Why chemists care about measurement
Every number in a laboratory comes from an instrument with limits. A balance that reads to the nearest 0.01 g cannot tell you the mass to 0.0001 g, and writing extra digits would claim knowledge you do not have. Significant figures (sig figs) are the digits in a value that carry real information: all the certain digits plus one estimated digit.
Counting significant figures
- Non-zero digits always count.
- Zeros between non-zero digits count (4.07 has three).
- Leading zeros never count; they only place the decimal point (0.0042 has two).
- Trailing zeros count when there is a decimal point (2.50 has three, 0.0430 has three).
- Trailing zeros in a number with no decimal point are ambiguous, so use scientific notation: 4.3 × 10^2 has two, 4.30 × 10^2 has three.
- Exact numbers, such as 12 eggs or the 100 centimetres in a metre by definition, have unlimited sig figs.
Calculating with sig figs
- Multiplying and dividing: the answer has as many sig figs as the factor with the fewest.
- Adding and subtracting: the answer has as many decimal places as the term with the fewest decimal places.
- Round only at the end, carrying one extra digit in intermediate steps.
Units and dimensional analysis
Always write units with numbers. To convert, multiply by conversion factors arranged so the unwanted unit cancels. For squares and cubes remember to square or cube the factor: 1 m = 100 cm, but 1 m^3 = (100)^3 cm^3 = 10^6 cm^3. Temperature in kelvin equals degrees Celsius plus 273.15, and a change of one degree Celsius equals a change of one kelvin.
Accuracy, precision and error
Accuracy is closeness to the true or accepted value. Precision is how closely repeated measurements agree with each other. A scale that always reads 0.5 g too high is precise but not accurate. Percent error = |measured − accepted| / accepted × 100.
Worked example
A metal block is measured as 2.5 cm by 1.2 cm by 3.04 cm and has a mass of 23.47 g. Find its density, then practise unit conversion by changing 90 km/h into metres per second.
- Volume = 2.5 × 1.2 × 3.04 = 9.12 cm^3. The least precise length has two sig figs, so volume is 9.1 cm^3.
- Density = 23.47 g / 9.1 cm^3 = 2.5792 g/cm^3. Two sig figs allowed, so 2.6 g/cm^3.
- Speed: 90 km/h × (1000 m / 1 km) × (1 h / 3600 s) = 25 m/s. The kilometres and hours cancel, leaving metres per second.
If you had reported 2.5792 g/cm^3 you would be claiming the length was known to many more digits than the ruler gave you. In an exam, check three things every time: the sig figs of the answer, the units, and whether the size of the answer is reasonable.