Question
Average velocity of molecules of a gas in a container, moving only in one dimension, is(a) $\sqrt{\frac{8 R T}{\pi M}}$(b) $\frac{1}{3} \cdot \sqrt{\frac{8 R T}{\pi M}}$(c) Zero(d) Infinite
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Step 1: The average velocity of gas molecules in a container is calculated by taking into account the velocities of all the molecules and dividing by the total number of molecules. Show more…
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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 $$ \bar{v}=\sqrt{\frac{8 R T}{\pi M}} $$
Techniques of Integration
Improper Integrals
The average speed of molecules in an ideal gas is $$ \overline{v}=\frac{4}{\sqrt{\pi}}\left(\frac{M}{2 R T}\right)^{3 / 2} \int_{0}^{\infty} v^{3} e^{-M v^{2} / 2 R T )} d v $$ 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 $$ \overline{v}=\sqrt{\frac{8 R T}{\pi M}} $$
The average speed of molecules in an ideal gas is $$\overline{v}=\frac{4}{\sqrt{\pi}}\left(\frac{M}{2 R T}\right)^{3 / 2}\int_{0}^{\infty} v^{3} e^{-M s^{2} / 2 R T_{1}} d v$$ where $M$ is the molecular weight of the gas, $R$ is the gas con- stant, $T$ is the gas temperature, and $v$ is the molecular speed. Show that $$\overline{v}=\sqrt{\frac{8 R T}{\pi M}}$$
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