Answer :
To find the molar mass of the unknown compound, we can use the concept of freezing point depression, which is a colligative property. The formula for freezing point depression is given by:
[tex]\Delta T_f = i \cdot K_f \cdot m[/tex]
Where:
- [tex]\Delta T_f[/tex] is the change in freezing point.
- [tex]i[/tex] is the van't Hoff factor (number of particles the solute breaks into, for non-electrolytes like urea and the unknown compound, [tex]i = 1[/tex]).
- [tex]K_f[/tex] is the cryoscopic constant of the solvent.
- [tex]m[/tex] is the molality of the solution.
First, let's calculate [tex]K_f[/tex] using the data for urea:
For urea (a non-electrolyte, so [tex]i=1[/tex]):
- Mass of urea = 0.02 g
- Molar mass of urea = 60.06 g/mol (assuming you have this value)
- Solvent mass = 98.5 g = 0.0985 kg
- [tex]\Delta T_f[/tex] for urea = 0.211 K
First, convert the mass of urea to moles:
[tex]moles\ of\ urea = \frac{0.02\ \text{g}}{60.06\ \text{g/mol}} \approx 0.000333\ \text{mol}[/tex]
Calculate the molality of the urea solution:
[tex]m = \frac{0.000333\ \text{mol}}{0.0985\ \text{kg}} \approx 0.00338\ \text{mol/kg}[/tex]
Now, use the freezing point depression formula to find [tex]K_f[/tex]:
[tex]0.211 = 1 \cdot K_f \cdot 0.00338\
\Rightarrow K_f = \frac{0.211}{0.00338} \approx 62.43\ \text{K kg/mol}[/tex]
Now that we have [tex]K_f[/tex], let's use the data for the unknown compound:
For the unknown compound:
- Mass of the unknown compound = 1.60 g
- (\Delta T_f = 0.34\ \text{K}
)
Let [tex]M[/tex] be the molar mass of the unknown compound. Find the molality of the solution:
[tex]m = \frac{\text{mass of solute (in moles)}}{\text{mass of solvent (in kg)}}[/tex]
[tex]0.34 = 1 \cdot 62.43 \cdot \frac{1.60 / M}{0.0985}[/tex]
Solving for [tex]M[/tex]:
[tex]0.34 \cdot 0.0985 = 62.43 \cdot \frac{1.60}{M}[/tex]
[tex]M \approx \frac{62.43 \times 1.60}{0.34 \times 0.0985} \approx 92.56\ \text{g/mol}[/tex]
Thus, the molar mass of the unknown compound is approximately [tex]92.56\ \text{g/mol}[/tex].
