Q 11-12-151JEE MainJEE Main 2019 (12 Apr, Shift 2)Medium
The number density of molecules of a gas depends on their distance $r$ from the origin as, $n(r) = n_0e^{-\alpha r^4}$. Then the number of molecules is proportional to:
Answer: (C) $n_0\alpha^{-3/4}$
$$N = \int_0^\infty n_0e^{-\alpha r^4}4\pi r^2dr$$
Substituting $u = \alpha^{1/4}r$, $r^2dr = \alpha^{-3/4}u^2du$:
$$N = 4\pi n_0\alpha^{-3/4}\int_0^\infty e^{-u^4}u^2du \propto n_0\alpha^{-3/4}$$
Solution by Sreeraj P, M.Sc Physics