Question

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 total number of molecules is proportional to :

• A. $n_0 \alpha^{1/4}$
• B. $n_0 \alpha^{-3}$
• C. $n_0 \alpha^{-3/4}$
• D. $\sqrt{n_0} \alpha^{1/2}$

Answer 1

Answer: (C)

$n_0 \alpha^{-3/4}$

Answer 2

Given number density of molecules of gas as a function of $r$ is $\, \, \, \, n(r) = n_0 e^{- \alpha r^4}$ $\therefore \,$ Total number of molecule = $\int_{0}^{\infty} n(r) dV$ $\, \, \, \, \, \, \, \, \, \, = \int_0^{\infty} e^{-\alpha x^4 } \, 4 \pi r^2$dr $\therefore$ Number of molecules is proportional to $n_0 \alpha^{-3/4}$