Question

Molarity of \( H_2SO_4 (aq.) \) solution is 4.9 M. If the density of the solution is 1.40 g/mL, then molality and mole fraction of solute in the solution is:
(Molar mass of \( H_2SO_4 = 98 \, \text{g mol}^{-1} \))

• A. \( m = 5.33 \), \( x_{\text{solute}} = 0.072 \)
• B. \( m = 5.33 \), \( x_{\text{solute}} = 0.087 \)
• C. \( m = 5.21 \), \( x_{\text{solute}} = 0.072 \)
• D. \( m = 5.21 \), \( x_{\text{solute}} = 0.087 \)

Hint:

To calculate molality and mole fraction, remember that molality involves the mass of the solvent in kilograms, and mole fraction is the ratio of moles of solute to the total moles of the solution.

Answer 1

The Correct Option is B

Step 1: Define the relationship for molarity and density.
Molarity (M) is defined as: \[ M = \frac{n_{\text{solute}}}{V_{\text{solution}}} \] where \( n_{\text{solute}} \) is the number of moles of solute and \( V_{\text{solution}} \) is the volume of the solution in liters. The mass of the solution is given by: \[ \text{Mass of solution} = \text{density} \times \text{volume} \] Given: - Molarity \( M = 4.9 \, \text{M} \) - Density \( \rho = 1.40 \, \text{g/mL} \) - Molar mass of \( H_2SO_4 = 98 \, \text{g/mole} \)
Step 2: Calculate the number of moles of solute.
First, calculate the moles of solute in the solution: \[ n_{\text{solute}} = M \times V_{\text{solution}} = 4.9 \, \text{mol/L} \times 1 \, \text{L} = 4.9 \, \text{mol} \]
Step 3: Calculate the mass of the solution.
Since the density is \( 1.40 \, \text{g/mL} \), and assuming 1 liter of solution: \[ \text{Mass of solution} = 1.40 \, \text{g/mL} \times 1000 \, \text{mL} = 1400 \, \text{g} \]
Step 4: Calculate molality.
Molality \( m \) is given by: \[ m = \frac{n_{\text{solute}}}{\text{mass of solvent in kg}} = \frac{4.9 \, \text{mol}}{1400 \, \text{g} - \text{mass of solute}} \] The mass of the solute is: \[ \text{Mass of solute} = 4.9 \, \text{mol} \times 98 \, \text{g/mol} = 480.2 \, \text{g} \] Thus: \[ \text{Mass of solvent} = 1400 \, \text{g} - 480.2 \, \text{g} = 919.8 \, \text{g} = 0.92 \, \text{kg} \] Now, calculate molality: \[ m = \frac{4.9 \, \text{mol}}{0.92 \, \text{kg}} = 5.33 \, \text{mol/kg} \]
Step 5: Calculate the mole fraction of solute.
The mole fraction of the solute \( x_{\text{solute}} \) is: \[ x_{\text{solute}} = \frac{n_{\text{solute}}}{n_{\text{solute}} + n_{\text{solvent}}} \] First, calculate the moles of solvent: \[ n_{\text{solvent}} = \frac{919.8 \, \text{g}}{18 \, \text{g/mol}} = 51.1 \, \text{mol} \] Now, calculate the mole fraction: \[ x_{\text{solute}} = \frac{4.9 \, \text{mol}}{4.9 \, \text{mol} + 51.1 \, \text{mol}} = 0.087 \] Final Answer: \( m = 5.33 \, \text{mol/kg}, x_{\text{solute}} = 0.087 \)