States of Matter 1 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) $T\left(\mathrm{H}{2}\right)=T\left(\mathrm{~N}{2}\right)$
(b) $T\left(\mathrm{H}{2}\right)>T\left(\mathrm{~N}{2}\right)$
(c) $T\left(\mathrm{H}{2}\right)<T\left(\mathrm{~N}{2}\right)$
(d) $T\left(\mathrm{H}{2}\right)=\sqrt{7} T\left(\mathrm{~N}{2}\right)$
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Solution:
- Root mean square speed $u_{\mathrm{rms}}=\sqrt{\frac{3 R T}{M}}$
$$ \Rightarrow \quad \frac{u_{\mathrm{rms}}\left(\mathrm{H}{2}\right)}{u{\mathrm{rms}}\left(\mathrm{N}{2}\right)}=\sqrt{7}=\sqrt{\frac{T\left(\mathrm{H}{2}\right)}{2} \times \frac{28}{T\left(\mathrm{~N}_{2}\right)}} $$
$$ \begin{array}{ll} \Rightarrow & 7=\frac{14 T\left(\mathrm{H}{2}\right)}{T\left(\mathrm{~N}{2}\right)} \ \Rightarrow & T\left(\mathrm{~N}{2}\right)=2 T\left(\mathrm{H}{2}\right) \text { i.e. } T\left(\mathrm{H}{2}\right)<T\left(\mathrm{~N}{2}\right) \end{array} $$
