Question

If at same pressure, the densities of two gases are ${\rho }_1$ and ${\rho }_2$ respectively,then the ratio of r.m.s. velocity of their molecules respectively are

• A. ${\rho }_1:{\rho }_2$
• B. $\sqrt{{\rho }_2}:\sqrt{{\rho }_1}$
• C. ${\rho }_2:{\rho }_1$
• D. $\sqrt{{\rho }_1}:\sqrt{{\rho }_2}$

Answer 1

The correct option is B $\sqrt{{\rho }_2}:\sqrt{{\rho }_1}$

We know that
Pressure $\left(P\right)=\frac{1}{3}\rho v_{\text{rms}}^2$
Where $\rho =$ density
$\therefore \rho =\frac{3P}{v_{\text{rms}}^2}$
$\Rightarrow \frac{{\rho }_1}{{\rho }_2}=\frac{\frac{3P_1}{v_{\text{rms(1)}}^2}}{\frac{3P_2}{v_{\text{rms(2)}}^2}}$
Given, pressure is same
$\therefore P_1=P_2$
$\Rightarrow \frac{{\rho }_1}{{\rho }_2}=\frac{v_{\text{rms(2)}}^2}{v_{\text{rms(1)}}^2}$
$\Rightarrow \frac{v_{\text{rms(2)}}}{v_{\text{rms(1)}}}=\sqrt{\frac{{\rho }_1}{{\rho }_2}}$
$\Rightarrow \frac{v_{\text{rms(1)}}}{v_{\text{rms(2)}}}=\frac{\sqrt{{\rho }_2}}{\sqrt{{\rho }_1}}$