(Solved): The distribution of molecules effusing through a small hole is proportional to \[ v^{3} e^{-\frac{m ...

The distribution of molecules effusing through a small hole is proportional to \[ v^{3} e^{-\frac{m v^{2}}{2 k n^{T}}} \] For molecules of mass \( m \) that effuse out of a container at temperature \( T \), find the ratio of the mean speed and most probable speed, i.e., \( \frac{\langle v\rangle}{v_{\max }} \) Round the answer to three decimal places. Hints: (i) Refer to Appendix C 2 for Gaussian integral identities. (ii) The most probable speed is the maximum of the distribution curve where the first derivative of the distribution function is zero.