Question

The root mean square velocity of hydrogen is $\sqrt{5}$ times than that of nitrogen. If $T$ is the temperature of the gas, then:

• A. $T_{H_2}=T_{N_2}$
• B. $T_{H_2}>T_{N_2}$
• C. $T_{H_2}<T_{N_2}$
• D. $T_{H_2}=\sqrt{7}{T}_{N_2}$

Answer 1

The correct option is C $T_{H_2}<T_{N_2}$

$V_{rns}=\sqrt{\frac{3RT}{M}}=\frac{(V_{rns}{)}_{H_2}}{(V_{rns}{)}_{N_2}}=\sqrt{\frac{T_{H_2}}{M_{H_2}}\times \frac{M_{N_2}}{T_{N_2}}}(V_{rns}{)}_{H_2}=\sqrt{5}(V_{rns}{)}_{N_2}\therefore \frac{\left(V_{rns}\right)H_2}{\left(V_{rns}\right)H_2}\times \sqrt{5}\sqrt{\frac{T_{H_2}}{T_{N_2}}\times \frac{28}{2}}\Rightarrow \frac{\sqrt{5}}{1}=\sqrt{\frac{T_{H_2}}{T_{N_2}}\times 14}\Rightarrow 5=\frac{T_{H_2}}{T_{N_2}}\times 14\Rightarrow T_{N_2}\times 5=T_{H_2}\times 14\therefore T_{N_2}>T_{H_2}$