Question

A container is filled with a sample of gas having n molecules with speed $\alpha ,2\alpha ,3\alpha ,......n\alpha$. The ratio of average speed to root mean square speed is :

• A. $\sqrt{\frac{3(n+1)}{2(2n+1)}}$
• B. $\sqrt{\frac{(n+1)}{2(2n+1)}}$
• C. $\sqrt{\frac{5(2n+2)}{7(3n+1)}}$
• D. $\sqrt{\frac{3(n+2)}{5(3n+2)}}$

Answer 1

The correct option is A $\sqrt{\frac{3(n+1)}{2(2n+1)}}$

Average speed is given as,

$V_{avg}=a+2a+3a+...../n$

$V_{avg}=a(1+2+\mathrm{3....})/n$

$V_{avg}=\frac{a}{n}\cdot \frac{n(n+1)}{2}$

$V_{avg}=\frac{a(n+1)}{2}$

Root mean square speed is given as,

$V_{rms}=\sqrt{\frac{a^2+4a^2+9a^2+.....}{n}}$

$V_{rms}=a\sqrt{\frac{(n+1)(2n+1)}{6}}$

$\frac{V_{avg}}{V_{rms}}=\frac{\frac{a(n+1)}{2}}{a\cdot \sqrt{\frac{(n+1)(2n+1)}{6}}}$

$\frac{V_{avg}}{V_{rms}}=\sqrt{\frac{6(n+1{)}^2}{4(n+1)(2n+1)}}$

$\frac{V_{avg}}{V_{rms}}=\sqrt{\frac{3}{2}\frac{(n+1)}{(2n+1)}}$.