Problem 47 - Chemistry: The Central Science

Question

A sulfuric acid solution containing \(571.6 \mathrm{~g}\) of \(\mathrm{H}_{2} \mathrm{SO}_{4}\) per liter 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.

Step-by-Step Solution

Answer

(a) Mass percentage of H2SO4 ≈ 43.00 % (b) Mole fraction of H2SO4 ≈ 0.1216 (c) Molality of H2SO4 ≈ 7.688 mol/kg (d) Molarity of H2SO4 ≈ 5.826 mol/L

Step 1: Calculate the mass of 1L solution

Given the density of the solution and the volume, we can calculate the mass: Density = mass/volume We have the density = 1.329 g/cm³ and volume = 1L (equivalent to 1000 cm³) To find the mass of the solution: Mass = Density × Volume = 1.329 g/cm³ × 1000 cm³ = 1329 g

Step 2: Calculate the mass of the solvent

Now that we know the mass of the solution, we can find the mass of the solvent by subtracting the mass of the solute from the mass of the solution. Mass of solvent = Mass of solution - Mass of solute Mass of solvent = 1329 g - 571.6 g = 757.4 g

Step 3: Calculate the mass percentage of H2SO4

The mass percentage can be found by dividing the mass of the solute by the mass of the solution, and then multiplying the result by 100 to get the percentage. Mass percentage = (Mass of solute / Mass of solution) * 100 Mass percentage = (571.6 g / 1329 g) * 100 ≈ 43.00 %

Step 4: Calculate the number of moles of H2SO4 and solvent

Moles of H2SO4 = mass / molar mass Molar mass of H2SO4 = 2(1.01) + 32.07 + 4(16.00) = 98.08 g/mol Moles of H2SO4 = 571.6 g / 98.08 g/mol ≈ 5.826 moles Assuming the solvent is water: Molar mass of water = 18.015 g/mol Moles of solvent (water) = 757.4 g / 18.015 g/mol ≈ 42.06 moles

Step 5: Calculate the mole fraction of H2SO4

To find the mole fraction, we divide the moles of \(H_2SO_4\) by the total number of moles. Mole fraction = Moles of H2SO4 / (Moles of H2SO4 + Moles of solvent) Mole fraction ≈ 5.826 moles / (5.826 moles + 42.06 moles) ≈ 0.1216

Step 6: Calculate molality of H2SO4

Molality is calculated by dividing the number of moles of the solute by the mass of the solvent in kilograms. Molality = Moles of H2SO4 / Mass of solvent (in kg) Molality ≈ 5.826 moles / 0.7574 kg ≈ 7.688 mol/kg

Step 7: Calculate molarity of H2SO4

Molarity is calculated by dividing the number of moles of the solute by the volume of the solution in liters. Molarity = Moles of H2SO4 / Volume of solution (in L) Molarity ≈ 5.826 moles / 1 L ≈ 5.826 mol/L

Step 8: Summary

(a) Mass percentage of H2SO4 ≈ 43.00 % (b) Mole fraction of H2SO4 ≈ 0.1216 (c) Molality of H2SO4 ≈ 7.688 mol/kg (d) Molarity of H2SO4 ≈ 5.826 mol/L

Key Concepts

Mass Percentage Calculation

When it comes to understanding the composition of solutions, the mass percentage is a handy concept. It tells us how much mass of a solute is present in a solution compared to the total mass of the solution. Here, in the context of sulfuric acid solutions, knowing the mass percentage helps us grasp the concentration level of the acid in the solution. To calculate the mass percentage, use the formula: \[ \text{Mass Percentage} = \left( \frac{\text{mass of solute}}{\text{mass of solution}} \right) \times 100 \]In our sulfuric acid solution, we know that the mass of the solute (\( H_2SO_4 \)) is 571.6 g, and the mass of the entire solution is 1329 g. Plugging these values into the formula gives us:\[ \text{Mass Percentage} = \left( \frac{571.6 \text{ g}}{1329 \text{ g}} \right) \times 100 \approx 43.00 \% \] This means 43% of the solution's mass is contributed by sulfuric acid.

Mole Fraction Determination

The mole fraction is a way of expressing concentration as a ratio of the number of moles of a component to the total number of moles in a solution. It's particularly useful because it doesn't require the concept of volume or mass units.To find the mole fraction of sulfuric acid in the solution, we calculate the number of moles of both the solute and the solvent: \( H_2SO_4 \) has 5.826 moles, and water, our solvent, has about 42.06 moles.The formula for mole fraction is given by:\[ \text{Mole Fraction} = \frac{\text{moles of solute}}{\text{total moles in solution}} \] Applying our data, the mole fraction of \( H_2SO_4 \) becomes:\[ \text{Mole Fraction of } H_2SO_4 = \frac{5.826}{5.826 + 42.06} \approx 0.1216 \] This indicates that out of the total number of moles present, about 12.16% are sulfuric acid.

Molality Calculation

Molality is a concentration unit that is expressed in terms of moles of solute per kilogram of solvent. It is especially useful in situations where temperature and pressure affect volume, as it relies purely on mass.To calculate molality, the formula is:\[ \text{Molality} = \frac{\text{moles of solute}}{\text{mass of solvent in kg}} \]In our problem, we have identified 5.826 moles of sulfuric acid and the mass of the solvent (water) as 757.4 g or 0.7574 kg.Thus, the calculation is as follows:\[ \text{Molality} = \frac{5.826 \text{ moles}}{0.7574 \text{ kg}} \approx 7.688 \text{ mol/kg} \] This result means that there are about 7.688 moles of sulfuric acid in every kilogram of water in the solution.

Molarity Calculation

Molarity is a measurement of concentration that depends on the volume of the solution. It is defined as the number of moles of a solute per liter of solution. Molarity is often used because it provides a straightforward way to relate volume to concentration, which is handy in many chemical applications.The formula for calculating molarity is:\[ \text{Molarity} = \frac{\text{moles of solute}}{\text{volume of solution in L}} \] In our sulfuric acid solution, we know the volume of the solution to be 1 liter. With 5.826 moles of \( H_2SO_4 \), we apply this simple formula:\[ \text{Molarity} = \frac{5.826 \text{ moles}}{1 \text{ L}} \approx 5.826 \text{ mol/L} \] This figure indicates that for every liter of the sulfuric acid solution, there are 5.826 moles of sulfuric acid.