Find 𝑣𝑟𝑚𝑠 = √⟨𝑣^2⟩ for a 3D ideal gas. Find the most probable speed of a 3D ideal gas molecule. Hint: one can maximize or minimize a function by equating its derivative with respect to the variable of interest to zero and solving the equation for that variable.
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Step 1
Step 1: The root mean square speed (vrms) of a gas molecule is given by the equation: vrms = √⟨v^2⟩ where ⟨v^2⟩ is the average of the square of the speed of the gas molecules. Show more…
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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 $v_{\mathrm{mas}}=\sqrt{3 R T / M}$.
Principles of Integral Evaluation
Improper Integrals
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
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 $$ \overline{v}=\sqrt{\frac{8 R T}{\pi M}} $$
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