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Answer :
To find the boiling point of a solution when urea is dissolved in substance X, we follow these steps:
1. Identify the Known Information:
- The normal boiling point of substance X is [tex]\(102.1^{\circ} \text{C}\)[/tex].
- The molal boiling point elevation constant ([tex]\(K_b\)[/tex]) for X is [tex]\(2.21^{\circ} \text{C} \cdot \text{kg} \cdot \text{mol}^{-1}\)[/tex].
- The mass of urea ([tex]\(\left(\text{NH}_2\right)_2\text{CO}\)[/tex]) is [tex]\(38.8 \text{ g}\)[/tex].
- The mass of the solvent (substance X) is [tex]\(400 \text{ g}\)[/tex].
2. Calculate the Molar Mass of Urea:
- The molar mass of urea:
[tex]\[
= 2 \times (\text{N: }14.01) + 4 \times (\text{H: }1.01) + (\text{C: }12.01) + (\text{O: }16.00) = 60.07 \text{ g/mol}
\][/tex]
3. Determine the Moles of Urea:
- Moles of urea:
[tex]\[
= \frac{\text{mass of urea}}{\text{molar mass of urea}} = \frac{38.8 \text{ g}}{60.07 \text{ g/mol}} = 0.6459 \text{ moles} \text{ (rounded to 4 significant digits)}
\][/tex]
4. Convert Mass of Solvent to Kilograms:
- Mass of substance X in kilograms:
[tex]\[
= \frac{400 \text{ g}}{1000} = 0.4 \text{ kg}
\][/tex]
5. Calculate Molality of the Solution:
- Molality:
[tex]\[
= \frac{\text{moles of urea}}{\text{mass of solvent in kg}} = \frac{0.6459 \text{ moles}}{0.4 \text{ kg}} = 1.615 \text{ mol/kg} \text{ (rounded to 4 significant digits)}
\][/tex]
6. Calculate Boiling Point Elevation ([tex]\(\Delta T_b\)[/tex]):
- Boiling point elevation:
[tex]\[
\Delta T_b = K_b \times \text{molality} = 2.21^{\circ} \text{C} \cdot \left(1.615 \text{ mol/kg}\right) = 3.569^{\circ} \text{C} \text{ (rounded to 4 significant digits)}
\][/tex]
7. Calculate the New Boiling Point of the Solution:
- New boiling point:
[tex]\[
= \text{normal boiling point} + \Delta T_b = 102.1^{\circ} \text{C} + 3.569^{\circ} \text{C} = 105.7^{\circ} \text{C} \text{ (rounded to 4 significant digits)}
\][/tex]
Thus, the boiling point of the solution is [tex]\(105.7^{\circ} \text{C}\)[/tex].
1. Identify the Known Information:
- The normal boiling point of substance X is [tex]\(102.1^{\circ} \text{C}\)[/tex].
- The molal boiling point elevation constant ([tex]\(K_b\)[/tex]) for X is [tex]\(2.21^{\circ} \text{C} \cdot \text{kg} \cdot \text{mol}^{-1}\)[/tex].
- The mass of urea ([tex]\(\left(\text{NH}_2\right)_2\text{CO}\)[/tex]) is [tex]\(38.8 \text{ g}\)[/tex].
- The mass of the solvent (substance X) is [tex]\(400 \text{ g}\)[/tex].
2. Calculate the Molar Mass of Urea:
- The molar mass of urea:
[tex]\[
= 2 \times (\text{N: }14.01) + 4 \times (\text{H: }1.01) + (\text{C: }12.01) + (\text{O: }16.00) = 60.07 \text{ g/mol}
\][/tex]
3. Determine the Moles of Urea:
- Moles of urea:
[tex]\[
= \frac{\text{mass of urea}}{\text{molar mass of urea}} = \frac{38.8 \text{ g}}{60.07 \text{ g/mol}} = 0.6459 \text{ moles} \text{ (rounded to 4 significant digits)}
\][/tex]
4. Convert Mass of Solvent to Kilograms:
- Mass of substance X in kilograms:
[tex]\[
= \frac{400 \text{ g}}{1000} = 0.4 \text{ kg}
\][/tex]
5. Calculate Molality of the Solution:
- Molality:
[tex]\[
= \frac{\text{moles of urea}}{\text{mass of solvent in kg}} = \frac{0.6459 \text{ moles}}{0.4 \text{ kg}} = 1.615 \text{ mol/kg} \text{ (rounded to 4 significant digits)}
\][/tex]
6. Calculate Boiling Point Elevation ([tex]\(\Delta T_b\)[/tex]):
- Boiling point elevation:
[tex]\[
\Delta T_b = K_b \times \text{molality} = 2.21^{\circ} \text{C} \cdot \left(1.615 \text{ mol/kg}\right) = 3.569^{\circ} \text{C} \text{ (rounded to 4 significant digits)}
\][/tex]
7. Calculate the New Boiling Point of the Solution:
- New boiling point:
[tex]\[
= \text{normal boiling point} + \Delta T_b = 102.1^{\circ} \text{C} + 3.569^{\circ} \text{C} = 105.7^{\circ} \text{C} \text{ (rounded to 4 significant digits)}
\][/tex]
Thus, the boiling point of the solution is [tex]\(105.7^{\circ} \text{C}\)[/tex].
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