Chapter 9: Problem 53

Using the average density of air at sea level (1.225 kilograms per cubic meter, or \(\mathrm{kg} / \mathrm{m}^{3}\) ) and the average mass of Earth's atmosphere above sea level per square meter \(\left(1.033 \times 10^{4} \mathrm{kg} /\right.\) \(\mathrm{m}^{2}\), calculate the total depth of Earth's atmosphere (in kilometers) if its density were the same at all altitudes. (This value is called a scale height, a useful quantity for comparing Earth's atmosphere with the atmospheres of other planets.)

Short Answer

Expert verified

The total depth of Earth's atmosphere is approximately 8.43 kilometers.

Step by step solution

- Understand the given data

Identify and note the given values: average density of air, \(\rho = 1.225 \ \text{kg/m}^3\), and average mass of Earth's atmosphere above sea level per square meter, \(m = 1.033 \times 10^4 \ \text{kg/m}^2\).

- Use the formula for volume and depth

We need to calculate the depth of the atmosphere. Use the relationship \(\text{mass} = \text{density} \times \text{volume}\). Since volume is area \(A \ \text{(m}^2\)) times depth \(h \ \text{(m)}\), we have: \(m = \rho \times A \times h\).

- Simplify the equation

Since the problem gives the mass per unit area, \(m/A = \rho \times h\), solve for \(h\).

- Calculate the depth

Rearrange and solve the equation: \[h = \frac{m}{\rho} = \frac{1.033 \times 10^4 \ \text{kg/m}^2}{1.225 \ \text{kg/m}^3} = 8,433.67 \ \text{m}\].

- Convert meters to kilometers

Convert the depth from meters to kilometers: \(h = \frac{8433.67 \ \text{m}}{1000} = 8.43 \ \text{km}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scale Height

To understand the depth of Earth's atmosphere, we use a concept known as 'scale height'. This is the height at which the atmospheric pressure decreases by a factor of approximately 2.718 (Euler's number). The scale height simplifies comparisons of atmospheric characteristics between different planets. For this exercise, if Earth's atmosphere had a uniform density of 1.225 kg/m³, the scale height can be calculated to provide a specific thickness of the atmosphere. By solving the equation using given values, we determined that the scale height for Earth in this hypothetical scenario is 8.43 kilometers. This helps us gain an intuitive understanding of atmospheric stratification.

Atmospheric Mass

Atmospheric mass plays a crucial role in various calculations related to Earth's atmosphere. It represents the weight of the air above a unit area at sea level. In our exercise, this value is given as 1.033 x 10^4 kg/m². This number helps us find the scale height by relating mass, density, and volume. Essentially, it tells us how much air is stacked above each square meter of Earth's surface. By understanding the total atmospheric mass, we can apply it to the equation: mass = density × volume, which further helps us understand atmospheric density variations with altitude.

Average Density

The average density of air at sea level is a fundamental quantity in atmospheric science. For our calculation, we use an average density of 1.225 kg/m³. Density is the mass per unit volume. Knowing the air's density helps in determining how the atmosphere's mass is distributed. In this exercise, assuming a uniform density, we simplify our calculations by using this average value throughout the entire column of the atmosphere. This is a general simplification, as in reality, the density decreases with altitude, but it provides a useful approximation for our specific calculation.

Kilometer Conversion

Converting units is a common step in many scientific calculations. In this exercise, we calculated the depth of the atmosphere in meters (8,433.67 m) and then converted it to kilometers for convenient comprehension. To convert from meters to kilometers, we divide the value by 1,000 (since 1 kilometer equals 1,000 meters). Thus, 8,433.67 meters become 8.43 kilometers. The conversion is straightforward but important for better understanding and visualizing the scale of the atmospheric height in a more familiar and manageable unit of measure.

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Short Answer

Step by step solution

- Understand the given data

- Use the formula for volume and depth

- Simplify the equation

- Calculate the depth

- Convert meters to kilometers

Key Concepts

Scale Height

Atmospheric Mass

Average Density

Kilometer Conversion

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