Calculate the mass of ascorbic acid (Vitamin C, C6H8O6) to be dissolved in 75 g of acetic acid to lower its melting point by 1.5°C. Kf = 3.9 K kg mol^-1.

Steps to Solve

1. Identify Key Variables

We know the following values:

• Mass of acetic acid ($m = 75 , \text{g}$)
• Freezing point depression ($\Delta T_f = 1.5 , \text{°C}$)
• Freezing point depression constant of acetic acid ($K_f = 3.9 , \text{K kg mol}^{-1}$)

2. Convert Mass of Acetic Acid to kg

To use the freezing point depression formula, convert the mass from grams to kilograms: $$ m_{\text{acetic acid}} = \frac{75 , \text{g}}{1000} = 0.075 , \text{kg} $$

3. Use the Freezing Point Depression Formula

The freezing point depression formula is given by: $$ \Delta T_f = K_f \cdot m_{\text{solute}} $$ Where:

• $\Delta T_f$ is the decrease in the freezing point
• $K_f$ is the freezing point depression constant
• $m_{\text{solute}}$ is the molality of the solute

4. Calculate Molality Required for the Depression

Rearranging the formula to find molality: $$ m_{\text{solute}} = \frac{\Delta T_f}{K_f} = \frac{1.5}{3.9} = 0.3846 , \text{mol/kg} $$

5. Calculate Moles of Ascorbic Acid Needed

We can relate molality to moles of solute ($n$) and mass of solvent: $$ m_{\text{solute}} = \frac{n}{m_{\text{solvent}}} $$ Where $m_{\text{solvent}} = 0.075 , \text{kg}$. Therefore: $$ n = m_{\text{solute}} \cdot m_{\text{solvent}} = 0.3846 , \text{mol/kg} \cdot 0.075 , \text{kg} = 0.0289 , \text{mol} $$

6. Calculate Mass of Ascorbic Acid

Now we need the molar mass of ascorbic acid (C₆H₈O₆): $$ \text{Molar mass} = 6(12.01) + 8(1.008) + 6(16.00) = 176.12 , \text{g/mol} $$ Using the number of moles to find the mass: $$ \text{mass} = n \cdot \text{molar mass} = 0.0289 , \text{mol} \cdot 176.12 , \text{g/mol} = 5.09 , \text{g} $$