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Chapter 14: Problem 3

Explain how the density of air varies with altitude.

Short Answer

Expert verified
The density of air decreases with increasing altitude due to the decrease in air pressure and temperature. As altitude increases, there are fewer air molecules above a given area, causing a decrease in air pressure. Additionally, temperature tends to decrease with increasing altitude. These factors together lead to a decrease in air density, which can be represented by the equation: \(\rho(h) = \frac{P_0 \cdot e^{(- \frac{mgh}{k(T_0 - Lh)})}}{R \cdot (T_0 - Lh)}\)

Step by step solution

01

Understand the relationship between altitude and air pressure

As altitude increases, air pressure decreases. This is because at higher altitudes, there are fewer air molecules above a given area, resulting in a decrease in air pressure. The relationship between altitude and air pressure can be described by the Barometric Formula: \( P(h) = P_0 \cdot e^{(- \frac{mgh}{kT})} \) where: - \(P(h)\) is the air pressure at altitude \(h\), - \(P_0\) is the air pressure at sea level, - \(m\) is the molar mass of Earth's air, - \(g\) is the acceleration due to gravity, - \(h\) is the altitude, - \(k\) is the Boltzmann constant, and - \(T\) is the temperature.
02

Understand the relationship between altitude and temperature

As altitude increases, temperature generally decreases. This occurs because air closer to Earth's surface absorbs more solar radiation and retains more heat than air at higher altitudes. The approximate relationship between altitude and temperature can be represented by the Vertical Temperature Profile: \( T(h) = T_0 - Lh \) where: - \(T(h)\) is the temperature at altitude \(h\), - \(T_0\) is the temperature at sea level, and - \(L\) is the temperature lapse rate.
03

Use the Ideal Gas Law to relate air pressure, temperature, and density

The Ideal Gas Law relates the pressure, temperature, and volume of a gas to its density. The equation is: \( PV = nRT \) where: - \(P\) is the pressure, - \(V\) is the volume, - \(n\) is the number of moles, - \(R\) is the universal gas constant, and - \(T\) is the temperature. Rearranging the equation for density (\(\rho\)): \(\rho = \frac{n}{V} = \frac{P}{RT} \)
04

Combine the Barometric Formula, Vertical Temperature Profile, and Ideal Gas Law

Using the relationship between altitude and air pressure (Barometric Formula) and the relationship between altitude and temperature (Vertical Temperature Profile), we can find the density of air at a given altitude. The Ideal Gas Law equation can be used to find the density: \(\rho(h) = \frac{P(h)}{R \cdot T(h)} \) Substituting the Barometric Formula and Vertical Temperature Profile into the Ideal Gas Law equation, we get the following equation for the density of air as a function of altitude: \(\rho(h) = \frac{P_0 \cdot e^{(- \frac{mgh}{k(T_0 - Lh)})}}{R \cdot (T_0 - Lh)} \)
05

Understand the effect of altitude on air density

As altitude increases, both air pressure and temperature typically decrease, which affects air density. The decrease in air pressure with increasing altitude usually has a greater impact on air density than the decrease in temperature. As a result, the density of air generally decreases with increasing altitude. The equation derived in the previous step can be used to calculate the specific density of air at a particular altitude.

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Key Concepts

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

Barometric Formula
The Barometric Formula is crucial in understanding how air pressure changes with altitude. This formula describes how, as we move higher above sea level, the air pressure decreases. This happens because there are fewer air molecules stacked above you in higher regions compared to sea level.

The mathematical expression for this is:
  • \( P(h) = P_0 \cdot e^{- \frac{mgh}{kT}} \)
  • In this equation, \(P(h)\) is the air pressure at height \(h\), \(P_0\) is the pressure at sea level, \(m\) is molar mass of air, \(g\) is gravitational acceleration, \(h\) is altitude, \(k\) is the Boltzmann constant, and \(T\) is temperature.
So, as you climb higher, the exponent becomes more negative, thus reducing air pressure exponentially. This barometric relationship is foundational in calculating how pressure affects other factors like air density.
Vertical Temperature Profile
Temperature doesn鈥檛 always behave intuitively with altitude. Generally, as you go higher, it gets colder. This change can be graphed with what's known as a Vertical Temperature Profile.

Think of it like a ladder where:
  • \( T(h) = T_0 - Lh \)
  • Here, \(T(h)\) is the temperature at altitude \(h\), \(T_0\) is the starting temperature at sea level, and \(L\) is the temperature lapse rate.
This profile provides a straightforward approximation because the rate \(L\) describes how quickly the temperature drops as you ascend. Usually, this rate is about \(6.5 \text{掳C/km}\). But be aware, actual temperature change can differ due to varying environmental conditions.
Ideal Gas Law
One of the key principles in physics is the Ideal Gas Law, which links pressure, volume, and temperature to gas density. When it comes to air density, this equation plays a huge role in understanding our atmosphere.

The main equation is:
  • \( PV = nRT \)
  • Here, \(P\) represents pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the universal gas constant, and \(T\) is the temperature.
If you rearrange it to solve for density \(\rho\), you get:
  • \( \rho = \frac{P}{RT} \)
When considering altitude, by substituting in expressions from the Barometric Formula and Vertical Temperature Profile, you can illustrate how air density changes. Hence, air density decreases as both the pressure and temperature fall with increased altitude.

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Most popular questions from this chapter

Water emerges straight down from a faucet with a \(1.80-\mathrm{cm}\) diameter at a speed of \(0.500 \mathrm{m} / \mathrm{s}\). (Because of the construction of the faucet, there is no variation in speed across the stream.) (a) What is the flow rate in \(\mathrm{cm}^{3} / \mathrm{s}\) ? (b) What is the diameter of the stream \(0.200 \mathrm{m}\) below the faucet? Neglect any effects due to surface tension.

Verify that the flow of oil is laminar for an oil gusher that shoots crude oil \(25.0 \mathrm{m}\) into the air through a pipe with a 0.100-m diameter. The vertical pipe is 50 m long. Take the density of the oil to be \(900 \mathrm{kg} / \mathrm{m}^{3}\) and its viscosity to $$\text { be } 1.00\left(\mathrm{N} / \mathrm{m}^{2}\right) \cdot \mathrm{s}(\text { or } 1.00 \mathrm{Pa} \cdot \mathrm{s})$$

The arterioles (small arteries) leading to an organ constrict in order to decrease flow to the organ. To shut down an organ, blood flow is reduced naturally to \(1.00 \%\) of its original value. By what factor do the radii of the arterioles constrict?

Some fish have a density slightly less than that of water and must exert a force (swim) to stay submerged. What force must an 85.0 -kg grouper exert to stay submerged in salt water if its body density is \(1015 \mathrm{kg} / \mathrm{m}^{3} ?\)

The left ventricle of a resting adult's heart pumps blood at a flow rate of \(83.0 \mathrm{cm}^{3} / \mathrm{s}\), increasing its pressure by \(110 \mathrm{mm} \mathrm{Hg},\) its speed from zero to \(30.0 \mathrm{cm} / \mathrm{s},\) and its height by \(5.00 \mathrm{cm} .\) (All numbers are averaged over the entire heartbeat.) Calculate the total power output of the left ventricle. Note that most of the power is used to increase blood pressure.
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