Measurement and Problem Solving: Structured Study Notes for Introductory Chemistry

Measurement and Problem Solving

Measured vs. Exact Numbers

In chemistry, it is important to distinguish between measured and exact numbers. Measured numbers are obtained using instruments and always carry some degree of uncertainty, while exact numbers are counted and have no uncertainty.

• Measured numbers: Inexact or approximate; uncertainty is always present. Example: 5.25 m, 500 L, 75 g.

• Exact numbers: Obtained by counting; no uncertainty. Example: 75 m&m’s in a bowl, 24 students in a class.

Uncertainty in Measurements

Every measurement has a degree of uncertainty due to limitations in the measuring instrument. The last digit in a measurement is always uncertain.

• Measurement: Comparison of a physical quantity with a unit (e.g., feet, miles/hour, seconds).

• Reporting measurements: Always include units. Example: 3.1 miles, not just 3.1.

• Uncertainty: In 19.257 mL, the '7' is uncertain, all other digits are certain.

Measurement and Units

Scientific measurements use two main systems: English and metric. The metric system is preferred for its simplicity and use of powers of ten.

• English system: Uses odd conversion factors.

• Metric system: Uses base units and prefixes for powers of ten.

Accuracy and Precision

Accuracy and precision are key concepts in measurement. Accuracy refers to how close a measurement is to the true value, while precision refers to how close repeated measurements are to each other.

• Accuracy: Example: Measurements of 4.20 cm, 4.25 cm, 4.30 cm for a nail whose actual length is 4.25 cm.

• Precision: Example: Three measurements of 4.10 cm (consistent, but not necessarily accurate).

• Both: Three measurements of 4.25 cm (accurate and precise).

Length Measurements

Length is measured using rulers, typically in meters (m) or centimeters (cm). The degree of uncertainty depends on the ruler's markings.

• Ruler A: Fewer hash marks, more uncertainty, less precision.

• Ruler B: More hash marks, less uncertainty, more precision.

Mass Measurements

Mass is the amount of matter in a substance, measured in grams (g) using a balance. The uncertainty depends on the balance's sensitivity.

• Degree of uncertainty: Can range from ±0.1 g to ±0.0001 g.

• Weight vs. Mass: On Earth, often used interchangeably. Weight is the force of gravity on an object.

Volume Measurements

Volume is the amount of space occupied by a substance, measured in liters (L), milliliters (mL), or cubic centimeters (cm3). Various laboratory equipment is used for measurement.

• Graduated cylinder: Used for measuring liquid volume; uncertainty ranges from ±0.1 mL to ±0.5 mL. Always read at the meniscus.

• Pipets: Used to deliver fixed volumes; uncertainty ranges from ±0.1 mL to ±0.01 mL.

• Buret: Used for titrations; uncertainty ranges from ±0.1 mL to ±0.01 mL.

Exponential Numbers and Scientific Notation

Scientific notation is a convenient way to express very large or very small numbers using powers of ten.

• Positive exponent: Number multiplied by 10 n times. Example: $100 = 1 \times 10^2$

• Negative exponent: Number divided by 10 n times. Example: $0.001 = 1 \times 10^{-3}$

Converting to Scientific Notation

• Move the decimal point to form a number between 1 and 9.

• Add "x 10n" where n is the number of places moved.

• Left movement: n is positive; right movement: n is negative.

Significant Figures (Sig Figs)

Significant figures reflect the precision of a measurement. The last digit is always uncertain.

• Rules:

  • All non-zero digits are significant.

  • Zeroes between non-zero digits are significant.

  • Zeroes to the right of a decimal point after a non-zero digit are significant.

  • Zeroes to the left of the first non-zero digit are not significant.

  • Zeroes at the end of a number before a decimal point are ambiguous; use scientific notation.

Rounding Significant Figures

• If the right-most non-significant digit is 5 or greater, round up.

• If 4 or less, leave unchanged.

Examples

• 2500 g: 2 sig figs ($2.5 \times 10^3$ g)

• 0.03070 g: 4 sig figs ($3.070 \times 10^{-2}$ g)

• 3,159.0 kg: 5 sig figs ($3.159 \times 10^3$ kg)

• 0.0000504 L: 3 sig figs ($5.04 \times 10^{-5}$ L)

• 340.0050 s: 7 sig figs ($3.40005 \times 10^2$ s)

Calculations with Significant Figures

Addition and Subtraction

Result should have the same number of decimal places as the measurement with the least decimal places.

Multiplication and Division

Result should have the same number of significant figures as the measurement with the least sig figs.

• Example: $4.0 \times 1.0009 \times 57826 = 2.3 \times 10^5$ (2 sig figs)

Unit Equations and Unit Factors

Unit conversion factors (unit factors) are ratios of two equivalent quantities, used to convert between units.

• Unit equation: 1 dollar = 10 dimes

• Unit factor: $\frac{1 \text{ dollar}}{10 \text{ dimes}}$ or $\frac{10 \text{ dimes}}{1 \text{ dollar}}$

Exactly Equivalent Relationships

• Exact equivalents have infinite significant figures. Example: 1 foot = 12 inches.

• English and metric units are not exact equivalents. Example: 1 mile = 1.609 km.

Unit Analysis Problem Solving

Unit analysis involves carrying units through calculations to ensure correct final units and prevent mistakes.

• Write units for the answer.

• Arrange unit factors as fractions so all units cancel except the desired unit.

• Check for correct units and sig figs in the final answer.

Metric Prefixes and Conversion Factors

Metric prefixes indicate powers of ten and are paired with base units. Conversion factors in the metric system are exact equivalents.

• Examples:

  • 1 mm = $10^{-3}$ m

  • 1 kg = $10^3$ g

  • 1 GL = $10^9$ L

  • 1 µs = $10^{-6}$ s

• Conversion factors: 1 km = $10^3$ m

Metric-Metric and Metric-English Conversions

Metric-metric conversions use exact equivalents; metric-English conversions do not and require attention to significant figures.

• Example (mass): Express 325 mg in grams.

• Example (volume): Convert 125 dL to mL.

• Example (length): Convert 375 nm to cm.

• Example (time): Convert $1.5 \times 10^{14}$ µs to Gs.

• Example (English-metric): Convert 300 m to yards.

Volume Calculations

Volume can be calculated for regular solids or determined by displacement for irregular solids.

• Rectangular solid: $\text{Volume} = \text{length} \times \text{width} \times \text{height}$

• Displacement: $\text{Volume}_{\text{solid}} = \text{Volume}_{\text{final}} - \text{Volume}_{\text{initial}}$

Density

Density is the amount of mass per unit volume and is a fundamental property in chemistry.

• Formula: $d = \frac{\text{mass}}{\text{volume}}$

• Units: Liquids: g/cm3 or g/mL; Gases: g/L

• Water: Density is 1.00 g/mL

• Floating and sinking: Objects with density greater than water sink; less than water float.

The Percent Concept

Percent expresses the amount of a single quantity in relation to the total sample.

• Formula: $\text{Percent} = \frac{\text{Sample quantity}}{\text{Total sample}} \times 100$

• Example: 1 dime is 10% of a dollar.

Practice Problems

Apply the concepts of measurement, significant figures, unit analysis, and conversions to solve problems.

• Exercises: 1-26, 32-40, 50-58, 65-68, 75-76 (see textbook for details)