Perform the following conversions. (a) \(8.4 \mathrm{ft}^{2}\) to \(\mathrm{cm}^{2}\) (c) \(231 \mathrm{ft}^{3}\) to \(\mathrm{cm}^{3}\) (b) \(223 \mathrm{mi}^{2}\) to \(\mathrm{km}^{2}\)

One of the fundamental concepts in chemistry and other sciences is the conversion factor. It is a number used to change from one unit to another via multiplication or division. In chemistry, conversion factors are often used to convert the units of measurements like mass, volume, and concentration. To perform a unit conversion, you need to know the equivalent values between different systems of units, such as metric to imperial or vice versa.

For example, to convert feet to centimeters, you can use the conversion factor of 30.48 (since 1 foot equals 30.48 centimeters). If you have a length of 2 feet, you can calculate the conversion to centimeters as follows: \[\begin{equation}2 \text{ ft} \times 30.48 \frac{\text{cm}}{\text{ft}} = 60.96 \text{ cm}d\text{.}\end{equation}\]Notice how the unit we want to convert from (feet) is in the denominator, which allows it to be 'canceled out', leaving us with the desired unit (centimeters). It's crucial to align the units so that they cancel properly when using conversion factors. Ensuring the correct alignment avoids errors and provides the correct conversion.

Converting square units, or units of area, involves a bit more than just converting linear measurements. This is because areas are two-dimensional. Therefore, when converting from one square unit to another, you must square the linear conversion factor. For instance, to convert from square feet to square centimeters, you would square the conversion factor between feet and centimeters.

Take the conversion factor from feet to centimeters, which is 30.48 cm per foot. To convert square feet to square centimeters:\[\begin{equation}(30.48 \text{ cm/ft})^2 = 929.0304 \text{ cm}^2/\text{ft}^2d\text{.}\end{equation}\]This means that 1 square foot is equivalent to 929.0304 square centimeters. To convert an area of 8.4 square feet into square centimeters, you multiply by the squared conversion factor:\[\begin{equation}8.4 \text{ ft}^2 \times 929.0304 \frac{\text{cm}^2}{\text{ft}^2} = 7803.85536 \text{ cm}^2d\text{.}\end{equation}\]As seen in the equation above, squaring the conversion while computing square unit conversion is vital.

Cubic unit conversion is necessary when dealing with volumes, which are three-dimensional. Similar to square unit conversion, you must cube the linear conversion factor when switching between cubic units. This extra step is essential because volume calculations involve three dimensions.
1 cubic foot is equivalent to 28,316.8466 cubic centimeters.

Therefore, when you want to convert cubic feet to cubic centimeters, you would use the following calculation:\[\begin{equation}(30.48 \text{ cm/ft})^3 = 28,316.8466 \text{ cm}^3/\text{ft}^3d\text{.}\end{equation}\]If you have a volume of 231 cubic feet, to convert it to cubic centimeters, you'll use this cubed conversion factor:\[\begin{equation}231 \text{ ft}^3 \times 28,316.8466 \frac{\text{cm}^3}{\text{ft}^3} = 6,541,191.38 \text{ cm}^3d\text{.}\end{equation}\]This calculation ensures accurate volume conversion, reflecting the fact that each linear dimension has been converted to the new unit, making it consistent in all three dimensions of volume.