Problem 85 - General Chemistry

Question

The total amount of fresh water on earth is estimated to be \(3.73 \times 10^{8} \mathrm{~km}^{3}\). What is this volume in cubic meters? in liters?

Step-by-Step Solution

Answer

Volume: \(3.73 \times 10^{17}\) m\(^3\), \(3.73 \times 10^{20}\) liters.

Step 1: Convert Cubic Kilometers to Cubic Meters

To convert the volume from cubic kilometers to cubic meters, recall that:1 ext{ km} = 1,000 ext{ m}.Since the conversion is cubed, we have:1 ext{ km}^3 = (1,000 ext{ m})^3 = 1,000,000,000 ext{ m}^3 = 10^9 ext{ m}^3.Therefore, the volume in cubic meters is given by:\[3.73 \times 10^8 \text{ km}^3 \times 10^9 = 3.73 \times 10^{17} \text{ m}^3.\]

Step 2: Convert Cubic Meters to Liters

To convert the volume from cubic meters to liters, note that:1 cubic meter ( ext{m}^3) = 1,000 liters.Therefore, the volume in liters is:\[3.73 \times 10^{17} \text{ m}^3 \times 1,000 = 3.73 \times 10^{20} \text{ liters}.\]

Key Concepts

Cubic Kilometers to Cubic Meters

Volume conversion from cubic kilometers to cubic meters is a vital process in understanding large-scale measurements such as those dealing with planetary resources. When converting cubic kilometers to cubic meters, it's essential to remember the basic conversion factor:

• 1 kilometer (km) is equal to 1,000 meters (m).
• Since we are working with volume, which is a three-dimensional measure, we cube the conversion factor: (1 km)³ = (1,000 m)³.
• This results in: 1 km³ = 1,000,000,000 m³ or more simply, 10⁹ m³.

For the Earth’s fresh water calculation: if we have 3.73 × 10⁸ km³ of fresh water, we need to multiply by 10⁹ to convert to cubic meters. This gives us a staggering 3.73 × 10¹⁷ m³ of fresh water when expressed in cubic meters. With these steps, you can easily convert any volume from cubic kilometers to cubic meters.

Cubic Meters to Liters

Once the volume is expressed in cubic meters, converting it to liters becomes straightforward. This conversion is particularly useful for scientific, industrial, and sometimes even everyday applications. The conversion factor here is:

• 1 cubic meter (m³) equals 1,000 liters.

In our given problem, we have already calculated that the total fresh water is 3.73 × 10¹⁷ m³.
To find out how many liters this corresponds to, multiply the cubic meters by 1,000. This conversion gives us:

• 3.73 × 10¹⁷ m³ × 1,000 = 3.73 × 10²⁰ liters.

Hence, the enormity of the Earth’s fresh water can also be visualized as 3.73 × 10²⁰ liters.

Volume Conversion Steps

Understanding and following the correct steps in volume conversion helps ensure accuracy in calculations. Here's a simple guide for converting between different units:

• Determine the units you are converting from and to. In this case, cubic kilometers to cubic meters and cubic meters to liters.
• Identify and apply the relevant conversion factor. Example: 1 km³ = 10⁹ m³ and 1 m³ = 1,000 liters.
• Multiply your initial volume by these factors step by step, ensuring dimensional consistency.

It’s also helpful to double-check each step as calculation errors in conversions can lead to significant misunderstandings, especially when dealing with large and small scales. Practicing these conversions with different numbers aids in solidifying these concepts, making them intuitive over time.