Metric Conversions and Dimensional Analysis
Dimensional analysis (also called the factor-label method) is a way of converting between units by multiplying a measurement by conversion factors written as fractions that equal one, so that unwanted units cancel. In brief: to convert 3.5 L to milliliters, multiply by 1,000 mL ÷ 1 L, which gives 3,500 mL; the liters cancel and the milliliters remain.
Chemistry problems almost always involve converting units, from grams to moles, from liters to milliliters, from kilometers per hour to meters per second. Dimensional analysis lets you do this with a single method, and it also acts as a check, since a wrong setup leaves the wrong units. This article explains the method, lists the metric conversion factors, shows multi-step problems, includes a table of useful non-metric conversions, works through examples and describes common errors.
How does dimensional analysis work?
A conversion factor is a ratio that expresses the same quantity in two units. Since 1 km = 1,000 m, we can write two factors: 1 km ÷ 1,000 m and 1,000 m ÷ 1 km. Each equals 1, so multiplying by it changes the units but not the value of the measurement. Choose the factor so that the unit you want to remove is in the denominator, which cancels it.
Example: 2.4 km → m is 2.4 km × (1,000 m ÷ 1 km) = 2,400 m. The kilometers cancel.
How do you convert within the metric system?
The metric system is based on powers of ten, so conversions only need prefix factors.
| Quantity | Conversion |
|---|---|
| Length | 1 km = 1,000 m; 1 m = 100 cm = 1,000 mm; 1 mm = 1,000 µm |
| Mass | 1 kg = 1,000 g; 1 g = 1,000 mg; 1 mg = 1,000 µg |
| Volume | 1 L = 1,000 mL; 1 mL = 1 cm3; 1 m3 = 1,000 L |
| Time | 1 min = 60 s; 1 h = 60 min = 3,600 s; 1 day = 86,400 s |
| Amount | 1 mol = 1,000 mmol |
What are the steps for a conversion problem?
- Write the given quantity with its unit.
- Write the unit you want to end with.
- Set up one or more conversion factors so that unwanted units cancel.
- Multiply the numbers, and divide by the numbers in the denominators.
- Check that the units are correct, and that the size of the answer makes sense.
- Give the answer with the right number of significant figures.
Worked examples
- Simple conversion. 0.750 L to mL: 0.750 L × 1,000 mL ÷ 1 L = 750 mL.
- Mass. 4,500 mg to g: 4,500 mg × 1 g ÷ 1,000 mg = 4.5 g.
- Speed. 90 km/h to m/s: 90 km/h × 1,000 m ÷ 1 km × 1 h ÷ 3,600 s = 25 m/s.
- Density. 13.5 g/cm3 to kg/m3: 13.5 g/cm3 × 1 kg ÷ 1,000 g × (100 cm ÷ 1 m)3 = 13.5 × 10−3 × 106 = 13,500 kg/m3.
- Grams to moles to particles. How many molecules are in 9.00 g of water? The molar mass of water is 18.0 g/mol. 9.00 g × 1 mol ÷ 18.0 g × 6.022 × 1023 molecules ÷ 1 mol = 3.01 × 1023 molecules.
- Reagent amount. A reaction needs 0.250 mol of NaCl (58.44 g/mol). The mass is 0.250 mol × 58.44 g ÷ 1 mol = 14.6 g.
- Dosage. A medicine is dosed at 15 mg per kg of body mass. For a person of 60.0 kg, the dose is 60.0 kg × 15 mg ÷ 1 kg = 900 mg, which is 0.90 g.
Useful conversions outside the metric system
| Quantity | Relationship | Exact? |
|---|---|---|
| Length | 1 inch = 2.54 cm | Exact |
| Length | 1 mile = 1.609344 km | Exact |
| Mass | 1 lb = 0.45359237 kg | Exact |
| Volume | 1 US gallon = 3.785 L | Rounded |
| Pressure | 1 atm = 101.325 kPa = 760 mmHg | Exact (atm to kPa) |
| Energy | 1 cal = 4.184 J | Exact (thermochemical) |
Example: a person is 6 ft 1 in tall. Convert to centimeters. 6 ft × 12 in ÷ 1 ft = 72 in, plus 1 in gives 73 in. 73 in × 2.54 cm ÷ 1 in = 185.42 cm, or about 185 cm.
How do you convert squared and cubed units?
When a unit is squared or cubed, the conversion factor must be too. Since 1 m = 100 cm, 1 m2 = (100 cm)2 = 10,000 cm2 and 1 m3 = (100 cm)3 = 1,000,000 cm3. For instance, 2.5 m3 = 2.5 × 1,000,000 = 2,500,000 cm3 = 2,500 L, using 1 L = 1,000 cm3.
Why did the metric system succeed?
The metric system, created in France in the 1790s, replaced many local units with a decimal system that scientists and traders could share. Its use of powers of ten made conversions easy, and it spread across Europe in the 19th century and then worldwide. The cost of mixed units can be high: in 1999 NASA’s Mars Climate Orbiter was lost, in part because one team used pound-force seconds while another software assumed newton seconds.
Which mistakes are common?
Mistake 1: Inverting the conversion factor
If the units do not cancel, flip the factor. The check on units catches the error.
Mistake 2: Forgetting to cube or square the factor
1 m3 is not 100 cm3. It is 1,000,000 cm3.
Mistake 3: Rounding in the middle of a calculation
Keep extra digits during the calculation, and round at the end.
Mistake 4: Treating exact conversion factors as limited digits
Defined relationships such as 1 m = 100 cm are exact, and do not limit the number of significant figures in the answer.