Chemistry and Measurements: Significant Figures and Rounding in GOB Chemistry

Chemistry and Measurements

Introduction

Accurate measurement and proper reporting of numerical data are fundamental skills in chemistry. This section introduces the concept of significant figures, their role in calculations, and the rules for rounding numbers to reflect the precision of measurements.

Significant Figures in Calculation

Definition and Importance

Significant figures (SFs) are the digits in a measured or calculated quantity that reflect the precision of the measurement. Reporting the correct number of significant figures ensures that calculated results do not imply greater accuracy than the measurements allow.

• Measured numbers are obtained from instruments and have uncertainty in the last digit.

• Calculated answers must be adjusted to reflect the correct number of significant figures.

Rules for Identifying Significant Figures

The following rules determine which digits in a number are significant:

• All nonzero digits are significant.

• Zeros between nonzero digits are significant.

• Zeros at the end of a decimal number are significant.

• Zeros at the beginning of a decimal number are not significant.

• Zeros used as placeholders in large numbers without a decimal point are not significant.

• Zeros in scientific notation are significant if they are part of the coefficient.

Table: Significant Figures in Measured Numbers

Measured Number	Number of Significant Figures
4.5 g	2
122.35 m	5
205 °C	3
5.008 kg	4
50 L	2
16.00 mL	4
4.8 × 105 m	2
5.70 × 10-3 g	3
0.0004 s	1
0.075 cm	2
850 000 m	2
1 250 000 g	3

Rounding Off

Purpose and Method

When performing calculations, the results often contain more digits than justified by the precision of the measurements. Rounding off is used to adjust the answer to the correct number of significant figures.

• Rounding rules ensure that the final answer does not imply greater precision than the data support.

Rules for Rounding Off

• If the first digit to be dropped is 4 or less, drop it and all following digits.

• If the first digit to be dropped is 5 or greater, increase the last retained digit by 1.

Table: Rounding Off and Significant Figures

Number to Round Off	Three Significant Figures	Two Significant Figures
8.4234	8.42 (drop 34)	8.4 (drop 234)
14.880	14.8 (drop 80, increase last retained digit by 1)	15 (drop 780, increase last retained digit by 1)
3260	3260* (drop 6, increase last retained digit by 1, add 0) (3.26 × 103)	3300* (drop 56, increase last retained digit by 1, add 00) (3.3 × 103)

Additional info: Placeholder zeros are used to maintain the magnitude of large numbers when digits are dropped.

Significant Figures in Calculations

Multiplication and Division

For multiplication and division, the final answer should have the same number of significant figures as the measurement with the fewest significant figures.

• Example: $2.8 imes 67.40 = 188.72$ (calculator answer)

• Since 2.8 has 2 SFs, the answer should be rounded to 2 SFs: $1.9 imes 10^2$

Addition and Subtraction

For addition and subtraction, the final answer should have the same number of decimal places as the measurement with the fewest decimal places.

• Example: $6.102 + 0.09 + 0.012 = 6.204$ (calculator answer)

• 0.09 has two decimal places, so the answer is rounded to two decimal places: $6.20$

Examples and Applications

Rounding Measured Numbers

• Round 3.145 g to three significant figures: 3.15 g

• Round 3.145 g to two significant figures: 3.1 g

Multiplication Example

• Calculate $5.00 ext{ cm} imes 3.408 ext{ cm} imes 2.0 ext{ cm}$

• Calculator answer: $34.08 ext{ cm}^3$

• Lowest SF is 2 (from 2.0 cm), so answer is rounded to 2 SFs: 34 cm3

Addition Example

• Calculate $6.102 + 0.09 + 0.012$

• Calculator answer: $6.204$

• Fewest decimal places is 2, so answer is rounded to 2 decimal places: 6.20

Summary Table: Significant Figures in Calculations

Operation	Rule	Example
Multiplication/Division	Answer has same number of SFs as measurement with fewest SFs	$2.8 imes 67.40 = 1.9 imes 10^2$ (2 SFs)
Addition/Subtraction	Answer has same number of decimal places as measurement with fewest decimal places	$6.102 + 0.09 + 0.012 = 6.20$ (2 decimal places)

Key Terms

• Significant Figures (SFs): Digits in a number that represent the precision of a measurement.

• Rounding Off: Adjusting a number to the correct number of significant figures or decimal places.

• Measured Number: A value obtained from a measurement instrument.

• Placeholder Zero: A zero used to maintain the magnitude of a number when digits are dropped.

Conclusion

Understanding significant figures and proper rounding is essential for reporting scientific data accurately. These rules ensure that calculated results reflect the true precision of the measurements and maintain the integrity of scientific communication.