Significant Figures

The Golden Rule for Rounding

• In chemistry calculations involving multiplication or division (like concentration or moles), you must provide an appropriate number of significant figures.
• This means your final calculated answer must be rounded to match the piece of raw data with the fewest (least) significant figures. This piece of data is known as the least significant data.
• For addition and subtraction (like calculating a temperature change), you round your answer to match the piece of data with the lowest number of decimal places.
• If an exam question does not specify a rounding requirement and the "least significant data" rule does not apply, the standard default accuracy is $3$ significant figures.
• Rounding is the process of reducing the digits in a number while keeping its value similar. If the first digit to be dropped is $5$ or greater, round up; if it is $4$ or less, round down.

Worked Example

Calculate the concentration of a solution when $0.052\,\text{mol}$ of sodium chloride is dissolved in $250.0\,\text{cm}^3$ of water. Give your answer to an appropriate number of significant figures.

Step 1: Identify the values and their significant figures.

• Moles = $0.052\,\text{mol}$ ($2$ s.f., leading zeros are not significant)
• Volume = $250.0\,\text{cm}^3$ ($4$ s.f., trailing zero with a decimal point is significant)
• The least significant data is the moles ($2$ s.f.), so the final answer must be to s.f.

Step 2: Convert the volume to $\text{dm}^3$.

• Volume = $250.0 / 1000 = 0.2500\,\text{dm}^3$

Step 3: State the formula and substitute the values.

• $\text{Concentration} = \frac{\text{moles}}{\text{volume}}$

• $\text{Concentration} = \frac{0.052}{0.2500}$

Step 4: Calculate and round the final answer.

• $\text{Concentration} = 0.208\,\text{mol/dm}^3$
• Final Answer = $0.21\,\text{mol/dm}^3$ (rounded up to $2$ s.f.)

Avoiding Intermediate Rounding Errors

• A major trap in multi-step calculations is premature rounding.
• You must never round numbers during the intermediate steps of a calculation.
• Always keep the full unrounded value in your calculator display, or write down intermediate values to at least one or two more significant figures than you need for the final answer.
• Rounding multiple times in a sequence (sequential rounding error) causes inaccuracies to multiply, leading to an incorrect final value.

Precision in Practical Data

• The resolution of a measuring instrument is the smallest change it can detect. You cannot claim your calculated answers have a higher degree of resolution than the raw data allows.
• In titration practicals, burette readings must always be recorded to exactly $2$ decimal places.
• The second decimal place must always be a $0$ or a $5$ (e.g., $23.40\,\text{cm}^3$ or $23.45{}^{}$).