Question

For a gas sample with $N_{0}$ number of molecules, the function $N\left(V\right)$ is given by, $N\left(V\right)=\frac{d N}{d V}=\left(\frac{3 N_{0}}{V_{0}^{3}}\right)V^{2}$ 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)

$N\left(V\right)=\frac{d N}{d V}=\left(\frac{3 N_{0}}{V_{0}^{3}}\right)V^{2}$
$0 < V < V_{0}$
$N\left(V\right)=0V>V_{0}$
$v_{rms}=\left[\frac{\int V^{2} d N}{\int d N}\right]^{1 / 2}$
$v_{rms}=\left[\frac{1}{N_{0}} \int \limits_{0}^{V_{0}} N \left(V\right) V^{2} d V\right]^{1 / 2}\left[d N = N_{0}\right]$
$v_{rms}=\left[\frac{1}{N_{0}} \int\limits _{0}^{V_{0}} \left(\frac{3 N_{0}}{V_{0}^{3}} V^{2}\right) V^{2} d V\right]^{1 / 2}$
$v_{rms}=\sqrt{\frac{3}{V_{0}^{3}} \int\limits _{0}^{V_{0}} \left(V^{2}\right) V^{2} d V}$
$v_{rms}=\sqrt{\frac{3}{V_{0}^{3}} \left[\frac{V^{5}}{5}\right]_{0}^{V_{0}}}$
$v_{rms}=\sqrt{\frac{3}{V_{0}^{3}} \left(\frac{\left(V_{0}\right)^{5}}{5}\right)}$
$v_{rms}=\sqrt{\frac{3}{5}}V_{0}$ .
Hence, the root mean squared velocity of the gas sample is $\sqrt{\frac{3}{5}}V_{0}$ .