Problem 70 - Chemistry Principles and Practice

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 of water, in liters, would cover these states with a rainfall of two inches?

Step-by-Step Solution

Answer

The volume of water would be approximately \(3.97 \times 10^{14}\) liters.

Step 1: Convert Inches to Meters

The first step is to convert the rainfall measurement from inches to meters. There are 0.0254 meters in an inch. Therefore, two inches of rainfall can be converted to meters by multiplying: \[ 2 ext{ inches} \times 0.0254 ext{ meters/inch} = 0.0508 ext{ meters}. \]

Step 2: Calculate Volume in Cubic Meters

Now we need to find the volume in cubic meters by multiplying the area by the rainfall depth in meters:\[3.02 \times 10^6 ext{ square miles} \times 0.0508 ext{ meters}.\]First, convert square miles to square meters. 1 square mile is equal to 2,589,988.11 square meters, so:\[3.02 \times 10^6 ext{ square miles} \times 2,589,988.11 ext{ m}^2/ ext{mi}^2 = 7.82 \times 10^{12} ext{ m}^2.\]Then,\[7.82 \times 10^{12} ext{ m}^2 \times 0.0508 ext{ meters} = 3.97 \times 10^{11} ext{ cubic meters}.\]

Step 3: Convert Cubic Meters to Liters

To convert cubic meters to liters, we use the conversion factor that 1 cubic meter equals 1,000 liters:\[3.97 \times 10^{11} ext{ cubic meters} \times 1,000 ext{ liters/cubic meter} = 3.97 \times 10^{14} ext{ liters}.\]

Key Concepts

Volume Calculation

Understanding how to calculate volume is an essential part of many scientific and mathematical tasks. In this exercise, we wanted to calculate the volume of water that would cover a large area, specifically the contiguous states of the US after a specified rainfall. The first step involves understanding how volume is defined. Volume is determined by multiplying the area of the base by the height (or depth, in this case). This formula is expressed as:\[ \text{Volume} = \text{Area} \times \text{Height} \]In the problem, the area is provided in square miles, and the height comes from the conversion of rainfall from inches to meters. It's important to ensure that all measurements are in compatible units before performing the calculation.
After converting all units appropriately, multiplying the area by the depth gives the volume in cubic meters, which represents the amount of space the water takes up.

Area Conversion

Area conversion is a fundamental skill when working with measurements from different systems. In our exercise, we started with an area given in square miles. However, to find the volume of water, it was necessary to convert this area to square meters.Here's a step-by-step on how to convert:- Understand the conversion rate: 1 square mile equals 2,589,988.11 square meters.- Multiply the number of square miles by this conversion rate to get the area in square meters.For our problem: \[ 3.02 \times 10^6 \text{ square miles} \times 2,589,988.11 \frac{\text{m}^2}{\text{mi}^2} = 7.82 \times 10^{12} \text{ m}^2 \] This step ensures that the area unit is compatible with the rain "height", allowing for accurate volume calculation. Standardizing units makes mathematical operations straightforward and reduces potential errors.

Rainfall Measurement

Rainfall measurement is crucial for understanding environmental impacts and planning. Typically, rainfall is measured in inches, which may be familiar on a household level but needs conversion for larger scientific computations.To convert rainfall from inches to meters, a key metric conversion is used:- **Conversion rate**: 1 inch equals 0.0254 meters.For the exercise, two inches of rain translates to:\[ 2 \text{ inches} \times 0.0254 \frac{\text{meters}}{\text{inch}} = 0.0508 \text{ meters} \]This metric depth is then multiplied by the area (in square meters) to get the water volume in cubic meters. Understanding rainfall in metric terms is essential for large-scale calculations such as calculating the volume of water after heavy rainfall in extensive areas.