 # 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

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## 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

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1. DO NOT confuse m/m with m/M
m/m refers to a mass fraction
m/M refers to the calculation of moles, moles = mass ÷ molar mass

2. DO NOT confuse percent by mass (or weight) concentration with weight ratio percentage concentration.
percent by mass concentration = mass(solute) ÷ mass(solution) × 100
weight ratio pecentage concentration = mass(solute) ÷ mass(solvent) × 100