Question

The number of molecules per unit volume of a gas depends on their distance r from the origin as, $n\left(\right.r\left.\right)=n_{0}e^{- \alpha r^{4}}$ , where $n_{0}$ and $\alpha $ are constants.The total number of molecules of the gas is proportional to:

• A. $$ $n_{0}\alpha ^{1 / 4}$
• B. $n_{0}\alpha ^{- 3 / 4}$
• C. $n_{0}\alpha ^{- 3}$
• D. $$ $\sqrt{n_{0}}\alpha ^{1 / 2}$

Answer 1

Answer: (B)

The number density or number of molecules per unit volume is given by equation $n\left(r\right)=n_{0}e^{- \alpha r^{4}}$ . Then, the total number of the molecule will be given by,
$\text{ Total molecule }N=\displaystyle \int _{0}^{\infty }n\left(\right.r\left.\right)\left(4 \pi r^{2} d r\right)$
$N=\displaystyle \int _{0}^{\infty }4\pi n_{0}e^{- \alpha r^{4}}r^{2}dr$
After integration,
$N \propto n_{0}\alpha ^{- 3 / 4}$ .