Final answer:
The mole fraction of the solute in the diluted urea solution, after diluting 1 kg of a solution with mole fraction 0.2 to 5 kg, is approximately 0.111. This calculation is based on the principle that the number of moles of solute remains constant during dilution, while the total number of moles increases due to the added solvent.
Step-by-step explanation:
To find the mole fraction of the solute in the diluted urea solution, we use the concept of mole fraction which is expressed as the ratio of the number of moles of the solute to the total number of moles in the solution. If we start with 1 kg of an aqueous urea solution with a mole fraction of 0.2 and dilute it to 5 kg, the amount of solute (urea) remains constant, while the amount of solvent has increased.
Using the ratio approach, the mole fraction (x) of urea after dilution will be:
- Initial moles of urea (n₁) = Total initial moles × mole fraction of urea = n₁ + n₂ (where n₂ is the initial moles of water)
- After dilution, total moles (n₃) = n₁ + 5n₂ (since the mass of the solution has been quintupled)
- The mole fraction of urea after dilution (x') = n₁ / n₃
- Substituting the values we get x' = n₁ / (n₁ + 5n₂)
- Substituting the mole fraction we get x' = (0.2 × (n₁ + n₂)) / ((n₁ + n₂) + 4n₂)
- Simplifying it further, x' = 0.2 / (1 + 4(1-0.2)) = 0.2 / 1.8
- Therefore, the mole fraction of urea in the diluted solution is 0.2/1.8, which is approximately 0.111.
Thus, the mole fraction of solute in the diluted urea solution is approximately 0.111.