The number density of molecules of a gas depends on their distance $r$ from the origin as,$n\left(r\right)=n_0{e}^{-\alpha r^4}$. Then the total number of molecules is proportional to

Given, number density, $n\left(r\right) =n_0{e}^{-\alpha r^4}$

Total number of molecules,

$N={\int }_0^{\infty }n\left(r\right)dV={\int }_0^{\infty }{n}_0{e}^{-\alpha r^4}dV={\int }_0^{\infty }{n}_0{e}^{-\alpha r^4}4{\mathrm{\pi r}}^{2}dr$

Take $\alpha r^4=t\Rightarrow 4\alpha r^{3}dr=dt$  or $dr=\frac{dt}{4\alpha }{\left(\frac{\alpha }{t}\right)}^{3/4}$

So, $\mathrm{N}={\mathrm{n}}_{0}4{\mathrm{\pi \alpha }}^{-3/4}{\int }_0^{\infty }{\mathrm{t}}^{-1/4}{\mathrm{e}}^{-\mathrm{t}}\mathrm{dt}$  

i.e., $N\propto n_0 {\alpha }^{-3/4}$