Problem 20 - Chemistry for Allied Health

Question

Complete each of the following conversions. a. \(342 \mathrm{~cm}^{3}\) to \(\mathrm{dm}^{3}\) b. \(2.70 \mathrm{~g} / \mathrm{cm}^{3}\) to \(\mathrm{kg} / \mathrm{L}\) c. \(34 \mathrm{mi} / \mathrm{hr}\) to \(\mathrm{km} / \mathrm{min}\) d. \(0.00722 \mathrm{~km}^{2}\) to \(\mathrm{m}^{2}\) e. \(4.9 \times 10^{5} \mathrm{mcm}^{3}\) to \(\mathrm{mm}^{3}\) f. \(80 . \mathrm{km} / \mathrm{hr}\) to \(\mathrm{mi} / \mathrm{hr}\)

Step-by-Step Solution

Answer

a. \(0.342 \, \mathrm{dm}^3\); b. \(2.70 \, \mathrm{kg/L}\); c. \(0.911 \, \mathrm{km/min}\); d. \(7220 \, \mathrm{m}^2\); e. \(4.9 \times 10^{5} \, \mathrm{mm}^3\); f. \(49.71 \, \mathrm{mi/hr}\).

Step 1: Converting Cubic Centimeters to Cubic Decimeters

To convert from cubic centimeters (\(\mathrm{cm}^3\)) to cubic decimeters (\(\mathrm{dm}^3\)), use the conversion factor: \(1 \mathrm{dm}^3 = 1000 \mathrm{cm}^3\). Divide the given volume by this factor: \(342 \mathrm{~cm}^{3} \times \frac{1 \mathrm{dm}^{3}}{1000 \mathrm{cm}^{3}} = 0.342 \mathrm{~dm}^{3}\).

Step 2: Converting Grams per Cubic Centimeter to Kilograms per Liter

The conversion factor is: \(1 \mathrm{g/cm}^3 = 1 \mathrm{kg/L}\). Multiply the given density by this factor: \(2.70 \mathrm{~g/cm}^3 \times \frac{1 \mathrm{kg/L}}{1 \mathrm{g/cm}^3} = 2.70 \mathrm{~kg/L}\).

Step 3: Converting Miles per Hour to Kilometers per Minute

First, convert miles to kilometers using the factor \(1 \mathrm{mi} = 1.60934 \mathrm{km}\). Then convert hours to minutes (1 hour = 60 minutes). Perform the conversion: \(34 \mathrm{mi/hr} \times \frac{1.60934 \mathrm{km}}{1 \mathrm{mi}} \times \frac{1 \mathrm{hr}}{60 \mathrm{min}} \approx 0.911 \mathrm{km/min}\).

Step 4: Converting Square Kilometers to Square Meters

To convert from square kilometers (\(\mathrm{km}^2\)) to square meters (\(\mathrm{m}^2\)), use \(1 \mathrm{km}^2 = 10^6 \mathrm{m}^2\). Multiply the area by this factor: \(0.00722 \mathrm{~km}^2 \times 10^6 \mathrm{m}^2/ ext{km}^2 = 7220 \mathrm{~m}^2\).

Step 5: Converting Micrometers Cubed to Millimeters Cubed

The conversion factor is: \(1 \text{micrometer}^3 = 10^{-9} \text{m}^3\) and \(1 \text{millimeter}^3 = 10^{-9} \text{m}^3\). They are equal volumes so \(4.9 \times 10^{5} \, \mathrm{mcm}^{3} = 4.9 \times 10^{5} \, \mathrm{mm}^{3}\).

Step 6: Converting Kilometers per Hour to Miles per Hour

To convert from kilometers per hour (\(\mathrm{km/hr}\)) to miles per hour (\(\mathrm{mi/hr}\)), use the factor \(1 \mathrm{km} = 0.621371 \mathrm{mi}\). Multiply the speed by this factor: \(80 \, \mathrm{km/hr} \times \frac{0.621371 \, \mathrm{mi}}{1 \, \mathrm{km}} \approx 49.71 \, \mathrm{mi/hr}\).

Key Concepts

Cubic Centimeters to Cubic Decimeters

Converting volumes from cubic centimeters (cm³) to cubic decimeters (dm³) is an essential skill in unit conversion. This conversion is straightforward because it involves a simple factor:

• 1 cubic decimeter (dm³) is equivalent to 1000 cubic centimeters (cm³).
  

To convert cm³ to dm³, simply divide the volume by 1000. For instance, if you have a volume of 342 cm³ and you wish to express it in dm³, perform the calculation as follows: \[ 342 \, \text{cm}^3 \times \frac{1 \, \text{dm}^3}{1000 \, \text{cm}^3} = 0.342 \, \text{dm}^3 \]This demonstrates that 342 cubic centimeters equals 0.342 cubic decimeters. Think of this process as shrinking a volume to a tenth of its size, an apt visualization for working between these metric units.

Grams per Cubic Centimeter to Kilograms per Liter

Density often needs to be expressed in various unit systems, and converting from grams per cubic centimeter (g/cm³) to kilograms per liter (kg/L) is common in science. Luckily, this conversion is directly one-to-one:

• 1 g/cm³ is exactly equal to 1 kg/L.
  

This direct relationship stems from the metric system’s design, where 1 liter is defined as 1000 cubic centimeters, and 1 kilogram as 1000 grams. Thus, density values do not change numerically when converting between these units. If your density is 2.70 g/cm³, it equally becomes 2.70 kg/L without any need for complex calculations. This makes interpreting density results fast and hassle-free across these measurements.

Miles per Hour to Kilometers per Minute

When converting speeds from miles per hour (mi/hr) to kilometers per minute (km/min), there are two conversions involved:

• Miles to kilometers (1 mile = 1.60934 kilometers).
  
• Hours to minutes (1 hour = 60 minutes).
  

Start by converting miles into kilometers using the given conversion factor, then convert hours into minutes. For example, to convert 34 mi/hr to km/min:\[ 34 \, \text{mi/hr} \times \frac{1.60934 \, \text{km}}{1 \, \text{mi}} \times \frac{1 \, \text{hr}}{60 \, \text{min}} \approx 0.911 \, \text{km/min} \]By doing these two conversions in sequence, you calculate that 34 miles per hour is approximately 0.911 kilometers per minute, enabling quicker interpretations of speed in contexts where minutes are more relevant, such as detailed travel plans.

Square Kilometers to Square Meters

Surface area conversions are essential for geography, engineering, and science. Squaring the metric conversion from kilometers to meters helps in this transformation. Know that:

• 1 square kilometer (km²) equals 1,000,000 square meters (m²).
  

This is because the linear conversion (1 km = 1000 m) is squared for two dimensions. Converting 0.00722 km² to m² involves simply multiplying by this factor:\[ 0.00722 \, \text{km}^2 \times 10^6 \, \text{m}^2/\text{km}^2 = 7220 \, \text{m}^2 \]This calculation shows that a seemingly small area in square kilometers translates to a larger value in square meters, emphasizing why different units might be more applicable depending on the context or scale of measurement.