Problem 137 - General Chemistry

Question

The largest of the Great Lakes is Lake Superior, which has a volume of \(12,230 \mathrm{~km}^{3}\). What is this volume in liters?

Step-by-Step Solution

Answer

Lake Superior's volume in liters is 12,230,000,000,000,000 liters.

Step 1: Understand the Units

The volume is originally given in cubic kilometers ( ^{3} ). To find the volume in liters, we should know that 1 ^{3} equals 1,000,000,000,000 liters since there are 1,000,000,000 cubic meters in a cubic kilometer and 1,000 liters in a cubic meter.

Step 2: Convert Cubic Kilometers to Cubic Meters

Multiply the volume of Lake Superior by the conversion factor from cubic kilometers to cubic meters. \[ 12,230 \text{ km}^3 = 12,230 \times 1,000,000,000 \text{ m}^3 \] Simplifying, we find:\[ 12,230 \text{ km}^3 = 12,230,000,000,000 \text{ m}^3 \]

Step 3: Convert Cubic Meters to Liters

Multiply the result from Step 2 by the conversion factor from cubic meters to liters. \[ 12,230,000,000,000 \text{ m}^3 = 12,230,000,000,000 \times 1,000 \text{ liters} \] Simplifying, we find:\[ 12,230,000,000,000,000 \text{ liters} \]

Key Concepts

Cubic Kilometers to Liters

Converting from cubic kilometers to liters involves understanding the relationship between these units. A cubic kilometer is a large volume, since the kilometer is a large unit of length.
To convert cubic kilometers to liters, we first convert kilometers to meters, since one kilometer is 1,000 meters. Then, because a cube's volume is calculated in terms of the cube of its length, we consider this in three dimensions:

• The length in meters is cubed to become cubic meters.
• Then, each cubic meter is converted to liters, knowing that there are 1,000 liters in one cubic meter.

Thus, the conversion from cubic kilometers to liters is done in two steps: first by converting to cubic meters, and then by converting cubic meters to liters. This results in a conversion factor wherein 1 cubic kilometer equals 1,000,000,000,000 liters. So, for Lake Superior's volume of 12,230 cubic kilometers, you multiply by this factor to get the volume in liters.

Volume Conversion

Volume conversion is a process of translating one volume of measurement to another within various systems. In this case, we are working within the metric system, which is very straightforward due to its use of base-10.
When converting volumes between different units within the metric system, it's helpful to know the equivalencies:

• 1 cubic kilometer (km³) = 1,000,000,000 cubic meters (m³).
• 1 cubic meter (m³) = 1,000 liters (L).

Understanding these relationships allows for seamless conversion. For Lake Superior, starting with 12,230 km³, the conversion involves multiplying by each respective step's conversion factor sequentially: to cubic meters, then to liters.
This layered conversion is crucial for accurately translating volumes for scientific calculations, engineering, and real-world applications.

Metric System

The metric system is a globally recognized decimal-based system of measurement. It is designed around basic units like meters for length, liters for volume, and grams for mass, scaled by powers of ten.
Its simplicity and consistency make it a preferred choice in scientific circles and most of the world.

• The basic prefixes include kilo- (1,000), centi- (1/100), and milli- (1/1,000), among others.
• Conversions within the metric system often only require shifting the decimal point because the system is structured around multiples of ten.

For volume, specifically, the metric system facilitates easy conversions between units like cubic kilometers, cubic meters, and liters.
This advantage is apparent when converting massive natural features like Lake Superior into more accessible units, allowing for better comprehension and utilization in areas like environmental studies or resource management.