Question

The area of the 48 contiguous states is \(3.02 \times 10^{6} \mathrm{mi}^{2}\). Assume that these states are completely flat (no mountains and no valleys). What volume, in liters, would cover these states with a rainfall of two inches?

Short answer

Answer: Approximately \(3.9759 \times 10^{14} \mathrm{L}\) of water would be needed.

Step-by-step solution

1. Convert the area from square miles to square meters

To convert the area of the states from square miles to square meters, we use the conversion factor 1 mile = 1609.34 meters. Since the area is in square miles, we will square the conversion factor. Area in square meters = \(3.02 \times 10^{6} \mathrm{mi}^{2} \times (1609.34 \mathrm{m/mi})^{2} = 7.8255 \times 10^{12} \mathrm{m^2}\)

2. Find the height of the rainfall in meters

Now, we need to convert the rainfall of two inches to meters. To do this, we use the conversion factor 1 inch = 0.0254 meters. Rainfall height in meters = \(2 \mathrm{in} \times 0.0254 \mathrm{m/in} = 0.0508 \mathrm{m}\)

3. Calculate the volume of water

To find the volume of water needed to cover the states with a rainfall of 0.0508 meters, we multiply the area with the height of the rainfall. Volume in cubic meters = \(7.8255 \times 10^{12} \mathrm{m^2} \times 0.0508 \mathrm{m} = 3.9759 \times 10^{11} \mathrm{m^3}\)

4. Convert the volume from cubic meters to liters

Finally, to convert the volume from cubic meters to liters, we use the conversion factor 1 cubic meter = 1000 liters. Volume in liters = \(3.9759 \times 10^{11} \mathrm{m^3} \times 1000 \mathrm{L/m^3} = 3.9759 \times 10^{14} \mathrm{L}\) The volume of water, in liters, that would cover the 48 contiguous states with a rainfall of two inches is approximately \(3.9759 \times 10^{14} \mathrm{L}\).

Key concepts

Unit Conversion

Unit conversion is a critical skill in chemistry and other sciences, acting as the foundation for analyzing and interpreting data. This is the process of converting a measure of physical quantity from one unit to another. For instance, if we have the area in square miles and we need it in square meters, we must apply a conversion factor. This factor is basically a ratio that represents the equivalent value of one unit in terms of another.

To apply a conversion factor, we simply multiply the original measurement by the factor. As illustrated in our example, the area of the 48 contiguous states is given in square miles, but the final volume is requested in liters, which requires multiple unit conversions. Stepping through each conversion cautiously ensures that we avoid errors and understand the relationship between different units. In daily life, these conversions can range from simple tasks like converting inches to centimeters on a ruler, to interpreting fuel efficiency of a vehicle in miles per gallon versus liters per 100 kilometers.

Volume Calculation

Volume calculation is essential when dealing with three-dimensional space in physical sciences like chemistry. To calculate volume, we usually multiply the measurements of length, width, and height. However, in cases where we're dealing with a particular depth of a substance spread over an area – such as rainfall over land – the calculation simplifies to multiplying the area by the depth.

In our exercise, we are asked to calculate the volume of rainfall needed to cover the 48 contiguous states, which are represented as a flat area with a uniform rainfall depth. This reduces our three-dimensional problem to a simpler one: area times height. Remember, the depth of the rainfall needs to be in the same unit of measure as the length and width of the area for accurate calculation. Calculating volumes in chemistry can be critical, for instance, when determining the amount of a reactant needed for a reaction or the capacity of a container necessary to hold a particular substance.

Dimensional Analysis

Dimensional analysis, sometimes called the factor-label method or the unit factor method, is a technique used widely in chemistry to convert one unit of measure to another, to calculate concentrations, and to interpret various kinds of rates. It involves using conversion factors arranged in such a way that unwanted units cancel out to give the desired units.

When performing dimensional analysis, we start with the quantity we aim to convert, include our conversion factors, and systematically cancel out units until we're left with the unit we need. Fundamental to this technique is ensuring that we have the correct conversion ratio and that the units we're aiming to cancel out are indeed cancelable. For instance, you cannot cancel out units of volume with units of area or length. For chemistry students, mastering dimensional analysis is vital as it provides a structured method for tackling complex conversions and calculations across various chemical calculations and equations.