Percent by Mass or Weight, and, Mass or Weight Fraction (w/w or m/m)

Key Concepts

• Mass fraction is also known as weight fraction.

  mass fraction is given the symbol m/m ¹
  weight fraction is given the symbol w/w

• Percent by mass is also known as percent by weight.

  percent by mass is given the symbol m/m % (or mass %)
  percent by weight is given the symbol w/w % (or wt %)

• Mass fraction (weight fraction) and percent by mass (percent by weight) are measurements of the concentration of a soluton.
• Mass fraction (weight fraction) is the ratio of the mass of one component of a solution to the total mass of the solution.²
• Mass fraction_(a) = mass_(a) ÷ (mass_(a) + mass_(b) + mass_(c) + ...)

  mass_(a) = mass of component a
  mass_(b) = mass of component b
  mass_(c) = mass of component c

  For a two component solution made up of a solute dissolved in a solvent:

  mass fraction (solute) = mass_(solute) ÷ (mass_(solute) + mass_(solvent))

• When added together, the mass fraction (weight fraction) of each component in the solution must equal one:

  mass fraction_(a) + mass fraction_(b) + mass fraction_(c) + .... = 1

  For a two component solution made up of a solute dissolved in a solvent:

  mass fraction _(solute) + mass fraction _(solvent) = 1

• Masses (weights) can be in any units as long as they are the same units for all components.
• Percent by mass (percent by weight) is the percent of the total mass of the solution that is one component.

  Percent by mass = mass fraction × 100
  OR
  Percent by weight = weight fraction × 100

  This can also be expressed as:

  mass % a = mass_(a) ÷ (mass_(a) + mass_(b) + mass_(c) + ...) × 100
  wt. % a = weight_(a) ÷ (weight_(a) + weight_(b) + weight_(c) + ...) × 100

• When added together, the percent by mass (or weight) of all components in the solution equals 100:

  mass % a + mass % b + mass % c + .... = 100
  OR
  wt % a + wt % b + wt % c + .... = 100

  For a two component solution made up of a solute dissolved in a solvent:

  mass % (solute) + mass % (solvent) = 100
  OR
  wt % (solute) + wt % (solvent) = 100

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Mass Fraction (Weight Fraction) Concentration Calculations

Question 1. What is the mass fraction (weight fraction) of sodium chloride in an aqueous solution containing 560 mg of sodium chloride and 10.4 g of water?

1. Extract the data from the question:

   Aqueous solution is made up of two components, a solute (NaCl) and a solvent (H₂O)
   mass_(NaCl) = 560 mg
   mass_(H2O) = 10.4 g

2. Convert all mass units to the same units (for example, all masses in grams):

   mass_(NaCl) = 560 mg = 560 × 10^-3 g = 0.560 g
   mass_(H2O) = 10.4 g

3. Write the equation for calculating the mass fraction (weight fraction) of sodium chloride:

   mass fraction_(NaCl) = mass_(NaCl) ÷ (mass_(NaCl) + mass_(H2O))

4. Substitute the values into the equation and solve:

   mass fraction_(NaCl) = 0.560 ÷ (0.560 + 10.4) = 0.051

Question 2. If 500 g of an aqueous solution of sucrose contains a mass fraction of water of 0.65, what mass of sucrose is present?

1. Extract the data from the question:

   Aqueous solution is made up of two components, a solute (sucrose) and a solvent (water)
   mass_{(aqueous solution)} = 500 g
   mass fraction_(water) = 0.65
   mass_(sucrose) = ? g

2. Calculate the mass fraction of sucrose:

   mass fraction_(sucrose) + mass fraction_(water) = 1

   mass fraction_(sucrose) = 1 - mass fraction_(water) = 1 - 0.65 = 0.35

3. Calculate the mass of sucrose present:

   mass fraction_(sucrose) = mass_(sucrose) ÷ total mass_(solution)
   mass_(sucrose) = mass fraction_(sucrose) × mass_(solution)
   mass_(sucrose) = 0.35 x 500 = 175 g

Percent by Mass (percent by weight) Concentration Calculations

Question 1. An aqueous solution of potassium chloride has a mass % water of 78%.
What is the mass percent potassium chloride?

1. Extract the data from the question:

   Aqueous solution is made up of two components, a solute, potassium chloride (KCl), and a solvent, water (H₂O)
   mass % (water) = 78 %
   mass % (KCl) = ? %

2. Write equation to calculate mass % (KCl):

   mass %_(KCl) + mass %_(H2O) = 100

   Rearrange the equation to find mass % KCl:
   mass %_(KCl) = 100 - mass %_(H2O)

3. Subsitute the values into the equation and solve:

   mass %_(KCl) = 100 - 78 = 22%

Question 2. An aqueous solution contains 700 g water and 45 g sodium hydroxide.
What is the percent by mass of sodium hydroxide?

1. Extract the data from the question:

   Aqueous solution is made up of two components, a solute (NaOH) and a solvent (H₂O)
   mass_(H2O) = 700 g
   mass_(NaOH) = 45 g
   mass % (NaOH) = ? %

2. Write the equation for finding mass %:

   mass % NaOH = mass_(NaOH) ÷ (mass_(NaOH) + mass_(H2O)) × 100

3. Substitute the values (all masses must have the same units) into the equation and solve:

   mass % NaOH = 45 ÷ (45 + 700) × 100 = 6.04%

Question 3. An aqueous solution contains 42% by mass ethanol.
What mass of ethanol is present in 250 g of solution?

1. Extract the data from the question:

   Aqueous solution is made up of two components, a solute (ethanol) and a solvent (water)
   mass % ethanol = 42 %
   mass_(solution) = mass_(ethanol) + mass_(water) = 250 g
   mass_(ethanol) = ? g

2. Write the equation for finding mass % ethanol:

   mass % ethanol = mass_(ethanol) ÷ mass_(solution) × 100

   Re-arrange the equation to find mass of ethanol:

   mass_(ethanol) = mass % ethanol × mass_(solution) ÷ 100

3. Substitute the values into the equation and solve:

   mass_(ethanol) = 42 × 250 ÷ 100 = 105 g