1.4

Introduction to Calculations and Conversions

• Calculations often involve basic algebraic operations:

  • Multiplication

  • Division

  • Addition

  • Subtraction

• Importance of unit conversions:

  • Convert from one unit to another (e.g., inches to feet, meters to kilometers)

  • Use exact values which do not affect significant figures in calculations

Exact Values and Their Implications

• Definition of exact values:

  • Defined values that are infinitely known

  • Have no effect on significant figures in calculations

• Examples of exact values:

  • 12 inches = 1 foot (American/English system)

  • 1,000 meters = 1 kilometer (Metric system)

Dimensional Analysis

• Process of unit conversion by expressing units as numerators and denominators

• Example of metric to metric conversion:

  • Convert 5.23 x 10^8 nanometers to centimeters

  • Step 1: Convert nanometers to meters using the conversion:
    1 nanometer = 1 x 10^-9 meters

  • Step 2: Convert meters to centimeters using the conversion:
    1 centimeter = 1 x 10^-2 meters

  • Units cancel during the conversion, leading to the final answer of 52.3 centimeters

  • Result retains three significant figures (from original measurement)

Example: American to Metric Conversions

• Conversion of 100 meters to feet:

  • Step 1: Convert meters to centimeters:
    1 meter = 100 centimeters

  • Step 2: Convert centimeters to inches using the conversion:
    1 inch = 2.54 centimeters

  • Step 3: Convert inches to feet:
    1 foot = 12 inches

  • Final calculation gives 328.0839 feet, rounded to 328.1 feet due to original measurement having four significant figures

Another Example: Velocity Conversion

• Conversion of sound velocity from miles per hour to meters per second:

  • Initial velocity: 767 miles/hour

  • Step 1: Convert hours to seconds:
    1 hour = 3600 seconds

  • Step 2: Convert miles to feet using the conversion:
    1 mile = 5,280 feet

  • Step 3: Convert feet to meters using the conversion:
    1 meter = 3.28 feet

  • Final answer is 343 meters/second, rounded to three significant figures

Multi-Dimensional Units

• Multi-dimensional units involve squared or cubed measurements:

  • Example: Volume conversion from cubic meters to liters

• Calculate the volume of a room measuring 8.28 m x 10.9 m x 4.44 m:

  • Volume in cubic meters: 400.7188 m³

  • Step 1: Convert cubic meters to cubic centimeters (requires cubing the conversion factor):
    1 m = 100 cm

  • Step 2: Convert cubic centimeters to milliliters:
    1 cm³ = 1 mL

  • Step 3: Convert milliliters to liters:
    1 liter = 1000 mL

  • Final answer is approximately 401,000 liters, rounded to three significant figures as 401 L or 4.01 x 10^5 L

Temperature Conversions

• Two main styles of temperature conversions: degrees Celsius and Fahrenheit

• Important equations:

  • Celsius to Fahrenheit: $F = \frac{9}{5}C + 32$

  • Celsius to Kelvin: $K = C + 273.15$

• Example of temperature conversion:

  • Convert -196°C to Fahrenheit yields -320.8°F, rounded to -321°F (due to 3 significant figures)

  • Convert -196°C to Kelvin gives 77.15 K, rounded to 77 K

Density as a Physical Property

• Definition of density: mass to volume ratio, given as small letter "d"

• Density of water: 1.00 g/mL (useful conversion)

• Different densities for different states:

  • Liquids: g/mL

  • Gases: g/L

  • Solids: g/cm³ (1 cm³ = 1 mL)

Density Conversion Example

• Problem: Calculate liters of benzene in a 25-pound container where benzene has a density of 0.874 g/mL

  • Step 1: Convert pounds to grams:
    1 pound = 454 grams

  • Step 2: Use density to convert grams to milliliters:
    Use density conversion

  • Step 3: Convert milliliters to liters

  • Final answer of approximately 13 liters

Conclusion

• Importance of unit conversions in scientific calculations to achieve accurate measurements

• Methods rely heavily on dimensional analysis and application of exact values and conversion factors to maintain integrity of significant figures in calculations.