Question

A sulfuric acid solution containing \(571.6 \mathrm{~g}\) of \(\mathrm{H}_{2} \mathrm{SO}_{4}\) per Jiter of solution has a density of \(1.329 \mathrm{~g} / \mathrm{cm}^{3}\). Calculate (a) the mass percentage, (b) the mole fraction, (c) the molality, (d) the molarity of \(\mathrm{H}_{2} \mathrm{SO}_{4}\) in this solution.

Short answer

The properties of the H2SO4 solution are: (a) Mass percentage: 43.0%, (b) Mole fraction: 0.122, (c) Molality: 7.70 mol/kg, and (d) Molarity: 5.83 mol/L.

Step-by-step solution

Determine mass of 1 liter of solution

Given the density of the solution, we can calculate the mass of 1 liter (1000 cm³) of the solution. The formula for this calculation is: mass = density × volume Density = 1.329 g/cm^3 Volume = 1000 cm³ mass = (1.329 g/cm³) × (1000 cm³) = 1329 g

Calculate mass of solvent

We are given the mass of H2SO4 in 1 liter of the solution (571.6 g). To find the mass of the solvent (H2O), we can subtract the mass of H2SO4 from the total mass of the solution calculated in Step 1. Mass of solvent (H2O) = mass of the solution - mass of H2SO4 Mass of solvent (H2O) = 1329 g - 571.6 g = 757.4 g

Calculate mass percentage of H2SO4

Now that we know the total mass of the solution and the mass of H2SO4, we can determine the mass percentage of H2SO4 using the following formula: Mass percentage of H2SO4 = (mass of H2SO4 / mass of the solution) × 100 Mass percentage of H2SO4 = (571.6 g / 1329 g) × 100 = 43.0 %

Calculate mole fraction of H2SO4

First, we need to determine the number of moles of H2SO4 and H2O. For this, we need the molar mass of H2SO4 and H2O: Molar mass of H2SO4 = 98.08 g/mol Molar mass of H2O = 18.02 g/mol Number of moles of H2SO4 = mass of H2SO4 / molar mass of H2SO4 = 571.6 g / 98.08 g/mol ≈ 5.83 moles Number of moles of H2O = mass of H2O / molar mass of H2O = 757.4 g / 18.02 g/mol ≈ 42.04 moles Now we can calculate the mole fraction of H2SO4 in the solution: Mole fraction of H2SO4 = moles of H2SO4 / (moles of H2SO4 + moles of H2O) = 5.83 / (5.83 + 42.04) ≈ 0.122

Calculate molality of H2SO4

Molality is defined as the number of moles of solute per kilogram of solvent. Thus, we can calculate the molality of H2SO4 in the solution using the following formula: Molality of H2SO4 = moles of H2SO4 / mass of solvent (in kg) Molality of H2SO4 = 5.83 moles / 0.7574 kg ≈ 7.70 mol/kg

Calculate molarity of H2SO4

Molarity is defined as the number of moles of solute per liter of solution. Using the data from Step 4, we can calculate the molarity of H2SO4 in the solution as follows: Molarity of H2SO4 = moles of H2SO4 / volume of solution (in L) Given that there are 571.6 g of H2SO4 in 1 L of solution: Molarity of H2SO4 = 5.83 moles / 1 L = 5.83 mol/L In summary, the following properties of the H2SO4 solution are: (a) Mass percentage of H2SO4: 43.0 % (b) Mole fraction of H2SO4: 0.122 (c) Molality of H2SO4: 7.70 mol/kg (d) Molarity of H2SO4: 5.83 mol/L

Key concepts

Mass Percentage

Mass percentage is a way to express the concentration of a solute in a solution. It represents the ratio of the mass of the solute to the total mass of the solution, multiplied by 100 to get a percentage. It's a straightforward measure that helps understand how much of a given substance is present in a solution relative to the entire mixture.
In the scenario provided, the mass percentage of sulfuric acid (H₂SO₄) is calculated by dividing the mass of the acid by the total mass of the solution. Given that the solution contains 571.6 grams of H₂SO₄ and the total mass of the solution is 1329 grams, the mass percentage formula is:\[\text{Mass percentage of } \mathrm{H_2SO_4} = \left(\frac{\text{mass of } \mathrm{H_2SO_4}}{\text{total mass of solution}}\right) \times 100 = \left(\frac{571.6 \text{ g}}{1329 \text{ g}}\right) \times 100 \approx 43.0\%\]
This means that 43% of the solution's mass comes from sulphuric acid.

Mole Fraction

The mole fraction is a way to express the concentration of a component in a mixture. It describes the ratio of the number of moles of a particular component to the total number of moles of all components in the mixture. This measure is particularly useful in calculating vapor pressures and for understanding colligative properties.To find the mole fraction of sulfuric acid in a solution, you first need to compute the number of moles of both the solute (H₂SO₄) and the solvent (H₂O). Using their respective molar masses, you can calculate:- Moles of H₂SO₄: 571.6 g ÷ 98.08 g/mol ≈ 5.83 moles- Moles of H₂O: 757.4 g ÷ 18.02 g/mol ≈ 42.04 molesThe mole fraction of H₂SO₄ is then determined by dividing the moles of H₂SO₄ by the total moles in the solution:\[\text{Mole fraction of } \mathrm{H_2SO_4} = \frac{5.83}{5.83 + 42.04} \approx 0.122\]
This value indicates that H₂SO₄ makes up 12.2% of the total mole content of the solution.

Molality

Molality is a measure of the concentration of solute in a solution in terms of moles per kilogram of solvent, not the overall solution. This is an advantageous measure when studying properties that are influenced by temperature changes, as molality doesn’t depend on temperature because it’s based on mass rather than volume.To calculate molality for the sulfuric acid solution, you use the number of moles of H₂SO₄ and the mass of the solvent in kilograms:- Moles of solute (H₂SO₄): 5.83 moles- Mass of solvent (H₂O): 757.4 g or 0.7574 kgThe formula for molality is then applied as follows:\[\text{Molality of } \mathrm{H_2SO_4} = \frac{5.83 \text{ moles}}{0.7574 \text{ kg}} \approx 7.70 \text{ mol/kg}\]
This means that there are 7.70 moles of sulfuric acid per kilogram of water in the solution.

Molarity

Molarity is a concentration term that specifies the number of moles of solute per liter of solution. It is a common and convenient measure used frequently in laboratory settings and chemical reactions as it relates directly to the volume of liquid.In order to calculate the molarity of the sulfuric acid solution, you simply take the number of moles of H₂SO₄ and divide it by the volume of the solution in liters (here, 1 liter is given):\[\text{Molarity of } \mathrm{H_2SO_4} = \frac{5.83 \text{ moles}}{1 \text{ liter}} = 5.83 \text{ mol/L}\]
This means there are 5.83 moles of H₂SO₄ in every liter of the solution. Molarity is particularly useful for preparing solutions and conducting titration experiments.