The r.m.s. velocity of $\mathrm{CO}_2$ at a temperature $T$ (in kelvin) is $x \mathrm{~cm} \mathrm{~s}^{-1}$…

The r.m.s. velocity of $\mathrm{CO}_2$ at a temperature $T$ (in kelvin) is $x \mathrm{~cm} \mathrm{~s}^{-1}$. At what temperature (in kelvin), the r.m.s. velocity of nitrous oxide would be $4 x \mathrm{~cm} \mathrm{~s}^{-1}$ ? (Atomic weights of $\mathrm{C}, \mathrm{N}$ and $\mathrm{O}$ are respectively 12,14 and 16 )
  1. $16 T$
  2. $2 T$
  3. $4 T$
  4. $32 T$

Solution

$r \mathrm{CO}_2=\sqrt{\frac{3 R T_{\mathrm{CO}_2}}{M_{\mathrm{CO}_2}}}$ $\begin{aligned} & r \mathrm{~N}_2 \mathrm{O}= \sqrt{\frac{3 R T_{\mathrm{N}_2 \mathrm{O}}}{M_{\mathrm{N}_2 \mathrm{O}}}} \\ & M_{\mathrm{CO}_2}= 44, M_{\mathrm{N}_2 \mathrm{O}}=44 \\ & \frac{r \mathrm{CO}_2}{r \mathrm{~N}_2 \mathrm{O}}=\sqrt{\frac{T_{\mathrm{CO}_2}}{T_{\mathrm{N}_2 \mathrm{O}}}} \\ & \frac{x}{4 x}=\sqrt{\frac{T_{\mathrm{C}_2}}{T_{\mathrm{N}_2 \mathrm{O}}}}\end{aligned}$ Squaring on both the sides, $T_{\mathrm{N}_2 \mathrm{O}}=16 T_{\mathrm{OO}_2}$

Asked in: AP EAMCET 2001

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