# What is the mole fraction of "Na"_2"SO"_4 in a solution which is 11.5% by weight "Na"_2"SO"_4 ?

Mar 15, 2018

$0.0162$

#### Explanation:

Your starting point here will be the solution's percent concentration by mass.

Your solution is said to have a percent concentration by mass of 11.5%, which means that for every $\text{100.0 g}$ of this solution, you get $\text{11.5 g}$ of sodium sulfate, the solute.

To make the calculations easier, let's pick a sample of this solution that has a mass of exactly $\text{100.0 g}$. Since you know how many grams of sodium sulfate it contains, you can calculate the mass of water present in the sample.

overbrace("mass of water")^(color(blue)("mass solvent")) = overbrace("100.0 g")^(color(blue)("mass solution")) - overbrace("11.5 g")^(color(blue)("mass solute"))

$\text{mass of water = 88.5 g}$

Now, use the molar mass of water to convert the mass to moles.

88.5 color(red)(cancel(color(black)("g"))) * ("1 mole H"_2"O")/(18.015color(red)(cancel(color(black)("g")))) = "4.9126 moles H"_2"O"

Do the same for sodium sulfate, i.e. use its molar mass to convert the mass to moles.

11.5 color(red)(cancel(color(black)("g"))) * ("1 mole Na"_2"SO"_4)/(142.04color(red)(cancel(color(black)("g")))) = "0.08096 moles Na"_2"SO"_4

To find the mole fraction of sodium sulfate, ${\chi}_{{\text{Na"_ 2"SO}}_{4}}$, you must divide the number of moles of sodium sulfate by the total number of moles present in the solution.

chi_ ("Na"_ 2"SO"_ 4) = (0.08096 color(red)(cancel(color(black)("moles"))))/((4.9126 + 0.08096)color(red)(cancel(color(black)("moles")))) = color(darkgreen)(ul(color(black)(0.0162)))

The answer is rounded to three sig figs.