The Formula to Master
All mass fraction calculations use the same core formula:
w = m(solute) ÷ m(solution)
Or as a percentage:
w% = [m(solute) ÷ m(solution)] × 100 %
Where m(solution) = m(solute) + m(solvent)
There are three types of calculation you may be asked to perform:
| Type | Given | Find |
|---|---|---|
| Type 1 | Masses of solute and solvent | Mass fraction (w or w%) |
| Type 2 | Mass fraction and mass of solution | Mass of solute |
| Type 3 | Mass fraction and mass of solute | Mass of solution |
Always check: are you given the mass of the solution, or the mass of the solvent? If you are given the solvent mass, you must first add the solute mass to find the solution mass before using the formula.
Type 1: Finding the Mass Fraction
Given the masses of solute and solvent, find w%.
Worked Example 1A
Problem: A student dissolves 12 g of potassium nitrate in 88 g of water. What is the mass fraction of potassium nitrate?
Step 1: Find the total mass of the solution. m(solution) = m(solute) + m(solvent) = 12 g + 88 g = 100 g
Step 2: Apply the mass fraction formula. w = m(solute) ÷ m(solution) = 12 ÷ 100 = 0.12
Step 3: Convert to percentage. w% = 0.12 × 100 % = 12 %
Answer: The mass fraction of potassium nitrate is 0.12 (12 %).
Worked Example 1B
Problem: A cook dissolves 45 g of salt in 705 g of water to make brine for pickling. What is the mass fraction of salt?
Step 1: Total solution mass = 45 g + 705 g = 750 g
Step 2: w = 45 ÷ 750 = 0.06
Step 3: w% = 0.06 × 100 % = 6 %
Answer: The brine is 6 % salt by mass.
Always add the solute and solvent masses to get the solution mass. Never divide the solute mass by the solvent mass — that is a very common mistake.
Type 2: Finding the Mass of Solute
Given the mass fraction and the total solution mass, find the solute mass.
Rearrange the formula: m(solute) = w × m(solution)
Or with percentage: m(solute) = (w% ÷ 100) × m(solution)
Worked Example 2A
Problem: You need 500 g of a 4 % glucose solution for a biology experiment. How many grams of glucose do you need?
Step 1: Convert percentage to decimal. w = 4 % ÷ 100 = 0.04
Step 2: Apply the rearranged formula. m(solute) = w × m(solution) = 0.04 × 500 g = 20 g
Answer: You need 20 g of glucose. Dissolve it in (500 − 20) = 480 g of water.
Worked Example 2B
Problem: A bottle of fruit juice contains 800 g of drink and has a sugar content of 11 %. How much sugar is in the bottle?
Step 1: w = 11 ÷ 100 = 0.11
Step 2: m(sugar) = 0.11 × 800 g = 88 g
Answer: The bottle contains 88 g of sugar.
Real world: Preparing a saline drip in a hospital
Hospital intravenous (IV) saline is made at exactly 0.9 % NaCl (physiological saline). A nurse preparing a 1,000 g bag of saline needs: m(NaCl) = 0.009 × 1,000 g = 9 g of sodium chloride. The remaining 991 g is sterile water. Getting this calculation precisely right is critical — too much salt could harm the patient.
Type 3: Finding the Mass of Solution
Given the mass fraction and the solute mass, find the total solution mass.
Rearrange the formula: m(solution) = m(solute) ÷ w
Or with percentage: m(solution) = m(solute) ÷ (w% ÷ 100) = m(solute) × (100 ÷ w%)
Worked Example 3A
Problem: How much of a 20 % copper sulfate solution can you make if you only have 50 g of copper sulfate?
Step 1: Convert percentage to decimal. w = 20 ÷ 100 = 0.20
Step 2: Find the solution mass. m(solution) = m(solute) ÷ w = 50 g ÷ 0.20 = 250 g
Step 3: Find the water needed. m(water) = m(solution) − m(solute) = 250 g − 50 g = 200 g
Answer: You can make 250 g of 20 % copper sulfate solution, using 50 g of copper sulfate dissolved in 200 g of water.
Worked Example 3B
Problem: A pharmacist needs to prepare a 2.5 % antiseptic solution using 10 g of antiseptic powder. What total mass of solution should be made?
Step 1: w = 2.5 ÷ 100 = 0.025
Step 2: m(solution) = 10 g ÷ 0.025 = 400 g
Step 3: Water needed = 400 g − 10 g = 390 g
Answer: The pharmacist should make 400 g of solution by dissolving 10 g of powder in 390 g of water.
Real world: Diluting bleach for cleaning
Household bleach is typically 5 % sodium hypochlorite. To make a milder 0.5 % cleaning solution, you can use the mass fraction principles. If you want 1,000 g of 0.5 % solution, you need m(bleach active) = 0.005 × 1,000 = 5 g of the active ingredient. Since the original bleach is 5 %, you need 5 g ÷ 0.05 = 100 g of bleach, diluted with 900 g of water. Mass fraction calculations are at the heart of safe chemical preparation every day.
Summary: Rearranging the Formula
| What to Find | Formula |
|---|---|
| Mass fraction (w%) | w% = [m(solute) ÷ m(solution)] × 100 % |
| Mass of solute | m(solute) = (w% ÷ 100) × m(solution) |
| Mass of solution | m(solution) = m(solute) ÷ (w% ÷ 100) |
| Mass of solvent | m(solvent) = m(solution) − m(solute) |
When converting between percentage and decimal, always divide by 100 to go from % to decimal (e.g. 15 % → 0.15), and multiply by 100 to go from decimal to % (e.g. 0.08 → 8 %). Forgetting this step is one of the most common sources of error.
Units and Rounding
- Mass can be in grams (g) or kilograms (kg) — but you must use the same unit throughout one calculation
- Mass fraction itself has no units (it is a ratio of mass to mass)
- Round to 2 or 3 significant figures unless told otherwise
- Double-check: the mass fraction should be between 0 and 1 (or 0 % and 100 %). If you get a number outside this range, you have made an error
Practice Problems
A student dissolves 30 g of sodium chloride in 270 g of water. Calculate the mass fraction as a percentage. Show all working.
Show answer
How many grams of sugar are needed to prepare 250 g of a 8 % sugar solution? How much water is needed?