The law of atmospheres states that the number density of...

The law of atmospheres states that the number density of molecules in the atmosphere depends on height $y$ above sea level according to $$n_{V}(y)=n_{0} e^{-m_{0} g_{0} / k_{B} T}$$ where $n_{0}$ is the number density at sea level (where $y=0 ) .$ The average height of a molecule in the Earth's atmosphere is given by $y_{\mathrm{avg}}=\frac{\int_{0}^{\infty} y n_{V}(y) d y}{\int_{0}^{\infty} n_{V}(y) d y}=\frac{\int_{0}^{\infty} y e^{-m_{0} g / k_{B} T} d y}{\int_{0}^{\infty} e^{-m_{0} g / k_{B} T} d y}$ (a) Prove that this average height is equal to $k_{\mathrm{B}} T / m_{0} g$ . (b) Evaluate the average height, assuming the temperature is $10.0^{\circ} \mathrm{C}$ and the molecular mass is $28.9 \mathrm{u},$ both uniform throughout the atmosphere.

The law of atmospheres states that the number density of molecules in the atmosphere depends on height $y$ above sea level according to $$n_{V}(y)=n_{0} e^{-m_{0} g_{0} / k_{B} T}$$
where $n_{0}$ is the number density at sea level (where $y=0 ) .$ The average height of a molecule in the Earth's atmosphere is given by
$y_{\mathrm{avg}}=\frac{\int_{0}^{\infty} y n_{V}(y) d y}{\int_{0}^{\infty} n_{V}(y) d y}=\frac{\int_{0}^{\infty} y e^{-m_{0} g / k_{B} T} d y}{\int_{0}^{\infty} e^{-m_{0} g / k_{B} T} d y}$
(a) Prove that this average height is equal to $k_{\mathrm{B}} T / m_{0} g$ .
(b) Evaluate the average height, assuming the temperature is $10.0^{\circ} \mathrm{C}$ and the molecular mass is $28.9 \mathrm{u},$ both uniform throughout the atmosphere.

Key Concepts

Exponential Distribution and Its Moments

In this context, the number density of molecules follows an exponential function of height, which is a common function used to model decay processes. The key mathematical property involves integrating exponential functions, particularly when calculating moments (like the average height) where integrals such as ??? y e^(–ay) dy and ??? e^(–ay) dy are evaluated. These standard integrals yield moment values that are inversely related to the decay parameter, linking directly to the average value of height in the distribution.

Calculation of Average Value

Determining the average height of the molecules involves computing the expected value of height using a weighted average approach. This requires setting up an integral where each height is weighted by its corresponding number density (given by the exponential distribution), and then normalizing by the total number density. This process is a fundamental method in statistical mechanics and probability theory for finding the mean of any continuous distribution.

Barometric Formula

The barometric formula describes how the atmospheric pressure or density decreases with increasing altitude under the influence of gravity. It is derived from the balance of gravitational forces and thermal energy, incorporating parameters such as the molecular mass, gravitational acceleration, temperature, and Boltzmann’s constant. This formula is key for understanding atmospheric structure and is closely related to the exponential form of the number density distribution.

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