Kinetic Theory

JEE Main level, single correct, medium

29 of 168

Question 29 of 168

At which temperature the r.m.s. velocity of a hydrogen molecule equal to that of an oxygen molecule at 47∘C47^{\circ}C ?

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Answer: (D) 20K20K

The r.m.s velocity of the molecule of a gas of molecular weight MM at Kelvin temperature TTis given by, Vrms=3RTM{V}_{rms}=\sqrt{\dfrac{3RT}{M}}.
Let MO{M}_{O} and MH{M}_{H} are molecular weight of oxygen and hydrogen and TO{T}_{O} and TH{T}_{H} the corresponding Kelvin temperature at which Vr.m.s{V}_{r.m.s} is same for both,
Vr.m.s=3RTOMO=3RTHMH{V}_{r.m.s}=\sqrt{\dfrac{3R{T}_{O}}{{M}_{O}}}=\sqrt{\dfrac{3R{T}_{H}}{{M}_{H}}}. Hence TOMO=THMH\dfrac{{T}_{O}}{{M}_{O}}=\dfrac{{T}_{H}}{{M}_{H}}
TO=273∘C+47∘C=320K{T}_{O}=273^{\circ}C+47^{\circ}C=320K,
MO=32{M}_{O}=32,
MH=2⇒TH=232×320=20K{M}_{H}=2 \Rightarrow {T}_{H}=\dfrac{2}{32}\times 320=20K

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