Final answer:
The mole fraction of urea in a 47% by mass aqueous solution is calculated by dividing the moles of urea by the total moles of all substances in the solution. The moles of urea are found by dividing its mass by its molar mass (60 g/mol), and the moles of water by dividing its mass by its molar mass (18 g/mol).
Step-by-step explanation:
The mole fraction of urea in an aqueous solution that is 47% by mass can be calculated by first determining the number of moles of urea and water in the solution. The mole fraction, X, is the ratio of the moles of urea, nurea, to the total moles of substances present in the solution, ntotal.
To calculate the moles of urea, you would divide the mass of urea by its molar mass (MM). Assuming we have 100 g of the solution (to make the calculation easier since it's a percentage), we would have 47 g of urea and 53 g of water. So, the moles of urea would be:
moles of urea = mass of urea / MM of urea = 47 g / 60 g/mol
And for water (MM = 18 g/mol):
moles of water = mass of water / MM of water = 53 g / 18 g/mol
Then, you would add the moles of urea and water together to find ntotal and divide the moles of urea by ntotal for Xurea.
The mole fraction of urea in the solution is:
Xurea = nurea / ntotal