Question

The velocity of sound in an ideal gas at temperature $T_1$ and $T_{2}K$ are found to be $V_1$ and $V_2$, respectively. If the root mean square velocity of the same gas at the same temperature are found to be $v_1$ and $v_2$, respectively then

• A. $v_2=v_1\frac{V_2}{V_1}$
• B. $v_2=v_1\frac{V_1}{V_2}$
• C. $v_2=v_1\frac{\sqrt{V_2}}{V_1}$
• D. $v_2=v_1\frac{\sqrt{V_1}}{V_2}$

Answer 1

The correct option is A $v_2=v_1\frac{V_2}{V_1}$

Root mean square velocity of ideal gas molecules at temperature $T$ is given by,

$v_{rms}=\sqrt{\frac{3RT}{M}}$, Here, $R$ is universal gas constant$,M$ is the molecular mass.

And velocity of sound in any gas at temperature is given as,

$V_s=\sqrt{\frac{\gamma RT}{M}}$. Here $\gamma =\frac{C_P}{C_V}$.

Since,

$v_{rms}\propto \sqrt{T}$ and $V_s\propto \sqrt{T}$, therefore,

Option 'A' is correct.