Question

Root mean square (rms) speed of $O_2$ is $500m/s$ at a constant temperature. Calculate the rms speed and the average kinetic energy of $H_2$ at the same temperature. (consider $R=8.33J{k}^{-1}mo{l}^{-1}$

• A. $500m/s$ and $4.0kJ/mol$
• B. $2000m/s$ and $4.0kJ/mol$
• C. $500m/s$ and $4.7kJ/mol$
• D. $2000m/s$ and $4.7kJ/mol$

Answer 1

Answer: (B)

Given, root mean square $(rms)$ speed of $O_2$ is $500m/s$ at a constant temperature.
Root mean square speed in given by the following expression
$u_{rms}=\sqrt{\frac{3RT}{Mw}}$
For $H_2$ gas, $u_{rms}\left(H_2\right)=\sqrt{\frac{3RT}{M{w}_{H_2}}}$
( $Mw$ of $H_2=2g/mol$ )
For $O_2$ gas, $u_{rms}\left(O_2\right)=\sqrt{\frac{3RT}{M{w}_{O_2}}}$
$(Mw$ of $O_2=32g/mol)$
$\frac{u_{rms\left(H_2\right)}}{u_{rms}\left(O_2\right)}=\sqrt{\frac{3RT}{M{w}_{H_2}}}\times \sqrt{\frac{M{w}_{O_2}}{3RT}}$
(Temperature constant, $R=8.33J{K}^{-1}mo{l}^{-1}$ )
$u_{rms\left(H_2\right)}=\sqrt{\frac{M{w}_{o_2}}{M{w}_{H_2}}}\times u_{rms\left(O_2\right)}=\sqrt{\frac{32}{2}}\times 500$
$=2000m/s$
Average kinetic energy of $H_2$ can be calculate as
$=\frac{1}{2}m{u}^2$
$\left[u=2000m/s,m=H_2=2g/mol=0.002kg/mol\right]$
$=\frac{1}{2}\times 0.002\times (2000{)}^2$
$=\frac{1}{2}\times \frac{2}{1000}\times 2000\times 2000$
$=4000Jmo{l}^{-1}$ or $4kJmo{l}^{-1}$