Unit Conversion Examples

Unit Conversion Examples

Converting units is like solving a riddle; the key is knowing the relationship between the units.

Converting Grams to Micrograms

• Problem: Convert 3.14 g to µg.

• Key Question: How many grams are in 1 microgram?

• Reference: Table 1.4 in the BLB book (perspective of the base unit, grams).

• Steps:

  1. Start with what you're given: $3.14 g = ?? µg$

  2. Establish the relationship: According to Table 1.4, $1 <br>lessimes 10^{-6} g = 1 µg$

  3. Rearrange the relationship into a ratio: Divide both sides of the equation by $1 <br>lessimes 10^{-6} g$ to get:
     $\frac{1 <br>lessimes 10^{-6} g}{1 <br>lessimes 10^{-6} g} = \frac{1 µg}{1 <br>lessimes 10^{-6} g}$
     $1 = \frac{1 µg}{1.0 <br>lessimes 10^{-6} g}$

  4. Ratio Interpretation: There are $1 <br>lessimes 10^{-6} g$ in $1 µg$. Converting units is like multiplying by 1, which doesn't change the physical quantity, only the numerical representation.

  5. Apply the ratio: Multiply the given value by the ratio:
     $3.14 g <br>lessimes \frac{1 µg}{1 <br>lessimes 10^{-6} g}$

  6. Cancel units: Grams (g) appear on both the top and bottom, so they cancel out.
     $\frac{3.14 \cancel{g} <br>lessimes 1 µg}{1 <br>lessimes 10^{-6} \cancel{g}} = 3140000 µg$

  7. Final Answer: $3.14 g = 3140000 µg$

  8. Sanity Check: Converting from a base unit (grams) to a smaller unit (micrograms) should result in a larger numerical value.

Converting Picometers to Nanometers

• Problem: Convert 140 pm to nm.

• Process: This conversion requires two steps since there is no direct relationship between picometers and nanometers: pm → m → nm.

  1. Start with what you're given: $140 pm = ?? nm$

  2. Convert pm to meters: How many meters are in 1 pm? From Table 1.4, $1 <br>lessimes 10^{-12} m = 1 pm$. Rearrange to have meters in the numerator:
     $\frac{1 <br>lessimes 10^{-12} m}{1 pm} = \frac{1 pm}{1 pm}$
     $\frac{1 <br>lessimes 10^{-12} m}{1 pm} = 1$

  3. Convert meters to nanometers: How many meters are in 1 nm? From Table 1.4, $1 <br>lessimes 10^{-9} m = 1 nm$
     $1 = \frac{1 nm}{1 <br>lessimes 10^{-9} m}$

  4. Combine the ratios:
     $140 pm <br>lessimes \frac{1 <br>lessimes 10^{-12} m}{1 pm} <br>lessimes \frac{1 nm}{1 <br>lessimes 10^{-9} m}$

  5. Cancel units: Picometers (pm) and meters (m) cancel out, leaving the answer in nm.
     $\frac{140 \cancel{pm} <br>lessimes 1 <br>lessimes 10^{-12} \cancel{m} <br>lessimes 1 nm}{1 \cancel{pm} <br>lessimes 1 <br>lessimes 10^{-9} \cancel{m}} = 0.140 nm$

  6. Final Answer: $140 pm = 0.140 nm$

  7. Sanity Check: Moving from pm to nm should result in a smaller magnitude for the number.

Shortcut Using Exponents

• Using properties of exponents:
  $\frac{1 <br>lessimes 10^{-12}}{1 <br>lessimes 10^{-9}} = 1 <br>lessimes 10^{-3} = \frac{1}{1000}$

• Dividing by 1000 corresponds to moving the decimal three places to the left, providing a direct conversion from pm to nm if the relationship to the base unit is understood.
  $\frac{1 <br>lessimes 10^{-12} \cancel{m}}{1 pm} <br>lessimes \frac{1 nm}{1 <br>lessimes 10^{-9} \cancel{m}} = \frac{1 <br>lessimes 10^{-3} nm}{1 pm}$