Question

The given figure shows a parabolic graph between $T$ and $\frac{1}{V}$ for a mixture of a gas undergoing an adiabatic process. The ratio of $v_{ \text{rms }}$ to the speed of sound in the mixture is $k$. Find $2 k^{2}$.

Answer 1

$v_{ \text{rms }}=\sqrt{\frac{3 R T}{M}}$
$v_{\text {sound }}=\sqrt{\frac{\gamma R T}{M}}$
$\therefore \frac{v_{\text {rms }}}{v_{\text {sound }}}=\sqrt{\frac{3}{\gamma}}$
Equation of the graph: $\left(\frac{1}{V}\right)=T^{2}$
$\Rightarrow T^{2} V=$ Constant (process equation)
$\Rightarrow P V^{3 / 2}=$ Constant
$\Rightarrow \gamma=\frac{3}{2}$