Question

For a gas sample with N₀ number of molecules, function N(V) is given by :

N(V) = $\frac{dN}{dV} = \left\lbrack \frac{3N_{0}}{V_{0}^{3}} \right\rbrack$V² for 0 £ V £ V₀ and N(V) = 0 for V > V₀ where dN is number of molecules in speed range V to V + dV. The rms speed of the molecules is –

• A. $\sqrt{\frac{2}{5}}$V₀
• B. $\sqrt{\frac{3}{5}}$V₀
• C. $\sqrt{2}V_{0}$
• D. $\sqrt{3}V_{0}$

Answer 1

Answer: (B)

$\sqrt{\frac{3}{5}}$V₀

Answer 2

$V_{rms}^{2}$= < V² > = $\frac{V_{1}^{2} + V_{2}^{2} + V_{3}^{2} + .......}{N}$

= $\frac{\int_{}^{}{V^{2}dN}}{\int_{}^{}{dN}}$ here $\frac{dN}{dV}$ = N(V)

$V_{rms}^{2}$ = $\frac{1}{N}\int_{0}^{\infty}{N(V)V^{2}dV}$

= $\frac{1}{N}\int_{0}^{V_{0}}\left\lbrack \frac{3N}{V_{0}^{3}}V^{2} \right\rbrack$ V² dV = $\frac{3}{5}V_{0}^{2}$̃ V_rms = $\sqrt{\frac{3}{5}}$ V₀