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 $2k^2$.

Answer 1

Answer: 4

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