Question
At what temperature is the $$r.m.s$$ velocity of a hydrogen molecule equal to that of an oxygen molecule at $${47^ \circ }C$$ ?
A.
$$80\,K$$
B.
$$-73\,K$$
C.
$$3\,K$$
D.
$$20\,K$$
Answer :
$$20\,K$$
Solution :
$$\eqalign{
& {v_{rms}} = \sqrt {\frac{{8RT}}{{\pi M}}} \cr
& {\text{For}}\,\,{v_{rms}}\,\,{\text{to}}\,{\text{be}}\,{\text{equal}}\,\,\frac{{{T_{{H_2}}}}}{{{M_{{H_2}}}}} = \frac{{{T_{{O_2}}}}}{{{M_{{O_2}}}}} \cr
& {\text{Here}}\,\,{M_{{H_2}}} = 2;\,\,{M_{{O_2}}} = 32; \cr
& {T_{{O_2}}} = 47 + 273 = 320\;K \cr
& \therefore \frac{{{T_{{H_2}}}}}{2} = \frac{{320}}{{32}} \Rightarrow {T_{{H_2}}} = 20\,K \cr} $$