Question

A solution is prepared by mixing 50.0 \(\mathrm{mL}\) toluene \(\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{CH}_{3}\right.\) \(d=0.867 \mathrm{g} / \mathrm{cm}^{3} )\) with 125 \(\mathrm{mL}\) benzene \(\left(\mathrm{C}_{6} \mathrm{H}_{6}, d=0.874 \mathrm{g} / \mathrm{cm}^{3}\right)\) Assuming that the volumes add on mixing, calculate the mass percent, mole fraction, molality, and molarity of the toluene.

Short answer

The mass percent of toluene in the solution is 28.4%, the mole fraction is 0.252, the molality is 4.30 mol/kg, and the molarity is 2.69 mol/L.

Step-by-step solution

Calculate the mass of toluene and benzene

Given, volume of toluene (V1) = 50.0 mL, density of toluene (d1) = 0.867 g/cm³ Given, volume of benzene (V2) = 125 mL, density of benzene (d2) = 0.874 g/cm³ To find the mass, we can use the formula: mass = volume × density Mass of toluene (m1) = V1 × d1 = 50.0 mL × 0.867 g/cm³ = 43.35 g Mass of benzene (m2) = V2 × d2 = 125 mL × 0.874 g/cm³ = 109.25 g

Calculate the moles of toluene and benzene

Next, we need to calculate the moles of toluene and benzene using their molar masses. Molar mass of toluene (M1) = 92.14 g/mol (C₆H₅CH₃) Molar mass of benzene (M2) = 78.11 g/mol (C₆H₆) To find the moles, we can use the formula: moles = mass ÷ molar mass Moles of toluene (n1) = m1 ÷ M1 = 43.35 g ÷ 92.14 g/mol = 0.470 mol Moles of benzene (n2) = m2 ÷ M2 = 109.25 g ÷ 78.11 g/mol = 1.398 mol

Calculate the mass percent of toluene

To find the mass percent of toluene, we can use the formula: mass percent = (mass of solute ÷ mass of solution) × 100 Mass of solution = m1 + m2 = 43.35 g + 109.25 g = 152.60 g Mass percent of toluene = (m1 ÷ (m1 + m2)) × 100 = (43.35 g ÷ 152.60 g) × 100 = 28.4%

Calculate the mole fraction of toluene

To find the mole fraction of toluene, we can use the formula: mole fraction = moles of solute ÷ total moles Total moles = n1 + n2 = 0.470 mol + 1.398 mol = 1.868 mol Mole fraction of toluene = n1 ÷ (n1 + n2) = 0.470 mol ÷ 1.868 mol = 0.252

Calculate the molality of toluene

To find the molality of toluene, we can use the formula: molality = moles of solute ÷ mass of solvent (in kg) The mass of benzene, m2, is the mass of the solvent in this case. Molality of toluene = n1 ÷ (m2 ÷ 1000) = 0.470 mol ÷ (109.25 g ÷ 1000) = 0.470 mol ÷ 0.10925 kg = 4.30 mol/kg

Calculate the molarity of toluene

To find the molarity of toluene, we can use the formula: molarity = moles of solute ÷ volume of solution (in L) Total volume of the solution (in L) = (V1 + V2) ÷ 1000 = (50.0 mL + 125 mL) ÷ 1000 = 0.175 L Molarity of toluene = n1 ÷ total volume = 0.470 mol ÷ 0.175 L = 2.69 mol/L The mass percent of toluene is 28.4%, the mole fraction is 0.252, the molality is 4.30 mol/kg, and the molarity is 2.69 mol/L.

Key concepts

Mass Percent

Mass percent is a way of expressing a concentration that specifies the mass of a solute divided by the total mass of the solution, multiplied by 100 to get a percentage. This provides a clear picture of how much of the solution is made up of the solute. In the given problem, toluene is the solute and the solution consists of toluene and benzene.

• First, calculate the total mass of the solution by adding the mass of toluene and the mass of benzene: \[\text{Mass}_{\text{solution}} = 43.35 \, \text{g (toluene)} + 109.25 \, \text{g (benzene)} = 152.60 \, \text{g}\]
• Then, find the mass percent using the formula: \[\text{Mass percent}_{\text{toluene}} = \left(\frac{43.35 \, \text{g}}{152.60 \, \text{g}}\right) \times 100 = 28.4\%\]

Knowing the mass percent is important for understanding how concentrated a solution is in terms of mass.

Mole Fraction

The mole fraction represents the ratio of the moles of one component to the total moles in a solution. It is a dimensionless quantity often used when dealing with gas mixtures or when relating different concentrations. To find the mole fraction of toluene:

• Calculate the total moles in the solution: \[\text{Total moles} = 0.470 \, \text{mol (toluene)} + 1.398 \, \text{mol (benzene)} = 1.868 \, \text{mol}\]
• Use the mole fraction formula: \[\text{Mole fraction}_{\text{toluene}} = \frac{0.470 \, \text{mol}}{1.868 \, \text{mol}} \approx 0.252\]

The mole fraction can help in calculating partial pressures in gas solutions or other reaction-specific calculations.

Molality

Molality is a measure of solute concentration defined by the number of moles of solute per kilogram of solvent. It is temperature-independent unlike molarity, making it particularly useful in scenarios where temperature varies.For calculating molality of toluene in the solution:

• Recognize benzene as the solvent, with a mass of 109.25 g or 0.10925 kg.
• Apply the molality formula: \[\text{Molality}_{\text{toluene}} = \frac{0.470 \, \text{mol (toluene)}}{0.10925 \, \text{kg (benzene)}} \approx 4.30 \, \text{mol/kg}\]

Molality is a useful concept especially in colligative properties calculations like boiling point elevation and freezing point depression.

Molarity

Molarity, symbolized as \(M\), is the number of moles of solute per liter of solution, making it volumetric in nature and sensitive to temperature changes.Steps to determine the molarity of toluene in the mixture:

• Total volume of the solution is determined by adding the volumes of toluene and benzene, resulting in 0.175 L:
• Using the molarity equation: \[\text{Molarity}_{\text{toluene}} = \frac{0.470 \, \text{mol}}{0.175 \, \text{L}} \approx 2.69 \, \text{mol/L}\]

Molarity is often used in titration calculations and is the most common concentration unit for laboratory solutions.