Measurement, Uncertainty, and Significant Figures in General Chemistry

Measurement and Uncertainty

Recording Measurements

Accurate measurement is fundamental in chemistry, as it ensures reliable experimental results. When recording measurements, it is important to use the correct number of decimal places and to account for uncertainty in the measurement process.

• Measurement Tools: Common laboratory tools include rulers, graduated cylinders, beakers, and burets. Each tool has a specific precision, determined by the smallest division marked on the instrument.

• Uncertainty in Measurement: Every measurement contains some degree of uncertainty, typically in the last digit recorded. This uncertainty arises from the limitations of the measuring device.

• Recording Rules:

  • Measurements should be recorded to one digit beyond the smallest division on the measuring device. This last digit is considered uncertain.

  • All measurements should be recorded to the same decimal place when comparing or combining them.

• Example: Measuring the length of an object with two rulers:

  • Ruler 1 (larger divisions): Measurement might be 12.3 cm (uncertainty in the tenths place).

  • Ruler 2 (smaller divisions): Measurement might be 12.34 cm (uncertainty in the hundredths place).

Correcting Measurements

Measurements that are improperly recorded (e.g., with too many or too few decimal places) should be corrected to reflect the proper uncertainty.

• Key Point: The uncertainty is always in the last digit recorded.

• Example: If a graduated cylinder measures to the nearest 0.1 mL, a reading of 10.2 mL is appropriate, not 10.23 mL.

Significant Figures

Definition and Importance

Significant figures (sig figs) are the digits in a measured number that include all certain digits plus one final digit that has some uncertainty. They communicate the precision of a measurement.

• Key Point: Significant digits are those digits in a measured number that include all certain digits plus one final digit having some uncertainty.

• Example: A measured volume of 15.4 mL has three significant figures; 15.40 mL has four significant figures.

Rules for Identifying Significant Figures

• All nonzero digits are significant.

• Zeros between nonzero digits are significant.

• Leading zeros (zeros before the first nonzero digit) are not significant.

• Trailing zeros:

  • Trailing zeros in a number with a decimal point are significant.

  • Trailing zeros in a number without a decimal point may or may not be significant; use scientific notation to clarify.

• Exact numbers: Numbers from counting or defined quantities (e.g., 24 students, 2.54 cm in an inch) have infinite significant figures.

Table: Significant Figure Rules

Type of Zero	Significant?	Example
Leading zeros	No	0.0025 (2 sig figs)
Trailing zeros (with decimal)	Yes	2.500 (4 sig figs)
Trailing zeros (no decimal)	Ambiguous	2500 (could be 2, 3, or 4 sig figs)
Zeros between nonzero digits	Yes	205 (3 sig figs)

Examples and Applications

• Measured values: 0.00560 (3 sig figs), 340 (2 or 3 sig figs, depending on context)

• Exact values: 24 students (infinite sig figs), 2.54 cm/inch (defined value)

Significant Figures in Calculations

Addition and Subtraction

When adding or subtracting, the result should be reported to the same decimal place as the least precise measurement.

• Rule: The answer should have the same number of decimal places as the measurement with the fewest decimal places.

• Example:

  • $15.356 + 2.70 = 18.06$ (answer rounded to two decimal places)

Multiplication and Division

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

• Rule: The answer should have the same number of significant figures as the measurement with the least number of significant figures.

• Example:

  • $0.013 imes 0.020 = 0.00026$ (two significant figures)

Multiple Operations

When performing calculations involving multiple steps, apply the appropriate significant figure rule at each step.

• Example: Calculating percent error: $\%\ error = \frac{|\text{theoretical} - \text{experimental}|}{\text{theoretical}} \times 100$ If the experimental value is 13.2 and the theoretical value is 12.5, calculate percent error and round to the correct number of significant figures.

Special Cases and Exceptions

Terminal Zeros and Scientific Notation

• Terminal (trailing) zeros: May be significant or not, depending on the presence of a decimal point. Use scientific notation to clarify.

• Example: 200 (1, 2, or 3 sig figs), 2.00 × 102 (3 sig figs)

Leading Zeros

• Leading zeros: Never significant; they only indicate the position of the decimal point.

• Example: 0.0045 (2 sig figs)

Exact Numbers

• Exact numbers: Have infinite significant figures and do not limit the precision of calculations.

• Example: 2.54 cm in an inch (defined value)

Practice Problems

Sample Calculations

• Addition/Subtraction: $6.543 - 6.4780 = 0.065$ (rounded to three decimal places)

• Multiplication/Division: $0.0654 / 0.000954 = 68.5$ (rounded to three significant figures)

• Multiple Operations: $(62.6 + 59.3) / (3.5529 \times 10^{-3}) = 34,400$ (rounded to three significant figures)

Summary Table: Significant Figure Rules

Operation	Rule	Example
Addition/Subtraction	Same decimal places as least precise measurement	$12.45 + 1.2 = 13.7$
Multiplication/Division	Same number of significant figures as least precise measurement	$2.5 \times 3.42 = 8.6$
Exact Numbers	Infinite significant figures	24 students

Conclusion

Understanding how to properly record measurements and apply significant figure rules is essential for accurate scientific work. Always consider the precision of your measuring device, record uncertainty in the last digit, and apply the correct significant figure rules in calculations. Use scientific notation to clarify ambiguous cases, and remember that exact numbers do not limit the precision of your results.