Mole Fraction and Mole Percent

Key Concepts

• Mole fraction and mole percent are ways of expressing the concentration of a solution.

Mole Fraction

• Mole fraction is the ratio of moles of one compound to the total number of moles present.
• Mole fraction is represented by the symbol Χ
• Χ_a = n_a ÷ (n_a + n_b + n_c + .....)

  Χ_a = mole fraction of component a
  n_a = moles of component a
  n_b = moles of component b
  n_c = moles of component c
  etc

• For a two component system, one component is the solute and the other is the solvent:

  Χ_(solute) = n _(solute) ÷ (n_(solute) + n_(solvent))
  Χ_(solvent) = n _(solvent) ÷ (n_(solvent) + n_(solute))

• The sum of the mole fractions for each component in a solution will be equal to 1.

  For a solution containing 2 components, solute and solvent,
  Χ_solute + Χ_solvent = 1

Mole Percent

• Mole percent is the percentage of the total moles that is of a particular component.
• Mole percent is equal to the mole fraction for the component multiplied by 100:

  mol % a = Χ_a × 100

• The sum of the mole percents for each component in a solution will be equal to 100.

  For a solution containing 2 components, solute and solvent,
  mol % solute + mol % solvent = 100

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Example: Calculating Mole Fraction of Solute and Solvent

Question: What is the mole fraction and mole percent of sodium chloride and the mole fraction and mole percent of water in an aqueous solution containing 25.0 g water and 5.0 g sodium chloride?

Solution:

1. Identify the components making up the solution:

   solute = sodium chloride = NaCl
   solvent = water = H₂O (because this is an aqueous solution)

2. Write the equation for finding the mole fraction of solute:

   Χ_(NaCl) = n_(NaCl) ÷ (n_(NaCl) + n_(H2O))

3. Calculate the moles of each component:

   moles of solute (NaCl): n(NaCl) = mass ÷ molar mass mass (NaCl) = 5.0 g molar mass (NaCl) = 22.99 + 35.45 = 58.44 g mol-1 n (NaCl) = 5.0 ÷ 58.44 = 0.08556 mol

   moles of solvent (H2O): n(H2O) = mass ÷ molar mass mass (H2O) = 25.0 g molar mass (H2O) = 2 × 1.008 + 16.00 = 18.016 g mol-1 n (H2O) = 25.0 ÷ 18.016 = 1.3877 mol

4. Calculate the mole fraction of sodium chloride:

   Χ_(NaCl) = n_(NaCl) ÷ (n_(NaCl) + n_(H2O))
   Χ_(NaCl) = 0.08556 ÷ (0.08556 + 1.3877)
   Χ_(NaCl) = 0.058

5. Calculate mole percent of NaCl:

   mole percent of NaCl = 100 × Χ_(NaCl)
   mole percent of NaCl= 100 × 0.058
   mole percent of NaCl= 5.8%

6. Write the equation for finding the mole fraction of solvent:

   Χ_(H2O) = 1 - Χ_(NaCl)

7. Calculate the mole fraction of solvent (water):

   mole fraction of solvent (water) = 1 - Χ_(NaCl)
   mole fraction of solvent (water) = 1 - 0.058
   mole fraction of solvent (water) = 0.942

8. Calculate mole percent of H₂O

   mole percent of H₂O = 100 × Χ_(H2O)
   mole percent of H₂O = 100 × 0.942
   mole percent of H₂O = 94.2%

Example: Calculating Mass of Solute or Solvent

Question: A particular aqueous solution contains 7.5 moles of water and a mole fraction of sodium chloride of 0.125.
How many grams of sodium chloride are present in the solution?

Solution:

1. Re-arrange the equation Χ_(NaCl) = n_(NaCl) ÷ (n_(NaCl) + n_(H2O)) to find n_(NaCl):

   Multiply both sides by (n_(NaCl) + n_(H2O)):

   Χ_(NaCl) × (n_(NaCl) + n_(H2O)) = n_(NaCl)

   Expand the expression on the left hand side:

   (Χ_(NaCl) × n_(NaCl)) + (Χ_(NaCl) × n_(H2O)) = n_(NaCl)

   Collect like terms (n_(NaCl)):

   Χ_(NaCl) × n_(H2O) = n_(NaCl) - (Χ_(NaCl) × n_(NaCl))

   Re-arrange the expression for n_(NaCl)

   Χ_(NaCl) × n_(H2O) = n_(NaCl)(1 - Χ_(NaCl))

   Divide throughout by (1 - Χ_(NaCl)) to get an expression for n_(NaCl):

   (Χ_(NaCl) × n_(H2O)) ÷ (1 - Χ_(NaCl)) = n_(NaCl)

2. Calculate moles of NaCl:

   n_(NaCl) = (Χ_(NaCl) × n_(H2O)) ÷ (1 - Χ_(NaCl))
   n_(NaCl) = (0.125 × 7.5) ÷ (1 - 0.125)
   n_(NaCl) = 1.071 mol

3. Calculate mass of NaCl:

   n(NaCl) = mass(NaCl) ÷ molar mass(NaCl)
   mass(NaCl) = n(NaCl) × molar mass(NaCl)
   mass(NaCl) = 1.071 × 58.44
   mass(NaCl) = 62.6 g