The r.m.s. velocity of CO_2 at a temperature | vedclass

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Question

(A) The formula for r.m.s. velocity is $v_{rms} = \sqrt{\frac{3RT}{M}}$.For $CO_2$: $v_{CO_2} = x = \sqrt{\frac{3RT}{M_{CO_2}}}$.For $N_2O$: $v_{N_2O}

Vedclass · Accepted Answer

$16$

Vedclass · Answer

(A) The formula for r.m.s. velocity is $v_{rms} = \sqrt{\frac{3RT}{M}}$.For $CO_2$: $v_{CO_2} = x = \sqrt{\frac{3RT}{M_{CO_2}}}$.For $N_2O$: $v_{N_2O} = 4x = \sqrt{\frac{3RT_{N_2O}}{M_{N_2O}}}$.Atomic weights are $C=12, N=14, O=16$. Thus,$M_{CO_2} = 12 + 2(16) = 44 \ g \ mol^{-1}$ and $M_{N_2O} = 2(14) + 16 = 44 \ g \ mol^{-1}$.Since $M_{CO_2} = M_{N_2O}$,we have $\frac{v_{N_2O}}{v_{CO_2}} = \frac{4x}{x} = \sqrt{\frac{T_{N_2O}}{T}}$.$4 = \sqrt{\frac{T_{N_2O}}{T}}$.Squaring both sides,$16 = \frac{T_{N_2O}}{T}$,so $T_{N_2O} = 16T$.

The r.m.s. velocity of $CO_2$ at a temperature $T$ (in kelvin) is $x \ cm \ s^{-1}$. At what temperature (in kelvin),the r.m.s. velocity of nitrous oxide would be $4 x \ cm \ s^{-1}$ (in $T$)? (Atomic weights of $C, N$ and $O$ are respectively $12, 14$ and $16$)

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