$\gamma=\frac{C_{p}}{C_{V}}$, where $p=$ pressure

and $\rho=$ density.

As, $\quad p=\frac{n R T}{V}=\frac{n R T}{\frac{4}{3} \pi r^{3}}$

and $\quad v=\frac{d r}{d t}$

So, by Eq. $(i)$, we have

$\frac{d r}{d t}=\sqrt{\frac{\gamma n R T}{\frac{4}{3} \pi r^{3} \cdot \rho}}$

$\Rightarrow \quad \frac{d r}{d t} =k \cdot r^{-3 / 2}$

$\Rightarrow \quad k d t=r^{3 / 2} d r$

Integrating above equation, we get

$\Rightarrow k \int d t =\int r^{3 / 2} d r$

$\Rightarrow k t =\frac{2}{5} r^{5 / 2}$

So, $\quad r \propto t^{\frac{2}{5}}$