Question

Calculate the temperature values at which the molecules of the first two members of the homologous
series, $C_n{H}_{2n+2}$ will have the same rms speed as $C{O}_2$ gas at 770 $K$. The normal b.p. of $n$- butane is 273 $K$.

This question has multiple answers.

• A. 280 $K$
• B. 525 $K$
• C. 280${}^{\circ }C$
• D. 525${}^{\circ }C$

Answer 1

Answer: (A), (B)

The correct options are
A 280 $K$
B 525 $K$

The temperature values at which the molecules of the first two members of the homologous
series, $C_n{H}_{2n+2}$ will have the same rms speed as $C{O}_2$ gas at 770 $K$ are 280 $K$ and 525 $K$.
The first two members of the homologous series $C_n{H}_{2n+2}$ include methane and ethane. They have molar masses 16 $g/mol$ and 30 $g/mol$ respectively. The molar mass of $C{O}_2$ is 44 $g/mol$ respectively.
The $rms$ speed is given by the expression $u=\sqrt{\frac{3RT}{M}}$
For methane and carbon dioxide
$\frac{u}{u^{\prime }}=\sqrt{\frac{T{M}^{\prime }}{T^{\prime }M}}$
Here, $u$ and $u^{\prime }$ are the rms speeds, $T$ and $T^{\prime }$ are the temperatures and $M$ and $M^{\prime }$ are the molar masses of methane and carbon dioxide respectively.
Substitute values in the above expression
$\frac{u}{u}=\sqrt{\frac{T\times 44}{770\times 16}}$
$T=280K$
For ethane and carbon dioxide
$\frac{u}{u^{\prime }}=\sqrt{\frac{T{M}^{\prime }}{T^{\prime }M}}$
Here, $u$ and $u^{\prime }$ are the rms speeds, $T$ and $T^{\prime }$ are the temperatures and $M$ and $M^{\prime }$ are the molar masses of ethane and carbon dioxide respectively.
Substitute values in the above expression
$\frac{u}{u}=\sqrt{\frac{T\times 44}{770\times 30}}$
$T=525K$