# What are the mole fractions of hydrochloric acid ("HCl") and water in a 20% "w/w" aqueous "HCl" solution?

Jan 29, 2018

$0.1$ and $0.9$, respectively.

#### Explanation:

You know that your solution is 20% hydrochloric acid by mass, which means that every $\text{100 g}$ of this solution contain $\text{20 g}$ of hydrochloric acid, the solute.

To make the calculations easier, pick a sample of this solution that has a mass of exactly $\text{100 g}$. Since you already know that this sample contains $\text{20 g}$ of hydrochloric acid, you can say that the mass of water, the solvent, will be

overbrace("mass H"_2"O")^(color(blue)("mass of solvent")) = overbrace("100 g")^(color(blue)("mass of solution")) - overbrace("20 g")^(color(blue)("mass of solute"))

$\text{mass H"_2"O" = "80 g}$

Nest, use the molar mass of hydrochloric acid and the molar mass of water to convert the masses to moles.

20 color(red)(cancel(color(black)("g"))) * "1 mole HCl"/(36.46color(red)(cancel(color(black)("g")))) = "0.5485 moles HCl"

80 color(red)(cancel(color(black)("g"))) * ("1 mole H"_2"O")/(18.015color(red)(cancel(color(black)("g")))) = "4.4407 moles H"_2"O"

Now, the mole fraction of hydrochloric acid is given by the ratio that exists between the number of moles of hydrochloric acid and the total number of moles present in the solution.

chi_ ("HCl") = "moles HCl"/"total moles"

In this case, the mole fraction of hydrochloric acid will be

chi_ ("HCl") = (0.5485 color(red)(cancel(color(black)("moles"))))/((0.5485 + 4.4407) color(red)(cancel(color(black)("moles")))) = color(darkgreen)(ul(color(black)(0.1)))

To find the mole fraction of water, use the fact that the mole fractions of the solute and of the solute, respectively, must add up to give $1$.

${\chi}_{\text{H"_ 2"O") = 1 - chi_ ("HCl}}$

You will find

${\chi}_{\text{H"_ 2"O}} = 1 - 0.1 = \textcolor{\mathrm{da} r k g r e e n}{\underline{\textcolor{b l a c k}{0.9}}}$

The answers are rounded to one significant figure.