Calculate Mole Fraction of Water in Aqueous Urea Solution

To find the mole fraction of water in a $10\%$ by weight ($\text{w/w}$) aqueous urea solution, we can assume a total mass of $100\text{ g}$ for the solution.

Step 1: Determine the masses of urea and water in the solution. For a $100\text{ g}$ aqueous urea solution:

• Mass of urea ($\text{NH}_2\text{CONH}_2$), $m_{\text{urea}} = 10\text{ g}$
• Mass of water ($\text{H}_2\text{O}$), $m_{\text{water}} = 100\text{ g} - 10\text{ g} = 90\text{ g}$

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Step 2: Calculate the molar masses.

• Molar mass of urea ($\text{NH}_2\text{CONH}_2$): $M_{\text{urea}} = 2 \times 14 + 4 \times 1 + 12 + 16 = 60\text{ g mol}^{-1}$
• Molar mass of water ($\text{H}_2\text{O}$): $M_{\text{water}} = 2 \times 1 + 16 = 18\text{ g mol}^{-1}$

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Step 3: Calculate the number of moles of each component.

• Moles of urea ($n_{\text{urea}}$): $n_{\text{urea}} = \frac{m_{\text{urea}}}{M_{\text{urea}}} = \frac{10\text{ g}}{60\text{ g mol}^{-1}} = \frac{1}{6}\text{ mol} \approx 0.167\text{ mol}$

• Moles of water ($n_{\text{water}}$): $n_{\text{water}} = \frac{m_{\text{water}}}{M_{\text{water}}} = \frac{90\text{ g}}{18\text{ g mol}^{-1}} = 5\text{ mol}$

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Step 4: Calculate the mole fraction of water ($x_{\text{water}}$). $x_{\text{water}} = \frac{n_{\text{water}}}{n_{\text{water}} + n_{\text{urea}}}$

Substituting the calculated values: $x_{\text{water}} = \frac{5}{5 + \frac{1}{6}} = \frac{5}{\frac{31}{6}} = \frac{30}{31} \approx 0.9677 \approx 0.967$

Thus, the mole fraction of water is approximately $0.967$.

Correct Option: D