Ch 14: Fluids and Elasticity

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 14: Fluids and Elasticity

Problem 48b

Chapter 14, Problem 48b

It's possible to use the ideal-gas law to show that the density of the earth's atmosphere decreases exponentially with height. That is, ρ = ρ₀ exp (-z/z₀), where z is the height above sea level, ρ₀ is the density at sea level (you can use the Table 14.1 value), and z₀ is called the scale height of the atmosphere. What is the density of the air in Denver, at an elevation of 1600 m? What percent of sea-level density is this?

Verified step by step guidance

Step 1: Recall the given formula for the density of the atmosphere as a function of height:

p = p₀ exp(-z/z₀)

. Here,

p₀

is the density at sea level,

z

is the height above sea level, and

z₀

is the scale height of the atmosphere.

Step 2: Look up the value of

p₀

, the density of air at sea level, from Table 14.1. This value is typically given as approximately

1.29 \, \(\text{kg/m}\)^3

Step 3: Determine the scale height

z₀

. The scale height is a constant that depends on the temperature and composition of the atmosphere. For Earth's atmosphere,

z₀

is approximately

8,400 \, \(\text{m}\)

Step 4: Substitute the elevation of Denver,

z = 1600 \, \(\text{m}\)

, along with the values of

p₀

and

z₀

, into the formula

p = p₀ exp(-z/z₀)

to calculate the density of air in Denver.

Step 5: To find the percentage of sea-level density, divide the calculated density of air in Denver by

p₀

and multiply by 100. This will give the percentage of sea-level density at Denver's elevation.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in thermodynamics that relates the pressure, volume, temperature, and number of moles of a gas. It is expressed as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature in Kelvin. This law helps in understanding how gases behave under different conditions and is essential for deriving the relationship between atmospheric pressure and density.

Exponential Decay

Exponential decay describes a process where a quantity decreases at a rate proportional to its current value. In the context of the atmosphere, the density of air decreases exponentially with height, which can be mathematically represented as p = p₀ exp(─z/z₀). This concept is crucial for understanding how atmospheric pressure and density change with altitude, leading to lower air density at higher elevations.

Scale Height

Scale height (z₀) is a measure of the rate at which the atmospheric pressure decreases with height. It is defined as the height over which the pressure decreases by a factor of e (approximately 2.718). The scale height varies with temperature and composition of the atmosphere, and it is a key parameter in calculating how density changes with altitude, particularly in applications like determining air density in locations such as Denver.

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