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Step-by-Step Solution
Step 1: Identify the Known Parameters
• Depression in freezing point, $ \Delta T_f = 3.82^\circ \text{C}$
• Freezing point depression constant for water, $K_f = 1.86^\circ \text{C} \, \text{m}^{-1}$
• Mass of solute (Na2SO4), $w_B = 5.00 \text{ g}$
• Molar mass of Na2SO4, $m_B = 142 \text{ g mol}^{-1}$
• Mass of solvent (water), $w_A = 45.0 \text{ g}$
Step 2: Write the Formula for Depression in Freezing Point
Depression in freezing point ($ \Delta T_f $) is related to the van’t Hoff factor ($i$), the freezing point depression constant ($K_f$), and the molality ($m$) of the solution by the relation:
$ \Delta T_f = i \times K_f \times m$
Step 3: Express Molality in Terms of Known Quantities
Molality $m$ is given by:
$ m = \dfrac{w_B \times 1000}{m_B \times w_A}$
where
• $w_B$ is the mass of solute in grams,
• $m_B$ is the molar mass of solute (g/mol),
• $w_A$ is the mass of solvent in grams.
Step 4: Combine and Rearrange to Solve for the Van’t Hoff Factor
Substituting $m$ into the expression for $ \Delta T_f $:
$ \Delta T_f = i \times K_f \times \dfrac{w_B \times 1000}{m_B \times w_A}$
So,
$ i = \dfrac{\Delta T_f \times m_B \times w_A}{K_f \times w_B \times 1000}$
Step 5: Substitute Numerical Values
Putting the given values into the formula:
$ i = \dfrac{3.82 \times 142 \times 45}{1.86 \times 5 \times 1000}$
Step 6: Calculate the Numerical Result
Perform the multiplication and division:
$ i = \dfrac{3.82 \times 142 \times 45}{1.86 \times 5 \times 1000} = 2.63
$
Step 7: Conclude the Van’t Hoff Factor
The van’t Hoff factor (i) for Na2SO4 under the given conditions is
2.63.
