Question

A mixture of ethanol and water contains $54\%$ water by mass. Calculate the mole fraction of ethanol in this solution.
A.$0.25$
B.$0.15$
C.$0.20$
D.$0.30$

Answer 1

To answer this question we must need to know the definition for the mass percent and formula for mole fraction. We will get the value of mass and hence number of moles from the mass percent. We know the total percentage is always 100.

Formula used: ${\text{number of mole (n) = }}\dfrac{{{\text{mass of substance}}}}{{{\text{Molar mass}}}}$
${\text{mole fraction of A ( }}{\chi _A}{\text{ ) = }}\dfrac{{{\text{no}}{\text{. of moles of A}}}}{{{\text{total number of moles}}}}$

Complete step by step solution:
We know the percentage is always calculated from 100, so if the mass percent of water is $54\%$ then the mass percent for ethanol will be $100 - 54\% = 46\%$.
Mass percentage is defined as the mass of substance present in the 100 g of solution.
So this means that $54\%$ water by mass means 54 g of water is present in the 100 g of solution and $46\%$ g of ethanol is present in 100 g water.
To calculate mole fraction we need to know the molar mass of water and ethanol.
${\text{Molar mass of }}{{\text{H}}_2}{\text{O = 2}} \times 1 + 16 = 18$
${\text{Molar mass of }}{{\text{C}}_2}{{\text{H}}_5}{\text{OH = 2}} \times 12 + {\text{ 5}} \times 1 + 16 + 1 = 46$
Now we we will calculate the number of moles in each case,
${{\text{n}}_{{{\text{H}}_2}{\text{O}}}} = \dfrac{{54}}{{18}} = 3$ And
${{\text{n}}_{{{\text{C}}_2}{{\text{H}}_5}{\text{OH}}}} = \dfrac{{46}}{{46}} = 1$
Mole fraction of ethanol will be:
${\text{mole fraction of ethanol ( }}{\chi _A}{\text{ ) = }}\dfrac{{{\text{no}}{\text{. of moles of ethanol}}}}{{{\text{total number of moles}}}}$
Putting the values,
${\chi _A} = \dfrac{1}{4} = 0.25$
Hence the correct option is A

Note: The sum of mole fraction of the entire component in a mixture will be 1. This is because a fraction has a maximum value of 1. For example if we have a mixture of two components water and ethanol then the sum of mole fraction of water and ethanol will be 1. So if we know the value of 1 the other can be calculated by subtracting the first one from 1. As we have calculated, the mole fraction of ethanol is 1, so the mole fraction of water will be $1 - 0.25 = 0.75$.