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, ( \mathbf{d} ) ~ t h e ~ m o l a r i t y ~ o f ~ \(\mathrm{H}_{2} \mathrm{SO}_{4}\) in this solution.

Short answer

(a) The mass percentage of H2SO4 in the solution is approximately 43%. (b) The mole fraction of H2SO4 in the solution is 0.122. (c) The molality of H2SO4 in the solution is 7.69 mol/kg. (d) The molarity of H2SO4 in the solution is 5.83 mol/L.

Step-by-step solution

(a) Calculate the mass of 1 liter of solution

Given that the density of the solution is 1.329 g/cm³, we need to find the mass of 1 liter of the solution. We know that 1 liter is equal to 1000 cm³. Now we can use the density formula mass = volume × density to calculate the mass of 1 liter of solution: mass = 1000 cm³ × 1.329 g/cm³ = 1329 g.

(a) Calculate mass percentage of H2SO4 in the solution

Now that we know the mass of 1 liter of the solution (1329 g) and the mass of H2SO4 in 1 liter of the solution (571.6 g), we can calculate the mass percentage using the formula: mass percentage = (mass of solute / mass of solution) × 100: mass percentage = (571.6 g / 1329 g) × 100 = 43% (approximately).

(b) Calculate the moles of H2SO4 and moles of solvent

To calculate the mole fraction, we need to find the moles of H2SO4 and moles of solvent. First, let's find the moles of H2SO4 using its given mass and molar mass: moles of H2SO4 = mass of H2SO4 / molar mass of H2SO4 = 571.6 g / 98.08 g/mol = 5.83 mol. Next, we need to find the moles of solvent (water). First, let's find the mass of water: mass of water = mass of solution - mass of H2SO4 = 1329 g - 571.6 g = 757.4 g. Now, let's find the moles of water using its molar mass (18.015 g/mol): moles of water = mass of water / molar mass of water = 757.4 g / 18.015 g/mol = 42.06 mol.

(b) Calculate the mole fraction of H2SO4

Now that we have the moles of H2SO4 and water, we can calculate the mole fraction of H2SO4 using the formula: mole fraction = moles of solute / (moles of solute + moles of solvent): mole fraction of H2SO4 = 5.83 mol / (5.83 mol + 42.06 mol) = 5.83 mol / 47.89 mol = 0.122.

(c) Calculate the molality of H2SO4

To calculate molality, we need to find the moles of the solute (H2SO4) and the mass of the solvent (water) in kilograms. We already have the moles of H2SO4 (5.83 mol) and the mass of water (757.4 g). Let's convert the mass of water to kilograms: mass of water = 757.4 g × (1 kg / 1000 g) = 0.7574 kg. Now, let's find the molality by using the formula: molality = moles of solute / mass of solvent (in kg): molality = 5.83 mol / 0.7574 kg = 7.69 mol/kg.

(d) Calculate the molarity of H2SO4

Lastly, we need to find the molarity of H2SO4. To do this, we need to know the moles of H2SO4 and the volume of the solution in liters. We know that there are 5.83 moles of H2SO4 and 1 liter of the solution. Now, we can find the molarity using the formula: molarity = moles of solute / volume of solution (in L): molarity = 5.83 mol / 1 L = 5.83 mol/L. #Results# (a) The mass percentage of H2SO4 in the solution is approximately 43%. (b) The mole fraction of H2SO4 in the solution is 0.122. (c) The molality of H2SO4 in the solution is 7.69 mol/kg. (d) The molarity of H2SO4 in the solution is 5.83 mol/L.

Key concepts

Mass Percentage

Mass percentage is a way to express the concentration of a component in a mixture or solution. It represents the mass of the solute (substance being dissolved) as a percentage of the total mass of the solution. Calculating the mass percentage involves dividing the mass of the solute by the total mass of the solution and multiplying by 100. For example, if you have 571.6 grams of sulfuric acid (H_2SO_4) in a solution with a total mass of 1329 grams, you would calculate the mass percentage using the formula:\[\text{Mass Percentage} = \left( \frac{\text{Mass of } H_2SO_4}{\text{Mass of Solution}} \right) \times 100\]Plugging in the numbers, we get\[(571.6 \text{ g} / 1329 \text{ g}) \times 100 \approx 43\%.\]This means sulfuric acid makes up about 43% of the mass of the solution. Remember: mass percentage can tell you how concentrated a solution is, with a higher percentage indicating a more concentrated solution.

Mole Fraction

Mole fraction is another way to express concentration, focusing on the proportion of each component's moles relative to the total moles in the solution. It is a dimensionless quantity and provides insights into the composition without involving mass or volume. To calculate the mole fraction of a solute like H_2SO_4, you need to know the number of moles of the solute and solvent. In our case, we have\[5.83 \text{ moles of } H_2SO_4\] and \[42.06 \text{ moles of water}\].The mole fraction (X_{solute}) can be found using the formula:\[X_{solute} = \frac{\text{Moles of Solute}}{\text{Moles of Solute} + \text{Moles of Solvent}}\]For H_2SO_4 in the solution, we calculate:\[X_{H_2SO_4} = \frac{5.83}{5.83 + 42.06} \approx 0.122.\]Thus, the mole fraction of sulfuric acid is 0.122. This means that 12.2% of the total moles in the solution are from H_2SO_4.

Molality

Molality is used to express the concentration of a solution, specifically accounting for the amount of solute per kilogram of solvent. Unlike molarity, molality doesn't change with temperature, making it useful for calculations involving temperature variations. To calculate molality, you need the moles of solute and the mass of solvent in kilograms.First, we find the moles of H_2SO_4, which is 5.83 moles, and the mass of water, the solvent, is 757.4 grams (or 0.7574 kg). Use the formula:\[\text{Molality} = \frac{\text{Moles of Solute}}{\text{Mass of Solvent in kg}}\]Substituting in our values:\[\text{Molality} = \frac{5.83}{0.7574} \approx 7.69 \text{ mol/kg}.\]This means there are 7.69 moles of sulfuric acid per kilogram of water. Molality is particularly handy in colligative properties calculations, like boiling point elevation and freezing point depression.

Molarity

Molarity measures the concentration of a solute in a solution, expressed as moles of solute per liter of solution. It's one of the most common ways to represent concentration, especially in chemistry labs. Molarity is straightforward to calculate using the formula: \[\text{Molarity (M)} = \frac{\text{Moles of Solute}}{\text{Volume of Solution in Liters}}\]In our exercise, there are 5.83 moles of H_2SO_4 in 1 liter of solution. So its molarity is:\[M = \frac{5.83}{1} = 5.83 \text{ M}\]This indicates a concentration of 5.83 moles of sulfuric acid per liter. When preparing solutions or performing reactions, knowing the molarity helps in determining the exact amount needed to achieve desired reactant concentrations.