The number density of molecules of a gas depends on their distance r from the origin as, n(r) = n _ 0 e ^ - α r ^ 4 . Then the numer of molecules is proportional to:

Question

Quantrex Academy · Accepted Answer

Correct answer: C. n _ 0 α ^ (- 3)/(4)

Density of molecules of gas as a of r is,

n ( r ) = {n}_{0}{e}^{- \alpha{r}^{4}}

Total no. of molecules

= \mathop{∫}_{0}^{\infty}n ( r ) dv

\mathop{∫}_{0}^{\infty}{n}_{0}{e}^{- \alpha{r}^{4}}4\pi{r}^{2}dr

To integrate this, substitute

u = {\alpha}^{3 / 4}{r}^{3}

this will give us

dr = \dfrac{1}{3{\alpha}^{3 / 4}{r}^{2}}du

Now, the integration will become,

\dfrac{4\pi{n}_{0}}{3{\alpha}^{3 / 4}}\mathop{∫}_{0}^{\infty}{e}^{- {u}^{4 / 3}}du

This shows that the numer of molecules is proportional to

{n}_{0}{\alpha}^{- 3 / 4}

The number density of molecules of a gas depends on their distance...

Options

Correct answer

Step-by-step solution

More from Kinetic Theory of Gases