Measurements and Problem Solving

Measurements

• Measurements determine a quantity.
• A unit is a standard of measure.
• Always include units with measurements.

Qualitative vs. Quantitative Measurements

• Qualitative descriptions are not sufficient for scientific testing.
• Scientific testing requires quantitative measurements.

Accuracy vs. Precision

• Accuracy reflects how close a measurement is to the accepted value.
• Precision indicates how well the numbers are grouped together based on the measurement process.
• Example: 3.756647 is considered more accurate than 3.75

Significant Figures

• Any non-zero number is significant.
• Any zero in front of a non-zero number never counts as a significant figure.
• Any zero between two non-zero numbers always counts.
• Zeroes behind a non-zero number only count if there is a decimal point.
  • 1.234 = 4 significant figures
  • 10001001 = 7 significant figures
  • 10. = 2 significant figures
  • 10 = 1 significant figure

Rounding

• When adding or subtracting, round to the smallest place value.
• Example: 114.15 + 1000 should be rounded based on the least precise number.

Multiplying and Dividing Significant Figures

• The result should have the same number of significant figures as the least significant measurement used in the calculation.
• Volume of a rectangular object: V = l \times w \times h
• Example: V = 24.55 \text{ cm} \times 12.22 \text{ cm} \times 2.34 \text{ cm} = 702.00234 \text{ cm}^3
  • Since 2.34 has the least number of significant figures (3), the volume should be rounded to 702 cm3

Scientific Notation

• Move the decimal behind the first non-zero number.
• Count the spaces moved and use that as the power of 10.
• If the original number is less than 1, use a negative power.
• If the original number is greater than 10, use a positive power.
• Round based on significant figures.
• Examples:
  • 1233 = 1.233 \times 10^3
  • 0.0034 = 3.4 \times 10^{-3}
  • 209 = 2.09 \times 10^2

Metric System

• Units are based on SI units.
• SI unit for length is the meter (m).
  • 1 \text{ cm} = 1 \times 10^{-2} \text{ m}
  • 1 \text{ mm} = 1 \times 10^{-3} \text{ m}
• SI unit for volume is m3.
  • 1 cubic decimeter = 1 liter
  • 1 \text{ L} = 1 \text{ dm}^3 = 1000 \text{ cm}^3 = 1000 \text{ mL}
• SI unit for mass is kg.
  • Gram is a more useful lab scale unit.
  • 1 \text{ cm}^3 = 1 \text{ mL}
• 10 \text{ cm} = 1 \text{ dm}

Density

• Density is mass divided by volume: \text{Density} = \frac{\text{mass}}{\text{volume}}
• Example: 250.0 mL of a liquid has a mass of 322.0 grams.
  • \text{Density} = \frac{322.0 \text{ grams}}{250.0 \text{ mL}} = 1.288 \text{ g/mL}

Conversion Factors

• Conversion factors are used to change units through equalities or ratios.
• Equality: 1 \text{ lb} = 453.6 \text{ g}
• Ratio: A balanced fraction with units.
• Examples:
  • \frac{60 \text{ s}}{1 \text{ min}}
  • \frac{60 \text{ miles}}{2 \text{ hr}}
  • 1 \text{ in} = 2.54 \text{ cm}
  • 1 \text{ gal} = 3.785 \text{ L}

Dimensional Analysis

• Find the desired unit to start.
• Identify any helpful conversion units.
• Write the conversion factor so the requested units are on top and the given units will cancel.
• Example: Convert 100 m to cm. Given unit = Meter, new unit = Centimeter
  • 1 \text{ cm} = 10^{-2} \text{ m}
  • 100 \text{ m} \times \frac{1 \text{ cm}}{10^{-2} \text{ m}} = 10000 \text{ cm}

Dimensional Analysis Examples

• 16 gal to liters: Use 1 \text{ gal} = 3.785 \text{ L}
• How many feet in 13 miles? Use 1 \text{ mile} = 5280 \text{ ft}