Chapter One: Matter, Measurement, and Problem Solving

Significant Figures and the Reliability of Measurements

In chemistry, the reliability of a measurement is communicated through the use of significant figures. These figures reflect the precision of the measuring instrument and the care taken during measurement. Every digit in a measurement is certain except the last, which is estimated.

• Precision depends on instrument quality, measurement technique, environmental conditions, and observer skill.

• Significant figures help preserve this precision during calculations and reporting.

Additional info: The last digit in a measurement is always an estimate, indicating the uncertainty in the measurement.

Reading Measurements: The Meniscus

When measuring the volume of a liquid in a graduated cylinder, always read the value at the bottom of the meniscus at eye level to ensure accuracy.

Counting Significant Figures

Rules for Identifying Significant Figures

Significant figures are determined by specific rules:

• Rule 1: All nonzero digits are significant.

• Rule 2: Interior zeroes (between nonzero digits) are significant.

• Rule 3: Leading zeroes (to the left of the first nonzero digit) are not significant; they only locate the decimal point.

• Rule 4: Trailing zeroes (at the end of a number):

  • After a decimal point: always significant.

  • Before a decimal point and after a nonzero number: always significant.

  • Before an implied decimal point: ambiguous; use scientific notation or a decimal point to clarify.

Additional info: For example, in 0.0032, only the '32' are significant; in 0.0540, all digits except the leading zeroes are significant.

Examples: Determining Significant Figures

• 1002.7 → 5 significant figures

• 1.00000 → 6 significant figures

• 0.0000000009 → 1 significant figure

• 0.00090 → 2 significant figures

• 5000 → ambiguous (could be 1, 2, 3, or 4 significant figures; use scientific notation to clarify)

• 5000.0 → 5 significant figures

Significant Figures in Calculations

Rules for Calculations

When performing calculations with measured quantities, the result must reflect the precision of the least precise measurement.

• Multiplication and Division: The result has the same number of significant figures as the factor with the fewest significant figures.

• Addition and Subtraction: The result has the same number of decimal places as the quantity with the fewest decimal places.

Additional info: For example, 2.345 + 0.07 + 2.9975 = 5.4125, but the answer is rounded to 5.41 to match the least number of decimal places.

Rounding Rules

• Round down if the last digit being dropped is four or less.

• Round up if the last digit being dropped is five or more.

• Only the leftmost digit being dropped determines rounding direction; ignore all digits to the right of it.

Examples:

• 5.37 rounds to 5.4 (since 7 > 5; rounded up)

• 5.34 rounds to 5.3 (since 4 < 5; rounded down)

• 5.35 rounds to 5.4 (rounded up)

• 5.349 rounds to 5.3 (since 4 < 5; rounded down)

Rounding in Multistep Calculations

• Follow the BODMAS/PEMDAS rule for order of operations.

• To avoid rounding errors, round only the final answer in multistep calculations.

• If writing intermediate answers, underline the least significant digit to keep track of significant figures.

Precision and Accuracy

Definitions

• Accuracy: How close a measured value is to the actual (true) value.

• Precision: How close or consistent a series of measurements are to one another.

Measurements are:

• Precise if they are consistent with one another.

• Accurate if they are close to the actual value.

Errors:

• Random error: Equal probability of being too high or too low; leads to inconsistency (imprecision).

• Systematic error: Tends to be consistently too high or too low; leads to inaccuracy.

Additional info: For example, if three students weigh a lead block (true mass 10.00 g): - Student A: Inaccurate and imprecise - Student B: Precise but inaccurate - Student C: Accurate and precise