The average speed of molecules in an ideal gas is...

The average speed of molecules in an ideal gas is $$ \bar{v}=\frac{4}{\sqrt{\pi}}\left(\frac{M}{2 R T}\right)^{3 / 2} \int_{0}^{\infty} v^{3} e^{-A b^{2} /(2 R T)} d v $$ where $M$ is the molecular weight of the gas, $R$ is the gas constant, $T$ is the gas temperature, and $v$ is the molecular speed. Show that $$ \bar{v}=\sqrt{\frac{8 R T}{\pi M}} $$

The average speed of molecules in an ideal gas is
$$
\bar{v}=\frac{4}{\sqrt{\pi}}\left(\frac{M}{2 R T}\right)^{3 / 2} \int_{0}^{\infty} v^{3} e^{-A b^{2} /(2 R T)} d v
$$
where $M$ is the molecular weight of the gas, $R$ is the gas constant, $T$ is the gas temperature, and $v$ is the molecular speed. Show that
$$
\bar{v}=\sqrt{\frac{8 R T}{\pi M}}
$$

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Key Concepts

Integration Techniques for Gaussian Integrals

Integrals of the form ??? v? exp(-a v²) dv are common in kinetic theory. Evaluating these integrals typically involves variable substitutions and employing properties of Gaussian integrals, which are essential for extracting mean values and moments from the distribution.

Gamma Function

The Gamma function generalizes the factorial function to non-integer values and is particularly useful in evaluating integrals with exponential and power terms. In the context of the ideal gas speed distribution, it simplifies the evaluation of integrals that arise when computing average speeds, connecting directly with the moments of the distribution.

Maxwell-Boltzmann Distribution

This statistical distribution describes the speeds of molecules in an ideal gas, derived from the principles of kinetic theory. It explains the probability of finding molecules with particular speeds at a given temperature, serving as the foundation for calculating macroscopic properties like average speed.

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