States of Matter - Result Question 22
####22. The rms velocity of hydrogen is $\sqrt{7}$ times the rms velocity of nitrogen. If $T$ is the temperature of the gas
(2000, 1M)
(a) $T\left(H _2\right)=T\left(N _2\right)$
(b) $T\left(H _2\right)>T\left(N _2\right)$
(c) $T\left(H _2\right)<T\left(N _2\right)$
(d) $T\left(H _2\right)=\sqrt{7} T\left(N _2\right)$
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Solution:
- Root mean square speed $u _{rms}=\sqrt{\frac{3 R T}{M}}$
$$ \Rightarrow \quad \frac{u _{rms}\left(H _2\right)}{u _{rms}\left(N _2\right)}=\sqrt{7}=\sqrt{\frac{T\left(H _2\right)}{2} \times \frac{28}{T\left(N _2\right)}} $$
$$ \begin{array}{ll} \Rightarrow & 7=\frac{14 T\left(H _2\right)}{T\left(N _2\right)} \ \Rightarrow & T\left(N _2\right)=2 T\left(H _2\right) \text { i.e. } T\left(H _2\right)<T\left(N _2\right) \end{array} $$
