Unit Conversions and Dimensional Analysis

Importance

• Essential for chemistry and problem-solving.
• Ensures all quantities are in the correct units for calculations.
• Provides a structured approach to solving problems.

Conversion Factors

• Definition: Ratios that express the equivalence between units.
• Example: 1 inch = 2.54 centimeters.
  • As a conversion factor: ( \frac{1 \text{ inch}}{2.54 \text{ cm}} = 1 ).
  • Can convert between inches and centimeters.

Example: Inches to Centimeters

• Convert 12.5 inches to centimeters:
  • Setup: ( \frac{12.5 \text{ inches}}{1} \times \frac{2.54 \text{ cm}}{1 \text{ inch}} ).
  • Calculation: ( 12.5 \times 2.54 = 31.8 \text{ cm} ).

Example: Centimeters to Inches

• Convert 31.8 cm to inches:
  • Setup: ( \frac{31.8 \text{ cm}}{1} \times \frac{1 \text{ inch}}{2.54 \text{ cm}} ).
  • Calculation: ( 31.8 \div 2.54 = 12.5 \text{ inches} ).

Multi-Step Calculations Example

• Convert 1.76 yards to centimeters:
  • Known: 1 meter = 1.094 yards, 100 cm = 1 meter.
  • Setup conversion factors:
    1. ( \frac{1 \text{ meter}}{1.094 \text{ yards}} ).
    2. ( \frac{100 \text{ cm}}{1 \text{ meter}} ).
  • Multiply all numerators and divide by all denominators.
  • Calculation: ( \frac{1.76 \times 100}{1.094} = 160.8775 \text{ cm} ).
  • Significant figures: Report as 161 cm.

Example: Volume Conversion

• Convert 5.70 liters to cubic inches:
  • Known conversions:
    • 1000 mL = 1 L.
    • 1 cubic cm = 1 mL.
    • 1 inch = 2.54 cm.
  • Setup conversions:
    1. ( \frac{1000 \text{ mL}}{1 \text{ L}} ).
    2. ( \frac{1 \text{ cubic cm}}{1 \text{ mL}} ).
    3. Use ( \left( \frac{1 \text{ inch}}{2.54 \text{ cm}} \right)^3 ) for cubic conversion.
  • Calculation: Result is 347.84 cubic inches.

Important Notes

• Remember to cube conversion factors when converting volumes.
• Ensure all units cancel appropriately to reach the desired unit.
• Check significant figures and report accordingly.