How To Calculate A Mole Fraction

When analyzing solutions, chemists measure concentrations of components in moles. The mole fraction of a solute is the ratio of the number of moles of that solute to the total number of moles of solute and solvent in solution. Because it's a ratio of moles to moles, the mole fraction is a dimensionless number, and of course, it's always less than one.

The mole fraction formula is straightforward. In any solution, the mole fraction of solute A is:

\(\text{mole fraction of A} = \frac{\text{moles of A}}{\text{total moles}}\)

and the mole fraction of the solvent:

\(\text{mole fraction of solvent} = \frac{\text{moles of solvent}}{\text{total moles}}\)

In some situations, you may not be given the number of moles directly. You can calculate it if you know the chemical formulas of the compounds and their weights or volumes. To do this, it helps to know what a mole is.

Each element in the periodic table has a characteristic mass, and by virtue of this, every compound also has a characteristic mass. At the atomic level, mass is measured in atomic mass units, but chemists need a way to express mass in macroscopic terms. To this end, they define a mole of any element or compounds as Avogadro's number (6.022 × 10²³) of atoms or molecules. The mass of this many particles, measured in grams, is the same number as the molecular mass, measured in atomic mass units.

The definition of a mole is thus the mass of any compound, measured in grams, that equals the masses of the component elements measured in atomic mass units. To calculate the number of moles of a compound you have on hand, you divide the mass by the mass of one mole of the compound, which you can calculate from the periodic table.

The mole fraction formula is particularly easy to understand and use if you happen to know the number of moles of all of the solutes and the solvent. For example, suppose you have 2 moles of carbon tetrachloride (CCl4), 3 moles of benzene (C₆H₆) and 4 moles of acetone (C₃H₆O). The total number of moles in solution is 9. The mole fraction equation tells you that the mole fraction of carbon tetrachloride is 2/9 = 0.22. Similarly, the mole fraction of benzene is 3/9 = 0.33 and the mole fraction of acetone is 4/9 = 0.44.

Things get more complicated if you only know the mass of one or more components of a solution, but only a little more. All you have to do is convert the mass of the component to number of moles, and that's a straightforward arithmetic problem, as long as you know the chemical formula.

Suppose you dissolve 77 grams of carbon tetrachloride (CCl₄) in 78 grams of acetone (C₃H₆O). What are the mole fractions of each compound in the solution?

Resist the urge to divide the mass of carbon tetrachloride by that of acetone. Since they are almost the same, the result would be 0.5 for each compound, and that would give an incorrect result for acetone. First, you have to convert the masses to the number of moles of each compound, and to do that, you have to look up the atomic masses of each of the elements in the periodic table.

The atomic mass of carbon is 12.0 amu (rounding to one decimal place) and that of chlorine is 35.5 amu, so one mole of carbon tetrachloride weighs 154 grams. You have 77 grams, which is 77/154 = 0.5 moles.

Noting that the atomic mass of hydrogen is 1 amu and that of oxygen is 16 amu, the molar mass of acetone is 58 grams. You have 78 grams, which is 1.34 moles. That means the total number of moles in solution is 1.84. Now you're ready to calculate mole fractions using the mole fraction equation.

\(\text{mole fraction of carbon tetrachloride}=\frac{0.5\text{ moles}}{1.84\text{ moles}}=0.27\text{ }\text{ mole fraction of acetone}=\frac{1.34\text{ moles}}{1.84\text{ moles}}=0.73\)