The average speed, v̅, of the molecules of an ideal gas is given by v̅=(4)/(√(π))((M)/(2 R T))^3

step 1 In the problem, we have two parts. So here in part A we have been given to find the integral tha

The average speed, v̅, of the molecules of an ideal gas is...

The average speed, $\bar{v},$ of the molecules of an ideal gas is given by $$\bar{v}=\frac{4}{\sqrt{\pi}}\left(\frac{M}{2 R T}\right)^{3 / 2} \int_{0}^{+\infty} v^{3} e^{-M v^{2} /(2 R T)} d v$$ and the root-mean-square speed, $v_{\mathrm{rms}},$ by $$v_{\mathrm{rms}}^{2}=\frac{4}{\sqrt{\pi}}\left(\frac{M}{2 R T}\right)^{3 / 2} \int_{0}^{+\infty} v^{4} e^{-M \tau^{2} /(2 R T)} d v$$ where $v$ is the molecular speed, $T$ is the gas temperature, $M$ is the molecular weight of the gas, and $R$ is the gas constant. (a) Use a CAS to show that $$\int_{0}^{+\infty} x^{3} e^{-a^{2} x^{2}} d x=\frac{1}{2 a^{4}}, \quad a>0$$ and use this result to show that $\bar{v}=\sqrt{8 R T / \pi M}$. (b) Use a CAS to show that $$\int_{0}^{+\infty} x^{4} e^{-a^{2} x^{2}} d x=\frac{3 \sqrt{\pi}}{8 a^{5}}, \quad a>0$$ and use this result to show that $v_{\mathrm{mas}}=\sqrt{3 R T / M}$.

The average speed, $\bar{v},$ of the molecules of an ideal gas is given by
$$\bar{v}=\frac{4}{\sqrt{\pi}}\left(\frac{M}{2 R T}\right)^{3 / 2} \int_{0}^{+\infty} v^{3} e^{-M v^{2} /(2 R T)} d v$$
and the root-mean-square speed, $v_{\mathrm{rms}},$ by
$$v_{\mathrm{rms}}^{2}=\frac{4}{\sqrt{\pi}}\left(\frac{M}{2 R T}\right)^{3 / 2} \int_{0}^{+\infty} v^{4} e^{-M \tau^{2} /(2 R T)} d v$$
where $v$ is the molecular speed, $T$ is the gas temperature, $M$ is the molecular weight of the gas, and $R$ is the gas constant.
(a) Use a CAS to show that
$$\int_{0}^{+\infty} x^{3} e^{-a^{2} x^{2}} d x=\frac{1}{2 a^{4}}, \quad a>0$$
and use this result to show that $\bar{v}=\sqrt{8 R T / \pi M}$.
(b) Use a CAS to show that
$$\int_{0}^{+\infty} x^{4} e^{-a^{2} x^{2}} d x=\frac{3 \sqrt{\pi}}{8 a^{5}}, \quad a>0$$
and use this result to show that $v_{\mathrm{mas}}=\sqrt{3 R T / M}$.

Key Concepts

Average Speed and Root-Mean-Square Speed

These are statistical measures used in kinetic theory to describe the motion of gas molecules. The average speed is the mean value of the speeds of all molecules, while the root-mean-square speed is a measure of the speed that accounts for the energy of the molecules. Their derivation using integrals over the Maxwell-Boltzmann distribution links the microscopic molecular dynamics to macroscopic thermodynamic quantities such as temperature.

Gamma Function

The Gamma function generalizes the factorial function to non-integer values and is key to evaluating integrals that involve exponentials and power law functions. In the evaluation of Gaussian integrals, the Gamma function provides a systematic way to express the result, which is crucial for deriving the exact forms of the average and root-mean-square speeds in an ideal gas.

Ideal Gas Kinetic Theory

This concept refers to the microscopic description of the behavior of gases, where the gas is modeled as a collection of molecules in random motion. The macroscopic properties such as pressure and temperature are understood in terms of these molecular motions. In the context of the problem, the speeds of the molecules are related to temperature and molecular mass, facilitating calculations of average and root?mean?square speeds.

Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann distribution describes the statistical spread of molecular speeds in an ideal gas. It characterizes how likely it is to find molecules at various speeds, allowing for the derivation of quantities such as average speed and root-mean-square speed. The integrals in the problem are directly related to the moments of this distribution, which encapsulate important statistical properties of molecular motion.

Gaussian Integrals

Gaussian integrals involve functions of the form exp(?ax²) and are ubiquitous in physics and mathematics. Evaluating integrals of the type ??? x? e^(?a²x²) dx is essential for computing the moments of distributions like the Maxwell-Boltzmann distribution. In the problem, these integrals lead to expressions that simplify the calculation of molecular speeds.

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