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: Establish the relationship between molarity and density.
Molarity (M) is defined as: \[ M = \frac{n_{\text{solute}}}{V_{\text{solution}}} \] where \( n_{\text{solute}} \) represents the number of moles of solute and \( V_{\text{solution}} \) is the volume of the solution in litres. The mass of the solution can be calculated using: \[ \text{Mass of solution} = \text{Density} \times \text{Volume} \] Given: - Molarity \( M = 4.9 \, \text{M} \) - Density \( \rho = 1.40 \, \text{g/mL} \) - Molar mass of \( H_2SO_4 = 98 \, \text{g/mol} \)
Step 2: Calculate the moles of solute in 1 litre of solution.
Assuming 1 litre of solution: \[ n_{\text{solute}} = M \times V \] \[ n_{\text{solute}} = 4.9 \times 1 = 4.9 \, \text{mol} \]
Step 3: Determine the mass of the solution.
Using the given density: \[ \text{Mass of solution} = 1.40 \, \text{g/mL} \times 1000 \, \text{mL} \] \[ \text{Mass of solution} = 1400 \, \text{g} \]
Step 4: Calculate the molality of the solution.
First, calculate the mass of solute: \[ \text{Mass of solute} = 4.9 \times 98 = 480.2 \, \text{g} \] Then, determine the mass of solvent: \[ \text{Mass of solvent} = 1400 - 480.2 = 919.8 \, \text{g} \] \[ \text{Mass of solvent} = 0.92 \, \text{kg} \] Now, molality \( m \) is: \[ m = \frac{n_{\text{solute}}}{\text{mass of solvent in kg}} \] \[ m = \frac{4.9}{0.92} = 5.33 \, \text{mol/kg} \]
Step 5: Calculate the mole fraction of the solute.
The mole fraction of the solute is given by: \[ x_{\text{solute}} = \frac{n_{\text{solute}}}{n_{\text{solute}} + n_{\text{solvent}}} \] First, calculate the moles of solvent (water): \[ n_{\text{solvent}} = \frac{919.8}{18} = 51.1 \, \text{mol} \] Now, substitute the values: \[ x_{\text{solute}} = \frac{4.9}{4.9 + 51.1} \] \[ x_{\text{solute}} = 0.087 \]
Final Answer: \( m = 5.33 \, \text{mol/kg}, \quad x_{\text{solute}} = 0.087 \)