Question

A mixture of alcohol and water contains {\text{54% }} water by mass. Calculate the mole fraction of alcohol.

Answer 1

In the above question we have to find out the mole fraction of alcohol. First we have to find out the number of moles of alcohol as well as water then we can find out the mole fraction of alcohol by dividing the number of moles of alcohol by the total number of moles present in the solution.
Formula used:
${{\chi = }}\dfrac{{{{\text{n}}_{\text{s}}}}}{{\text{n}}}$
where ${{\chi }}$ = mole fraction of the substance.
${{\text{n}}_{\text{s}}}$= number of mole of that substance whose mole fraction we want to find.
n = total moles present in the solution.

Complete step by step answer:
Since, the percentage of water is given in the question. Suppose we have 100g of mixture which implies that 54g of water is present
And weight of alcohol present is ${\text{(100}} - {\text{54)g = 46g}}$
Now we want find the number of moles present in the mixture by the formula:
${\text{n = }}\dfrac{{\text{m}}}{{\text{M}}}$
where n= number of moles
m = given mass
M = molar mass
So, first we have to find out the molar mass of alcohol and water.
Molar mass of alcohol (${{\text{C}}_{\text{2}}}{{\text{H}}_{\text{5}}}{\text{OH}}$), $M_1$ = 2 $\times$atomic mass of carbon + 6 $\times$ atomic mass of hydrogen + 1$\times$ atomic mass of oxygen
So, $M_1$ = ${{2 \times 12 + 6 \times 1 + 1 \times 16 = 46g}}$
Given mass, $m_1$ = 46g
So number of moles of alcohol ($n_1$) is $\dfrac{{{{m_1}}}}{{{{M_1}}}} = \dfrac{{46}}{{46}}{\text{ = 1mole}}$
Molar mass of alcohol (${{\text{H}}_2}{\text{O}}$), $M_1$ = 2 $\times$ atomic mass of hydrogen + 1$\times$ atomic mass of oxygen
So, $M_2$ = ${{2 \times 1 + 1 \times 16 = 18g}}$
Given mass, m2 = 54g
So number of moles of alcohol ($n_2$) is $\dfrac{{{{m_2}}}}{{{{M_2}}}}{\text{ = }}\dfrac{{{\text{54}}}}{{{\text{18}}}}{\text{ = 3moles}}$
Mole fraction alcohol = $\dfrac{{{{n_1}}}}{{{{n_1 + n_2}}}}{\text{ = }}\dfrac{1}{{1 + 3}}{\text{ = 0}}{\text{.25}}$

$\therefore$ Mole fraction of alcohol is ${\text{0}}{\text{.25}}$.

Note:
In these types of questions where you were given a percentage of substance present, take the amount of mixture taken as ${\text{100g}}$ which makes calculation easier.
Sometimes, we get confused by mole fraction and number of moles. Remember, when the total number of moles is equal to one, then only mole fraction is equal to the number of moles of that substance. Normally, mole fraction is always less than 1.