Question

For a gas sample with $N_0$ number of molecules, the function $N\left(V\right)$ is given by, $N\left(V\right)=\frac{dN}{dV}=\left(\frac{3N_0}{V_0^3}\right)V^2$ for $0<V<V_0$ and $N\left(V\right)=0$ for $V>V_0$ . Where $dN$ is a number of molecules in speed range $V$ to $V+dV$ . The rms speed of the molecules is

• A. $\sqrt{\frac{3}{5}}{V}_0$
• B. $\sqrt{\frac{2}{5}}{V}_0$
• C. $\sqrt{2}{V}_0$
• D. $\sqrt{3}{V}_0$

Answer 1

Answer: (A)

$N\left(V\right)=\frac{dN}{dV}=\left(\frac{3N_0}{V_0^3}\right)V^2$
$0<V<V_0$
$N\left(V\right)=0V>V_0$
$v_{rms}={\left[\frac{\int V^{2}dN}{\int dN}\right]}^{1/2}$
$v_{rms}={\left[\frac{1}{N_0}\underset{0}{\overset{V_0}{\int }}N\left(V\right)V^{2}dV\right]}^{1/2}\left[dN=N_0\right]$
$v_{rms}={\left[\frac{1}{N_0}\underset{0}{\overset{V_0}{\int }}\left(\frac{3N_0}{V_0^3}{V}^2\right)V^{2}dV\right]}^{1/2}$
$v_{rms}=\sqrt{\frac{3}{V_0^3}\underset{0}{\overset{V_0}{\int }}\left(V^2\right)V^{2}dV}$
$v_{rms}=\sqrt{\frac{3}{V_0^3}{\left[\frac{V^5}{5}\right]}_0^{V_0}}$
$v_{rms}=\sqrt{\frac{3}{V_0^3}\left(\frac{{\left(V_0\right)}^5}{5}\right)}$
$v_{rms}=\sqrt{\frac{3}{5}}{V}_0$ .
Hence, the root mean squared velocity of the gas sample is $\sqrt{\frac{3}{5}}{V}_0$ .