Find the temperatures at which methane and ethane will have the same rms speed as carbon dioxide at `400^(@)C`. Also calculate the mean velocity and most probable velocity of methane molecules at `400^(@)C`.

Explanation

`C_(rms)=sqrt((3RT)/(M))`
Let `1: CO_(2)` and `2:Ethane`
For `C_(rms)` to be same for `1` and `2`
`(T_(1))/(M_(1))=(T_(2))/(M_(2))`
(`a`) `T_(CH_(4))=(673)/(44)xx16=244.73 K`
(`b`) `T_(C_(2)H_(6))=(673)/(44)xx30=458.86 K`
Mean speed `(CH_(4))=sqrt((8RT)/(piM))=sqrt((8xx8.31xx673)/(3.14xx16xx10^(-3))`
`=943.68 m s^(-1)`
Most probable speed `(CH_(4))`
`=sqrt((2RT)/(M))=sqrt((2xx8.31xx673)/(16xx10^(-3)))`
`=836.11 m s^(-1)`