(Solved): The velocity distribution, \( \mathrm{f}(\mathrm{v}) \), of molecules in a box is governed by the ...

The velocity distribution, \( \mathrm{f}(\mathrm{v}) \), of molecules in a box is governed by the following distribution: \( \mathrm{f}(\mathrm{v})=\frac{\mathrm{a}}{\mathrm{v}} e^{-\left(\frac{\ln (\mathrm{v})}{a}\right)^{2}} \) where \( \mathrm{v} \) is the velocity in \( \frac{\mathrm{m}}{\mathrm{s}} \) and \( \ln (\mathrm{v}) \) is the natural logarithm or logarithm to base e: \( \ln (\mathrm{v})=\log _{e} \mathrm{v}(\ln (2)=0.693 \) and \( \ln (\mathrm{e})=1) \) (a) Assuming that the velocity of the molecules can have any positive value, determine the value for \( a \) that normalizes the distribution function, \( f(\mathrm{v}) \). (b) Determine the mean or average velocity of molecules in the box. (c) Provide a plot of the normalized function, \( f(v) \).