Chemistry for Liberal Studies - Forensic Academy / Dr. Stephanie R. Dillon

So far we have only talked about converting between units in the metric system of prefixes. This unfortunately is not the only conversions we will need to practice especially if you live in the United States. Why do I say this? Well in the US along with a few other countries around the world we use a lot of English system units like the inch, foot, mile, pound etc. Some of the conversion factors commonly used to convert units of length are shown in the table. Note that unlike our previous values, most of these are not based on the number 10.

Conversion Chart

\begin{align} 1 \text{ centimeter} & = 0.3937 \text{ inches} \\ 12 \text{ inches} & = 1 \text{ foot} \\ 1 \text{ inch} & = 2.54 \text{ centimeters} \\ 1 \text{ foot} & = 0.3048 \text{ meters} \\ 3 \text{ feet} & = 1 \text{ yard} \\ 1760 \text{ feet} & = 1 \text{ mile} \\ 5280 \text{ feet} & = 1 \text{ mile} \\ 1 \text{ yard} & = 0.9144 \text{ meters} \\ 1 \text{ meter} & = 3.28083 \text{ feet} \\ 1 \text{ kilometer} & = 3281 \text{ feet} \\ 1 \text{ kilometer} & = 0.6214 \text{ miles} \end{align}

Let's practice converting a common measurement in English units into metric units. A typical female student is 5 feet and 6 inches tall. How many centimeters tall would she be? This question actually takes two steps to calculate because we have a total of 3 units to deal with: Feet, inches and centimeters. According to the table provided, 1 foot equals 12 inches and 2.54 centimeters equals 1 inch. We can use these conversion factors to complete our calculation.

First we need to get the feet converted into inches using the conversion factor of 12 inches equals 1 foot. That will change the 5 feet given into 60 inches. If we add the 60 inches to the 6 inches we were given in the problem this tell us that we have 66 inches overall. Now it is a simple matter of multiplication by the conversion factor for inches to centimeters to get our value into the units we want. It turns out that a female student that is 5'6" tall will be 168 cm tall in metric units.

Example

A typical female student is 5.00 feet and 6.00 inches tall. How many centimeters tall would she be?

$$ 1 \text{ foot} = 12 \text{ inches, } 2.54 \text{ cm} = 1 \text{ inch} \\ 5 \text{ feet} \times 12 \text{in}/1 \text{ft} = 60 \text{in} \\ 60 \text{in} + 6 \text{in} = 66 \text{in} \\ 66 \text{in} \times 2.54 \text{cm}/1 \text{in} = 167.64 \text{cm} = 168 \text{cm (in correct sf)} $$

Volume

The volume of any solid, plasma, vacuum or theoretical object is how much three-dimensional space it occupies, often quantified numerically. Volume is commonly presented as gallons, milliliters, or cubic centimeters.

\begin{align} \text{Volume of a Cube } & = a^3 \\ \text{Volume of a Rectangular Prism } & = abc \\ \text{Volume of a Cylinder } & = b \ast h = \pi r^2h \\ \text{Volume of a Sphere } & = (4/3) \pi r^3 \end{align}

In addition to calculations for converting single unit values, there are often times when a unit may be squared or even cubed. This kind of calculation is often necessary when determining the volume of an object. The important thing to remember is that whatever mathematical process you apply to the number value, you must apply that same process to the unit and vice versa.

For example, if you had a cube with a volume of 2.0 ft^{3} but you needed its volume in the metric unit of cm^{3}, how would you complete this conversion? Start by collecting the conversion factors that you need. We know that 2.54 cm = 1 inch and 12 inches = 1 foot. The first conversion should be from feet to inches and then we can convert the inches to centimeters. But note that since the original units are cubed, we must maintain that cubing throughout the conversion process and whatever we do to the units to keep them the same we must also apply to the numbers involved.

Example

What would be the volume in cm^{3} of a 2.0 ft^{3} cube?

2.54 cm = 1 inch, 12 inches = 1 foot

$$2.0 ft^3 \times \left(12 in \over 1 ft\right)^3 \times \left(2.54 cm \over 1 in\right)^3 =$$

$$2.0 ft^3 \times {1728in^3 \over 1ft} \times {16.4cm^3 \over 1in} = 5.67 \times 10^4cm^3$$

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