2.1 Significant figures and measurement uncertainty

Significant figures and measurement uncertainty are crucial concepts in analytical chemistry. They help scientists communicate the precision and reliability of their measurements, ensuring data integrity and reproducibility.

Understanding these concepts allows chemists to report results accurately and interpret data correctly. By mastering significant figures and measurement uncertainty, you'll be better equipped to analyze and present chemical measurements effectively.

Significant Figures in Measurement

Importance of Significant Figures

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• Significant figures are the meaningful digits in a measured or calculated quantity that indicate the precision and uncertainty of the value
• The number of significant figures in a reported value reflects the precision of the measuring instrument and the certainty of the measurement
  • More significant figures imply higher precision and less uncertainty
  • Fewer significant figures suggest lower precision and greater uncertainty
• Reporting the appropriate number of significant figures is crucial for maintaining the integrity and reproducibility of scientific data
  • Overreporting can imply a false sense of precision
  • Underreporting can lead to a loss of valuable information
• Significant figures are essential for communicating the limitations and reliability of experimental results
  • They allow other researchers to assess the quality and reproducibility of the data
  • They help in determining the appropriate level of precision for subsequent calculations and analyses

Rules for Determining Significant Figures

• All non-zero digits (1-9) are always significant
  • Example: In the number 123.45, all five digits are significant
• Zeros between non-zero digits are always significant
  • Example: In the number 1002.05, all six digits are significant
• Leading zeros (to the left of the first non-zero digit) are never significant, as they merely indicate the position of the decimal point
  • Example: In the number 0.0123, only the last three digits are significant
• Trailing zeros (to the right of the last non-zero digit) are significant only if the decimal point is explicitly shown
  • Example: In the number 12.3000, all six digits are significant because the decimal point is shown
  • Example: In the number 12300 (without a decimal point), only the first three digits are significant

Determining Significant Figures in Calculations

Addition and Subtraction

• In addition and subtraction, the result should have the same number of decimal places as the least precise measurement
  • Example: 12.3 + 1.456 = 13.8 (rounded to one decimal place, as 12.3 has only one decimal place)
  • Example: 10.1 - 9.78 = 0.3 (rounded to one decimal place, as 10.1 has only one decimal place)
• When adding or subtracting measurements with different units, convert them to the same unit before determining the number of significant figures
  • Example: 5.2 cm + 12.34 mm = 5.2 cm + 1.234 cm = 6.4 cm (rounded to one decimal place)

Multiplication and Division

• In multiplication and division, the result should have the same number of significant figures as the quantity with the least number of significant figures
  • Example: 2.3 × 1.456 = 3.3 (rounded to two significant figures, as 2.3 has only two significant figures)
  • Example: 12.34 ÷ 2.1 = 5.9 (rounded to two significant figures, as 2.1 has only two significant figures)
• When multiplying or dividing measurements with different units, the result should have the appropriate unit derived from the input units
  • Example: 5.2 cm × 3.1 cm = 16 cm² (rounded to two significant figures)

Measurement Uncertainty and Its Sources

Concept of Measurement Uncertainty

• Measurement uncertainty is the doubt that exists about the result of any measurement, which is an estimate of the range of values within which the true value is expected to lie
• No measurement is perfect, and there is always some level of uncertainty associated with the measured value
• Measurement uncertainty arises from various sources, including instrument limitations, environmental factors, operator error, and the inherent variability of the measured quantity
• Understanding and quantifying measurement uncertainty is essential for assessing the reliability and comparability of experimental results

Sources of Measurement Uncertainty

• Instrument limitations, such as the resolution of the measuring device and its calibration, contribute to systematic errors in measurements
  • Example: A ruler with a resolution of 1 mm cannot accurately measure lengths smaller than 1 mm
  • Example: An improperly calibrated balance may consistently give readings that are higher or lower than the true value
• Environmental factors, like temperature, humidity, and pressure, can affect the performance of measuring instruments and introduce uncertainties
  • Example: Changes in temperature can cause thermal expansion or contraction of materials, affecting length measurements
  • Example: Variations in humidity can influence the mass of hygroscopic substances during weighing
• Operator errors, including parallax errors and inconsistencies in measurement techniques, can lead to random errors and increased uncertainty
  • Example: Parallax errors occur when the observer's eye is not aligned properly with the measuring scale, leading to inaccurate readings
  • Example: Inconsistent placement of the measuring instrument or variations in the applied force during measurements can introduce random errors

Estimating and Reporting Uncertainty

Expressing Uncertainty

• Uncertainty in measured values is typically reported as an absolute or relative uncertainty, depending on the context and the desired level of precision
• Absolute uncertainty is expressed in the same units as the measured value and indicates the range within which the true value is expected to lie
  • Example: A length measurement of 5.2 cm with an absolute uncertainty of ±0.1 cm would be reported as 5.2 ± 0.1 cm
• Relative uncertainty is the ratio of the absolute uncertainty to the measured value, often expressed as a percentage
  • Example: A length measurement of 5.2 cm with an absolute uncertainty of ±0.1 cm has a relative uncertainty of (0.1 cm / 5.2 cm) × 100% ≈ 2%
• When reporting uncertainty, the number of significant figures in the uncertainty should match the number of significant figures in the measured value
  • Example: 5.2 ± 0.1 cm (both the measured value and the uncertainty have two significant figures)

Propagation of Uncertainty

• Propagation of uncertainty rules are used to estimate the uncertainty in calculated values based on the uncertainties of the input measurements
• For addition and subtraction, the absolute uncertainties are added in quadrature (square root of the sum of squares)
  • Example: If $A = 5.2 \pm 0.1$ cm and $B = 3.4 \pm 0.2$ cm, then for $C = A + B$, the absolute uncertainty in $C$ is $\sqrt{(0.1)^2 + (0.2)^2}$ cm $\approx 0.2$ cm
• For multiplication and division, the relative uncertainties are added in quadrature
  • Example: If $A = 5.2 \pm 0.1$ cm and $B = 3.4 \pm 0.2$ cm, then for $D = A \times B$, the relative uncertainty in $D$ is $\sqrt{(0.1/5.2)^2 + (0.2/3.4)^2} \times 100\%$ $\approx 3\%$
• These propagation rules allow for the estimation of uncertainty in derived quantities based on the uncertainties of the input measurements