Question

Figure shows a parabolic graph between T and I/V for a mixture of a gas undergiong an adiabatic process. What is the ratio of $V_{rms}$ of molecules and speed of sound in mixture:

• A. $\sqrt{3/2}$
• B. $\sqrt{2}$
• C. $\sqrt{2/3}$
• D. $\sqrt{3}$

Answer 1

The correct option is B $\sqrt{2}$

According the graph, $T^2\propto \frac{1}{V}$ $⟹T{V}^{1/2}=constant$

Comparing with $T{V}^{\gamma -1}=constant$, we get $\frac{1}{2}=\gamma -1$

$⟹$ $\gamma =\frac{3}{2}$

Now $V_{rms}=\sqrt{\frac{3P}{\rho }}$ where all the symbols have their usual meaning

Also $V_{sound}=\sqrt{\frac{\gamma P}{\rho }}$

$⟹$ $\frac{V_{rms}}{V_{sound}}=\sqrt{\frac{3}{\gamma }}=\sqrt{\frac{3}{3/2}}=\sqrt{2}$