Question

The root mean square velocity of one mole of a monatomic gas having molar mass M is $u_{rms}$. The relation between the average kinetic energy $E$ per mole of the gas and $u_{rms}$ is.

• A. $u_{rms}={\left(\frac{2E}{m}\right)}^{\frac{1}{2}}$
• B. $u_{rms}={\left(\frac{3E}{2m}\right)}^{\frac{1}{2}}$
• C. $u_{rms}={\left(\frac{E}{2m}\right)}^{\frac{1}{2}}$
• D. $u_{rms}={\left(\frac{E}{3m}\right)}^{\frac{1}{2}}$

Answer 1

The correct option is A $u_{rms}={\left(\frac{2E}{m}\right)}^{\frac{1}{2}}$

Kinetic energy $\left(E\right)=\frac{3}{2}kT$.......(i)
RMS speed $\left(u_{rms}\right)=\sqrt{\frac{3kT}{M}}$.......(ii)
Multiply and divide 2 in equation (ii)
RMS speed $\left(u_{rms}\right)=\sqrt{\frac{3kT\times 2}{M\times 2}}....\left(iii\right)$
Substituting equation (ii) in (iii) we get,
$\Rightarrow u_{rms}=\sqrt{\frac{2E}{M}}$