Assume the Earth's atmosphere has a uniform temperature of...

Assume the Earth's atmosphere has a uniform temperature of $20.0^{\circ} \mathrm{C}$ and uniform composition, with an effective molar mass of $28.9 \mathrm{~g} / \mathrm{mol} .$ (a) Show that the number density of molecules depends on height $y$ above sea level according to $$n_{V}(y)=n_{0} e^{-m_{0} \sigma / k_{\mathrm{s}} T}$$ where $n_{0}$ is the number density at sea level (where $y=0$ ). This result is called the law of atmospheres. (b) Commercial jetliners typically cruise at an altitude of $11.0 \mathrm{~km} .$ Find the ratio of the atmospheric density there to the density at sea level.

Assume the Earth's atmosphere has a uniform temperature of $20.0^{\circ} \mathrm{C}$ and uniform composition, with an effective molar mass of $28.9 \mathrm{~g} / \mathrm{mol} .$ (a) Show that the number density of molecules depends on height $y$ above sea level according to
$$n_{V}(y)=n_{0} e^{-m_{0} \sigma / k_{\mathrm{s}} T}$$
where $n_{0}$ is the number density at sea level (where $y=0$ ). This result is called the law of atmospheres. (b) Commercial jetliners typically cruise at an altitude of $11.0 \mathrm{~km} .$ Find the ratio of the atmospheric density there to the density at sea level.

Key Concepts

Hydrostatic Equilibrium

This principle states that at any given height within a fluid under the influence of gravity, the vertical pressure gradient is exactly balanced by the gravitational force acting on the fluid. In the context of planetary atmospheres, it leads to a differential equation that relates the change in pressure with altitude to the local density and the gravitational acceleration. This concept is foundational in deriving the vertical distribution of properties like pressure and density in the atmosphere.

Ideal Gas Law

The ideal gas law relates the pressure, volume, temperature, and number density (or mole density) of a gas through the equation P = n k_B T. When applied to an atmospheric context, it provides a way of connecting the macroscopic properties of the gas (such as pressure and temperature) with the microscopic property of number density. This law is essential for expressing density in terms of pressure and temperature, which is used in the derivation of the atmospheric density profile.

Barometric Formula

The barometric formula describes how the pressure and corresponding number density of an isothermal atmosphere decrease exponentially with altitude. It is derived by combining the principles of hydrostatic equilibrium and the ideal gas law. This formula, also known as the law of atmospheres, quantifies the vertical distribution of atmospheric molecules and is pivotal in understanding phenomena such as the decrease in weather, air travel conditions, and the behavior of gases in planetary atmospheres.

Scale Height

The scale height is a measure of the characteristic vertical distance over which the atmospheric pressure (or density) decreases by a factor of e. It is defined by the ratio of the thermal energy (k_B T) to the gravitational potential energy per molecule (m g). This concept not only simplifies the exponential form of the density profile but also provides insights into the effects of temperature and gravity on an atmosphere’s structure.

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