{8} = \frac{16.5 \text{ g}}{128 \text{ g/mol}} = 0.129 \text{ moles} \).

Convert the mass of the solvent to kilograms

For water (HCl solution): \(757 \text{ g} \to 0.757 \text{ kg}\).For benzene (naphthalene solution): \(53.3 \text{ g} \to 0.0533 \text{ kg}\).

Calculate the molality of the solution

Molality \( (m) \) is defined as the number of moles of solute per kilogram of solvent.For HCl solution: \( \text{Molality of HCl} = \frac{4.77 \text{ moles}}{0.757 \text{ kg}} = 6.3 \text{ mol/kg} \).For naphthalene solution: \( \text{Molality of C}_{10} \text{H}_{8} = \frac{0.129 \text{ moles}}{0.0533 \text{ kg}} = 2.42 \text{ mol/kg} \).

Key concepts

molar mass

In chemistry, the concept of molar mass is crucial. It relates to the mass of one mole of a substance. Molar mass is measured in grams per mole (g/mol).

To find the molar mass of a compound, you need to sum up the relative atomic masses of all the elements in that compound.

• For HCl: Hydrogen (H) has an atomic mass of 1 g/mol and Chlorine (Cl) has an atomic mass of 35.5 g/mol. Thus, the molar mass of HCl is \[ 1 + 35.5 = 36.5 \text{ g/mol} \]
  
• For naphthalene (\text{C}_{10} \text{H}_{8}): Carbon (C) has an atomic mass of 12 g/mol and Hydrogen (H) has an atomic mass of 1 g/mol. Hence, the molar mass of naphthalene is \[ 10 \times 12 + 8 \times 1 = 128 \text{ g/mol} \]
  

This is the first key step in solving many chemistry problems involving substances in different quantities.

number of moles

The number of moles links the mass of a sample to the amount of substance it contains. This quantity is central in stoichiometry, helping you compare amounts of different substances. To determine the number of moles (n), use the formula:

\[ n = \frac{\text{mass of the solute}\text{ (g)}}{\text{molar mass} \text{ (g/mol)}} \]

• For example, in the problem, to find the moles of HCl, you use the mass of HCl (174 g) divided by its molar mass (36.5 g/mol): \[ n(\text{HCl}) = \frac{174}{36.5} = 4.77 \text{ moles} \]
• Similarly, for naphthalene (\text{C}_{10} \text{H}_{8}), the number of moles is: \[ n(\text{C}_{10} \text{H}_{8}) = \frac{16.5}{128} = 0.129 \text{ moles} \]

This calculation is necessary before determining the solution's molality.

convert mass to kilograms

Before calculating molality, you must convert the mass of the solvent from grams to kilograms. This step is crucial because molality (m) is defined in terms of moles of solute per kilogram of solvent. The conversion is straightforward:

• For water in the HCl solution: \[ 757 \text{ g} = 0.757 \text{ kg} \]
• For benzene in the naphthalene solution: \[ 53.3 \text{ g} = 0.0533 \text{ kg} \]

Accurately converting these values ensures you calculate the molality correctly later.

molality calculation

Finally, let's calculate the molality of the given solutions. Molality (m) is defined as the number of moles of solute per kilogram of solvent:

• \textbf{Formula}: \[ m = \frac{\text{moles of solute}}{\text{kilograms of solvent}} \]
• For the HCl solution, where you have 4.77 moles of HCl in 0.757 kg of water:
  \[ m(\text{HCl}) = \frac{4.77}{0.757} = 6.3 \text{ mol/kg} \]
• For the naphthalene solution, with 0.129 moles of \text{C}_{10} \text{H}_{8} in 0.0533 kg of benzene:
  \[ m(\text{C}_{10} \text{H}_{8}) = \frac{0.129}{0.0533} = 2.42 \text{ mol/kg} \]

Molality provides a concentration measure that is temperature-independent since it relies on mass, not volume, making it vital in various scientific calculations.