Physics - Heat & Thermo Question with Solution | TestHub

Density of molecules of gas as a of r is,$\mathrm{n}\left(\mathrm{r}\right)={\mathrm{n}}_0{\mathrm{e}}^{-\mathrm{\alpha }{\mathrm{r}}^4}$
Total no. of molecules$=\underset{0}{\overset{\mathrm{\infty }}{\int }}\mathrm{n}\left(\mathrm{r}\right)\mathrm{d}\mathrm{v}$
$\underset{0}{\overset{\mathrm{\infty }}{\int }}{\mathrm{n}}_0{\mathrm{e}}^{-\mathrm{\alpha }{\mathrm{r}}^4}4\mathrm{\pi }{\mathrm{r}}^2\mathrm{d}\mathrm{r}$
To integrate this, substitute$u={\alpha }^{3/4}{r}^3$this will give us$dr=\frac{1}{3{\alpha }^{3/4}{r}^2}du$
Now, the integration will become,$\frac{4\mathrm{\pi }{\mathrm{n}}_0}{3{\mathrm{\alpha }}^{3/4}}\mathrm{ }\underset{0}{\overset{\mathrm{\infty }}{\int }}{\mathrm{e}}^{-{\mathrm{u}}^{4/3}}\mathrm{d}\mathrm{u}$
This shows that the numer of molecules is proportional to${\mathrm{n}}_0{\mathrm{\alpha }}^{-3/4}$