$V=\sqrt{\frac{\gamma R T}{M}}$
Volume before expansion $=V_{b}$
Volume after expansion $=V_{a}$
Temp before expansion $=T_{b}$
Temp after expansion $=T_{a}$.
As, Volume os Temperature,
$\frac{V_{b}}{V_{a}}=\frac{T_{b}}{T_{a}}=\frac{V_{b}}{2 V_{b}}=\frac{1}{2}$
$\therefore \frac{T_{b}}{T_{a}}=\frac{1}{2}$
Whow, Before expansion,
$C_{b}=\sqrt{\frac{\gamma R T_{b}}{I M}}$ $\dots \; (1)$
After exparsion,
$C_{a}=\sqrt{\frac{\gamma R T_{a}}{M}}$ $\dots \; (2)$
eq. $1/2$
$\frac{c_{b}}{c_{a}}=\sqrt{1 / 2} \quad \therefore \quad c_{a}=\sqrt{2} c_{b}$.
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