Figure shows a parabolic graph between $T$ and $\frac{1}{V}$ ( $T =$ temperature, $V =$ volume) for a mixture of gases undergoing an adiabatic process. The ratio of rms velocity of molecules and speed of sound in the mixture of gases at the same temperature is given by $\sqrt{n}$ . Find $n$ .

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Solution

Let the relation between $T$ and $\frac{1}{V}$ be
$T = K {(\frac{1}{V})}^{n}$
$\Rightarrow 2 T_{0} = K {(\frac{4}{V_{0}})}^{n}$ and $T_{0} = K {(\frac{1}{V_{0}})}^{n}$
$\Rightarrow 2 = 4^{n} \Rightarrow n = \frac{1}{2}$
Therefore $T = K {(\frac{1}{V})}^{\frac{1}{2}}$ or $T V^{1 / 2} = constant$
But for adiabatic process, $T V^{γ - 1} = constant \Rightarrow γ - 1 = \frac{1}{2}$ or $γ = \frac{3}{2}$
$\frac{V_{m i x}}{V_{s o u n d}} = \sqrt{\frac{3}{γ}} = \sqrt{\frac{3}{3 / 2}} = \sqrt{2}$

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