The mass of an oxgyen molecule differs from that of a nitrogen molecule . Do the oxygen molecules and the nitrogen molecules in air have the same rms speed `v_(rms)` ?

Explanation

we know that ` V_(rms)` is related to Kavg by equation .
` K_(Avg) = 1/2 mv_(rms )^(2)`
Although the two types of molecules have the same `K_(Avg) ` they have different masses m and thus their rms speeds must differ .
the find their rms speeds , we solve Eq. 20-15 for ` v_(rms ) ` and then substitute for molecular mass m with equation `M = m N_A` because molar masses M can be found in appendix F. we obtain
` V_(rms) = sqrt((2K _(avg) N_A)/(M ))`
Calculations : in the air that we breathe oxygen and nitrogen are diatomic molecules `(O_2 and N_2)` -each molecule consists of two atoms . the molar mass `M_(O_2)` of diatomic oxygen is 1 ( 15.9994 g / mol ) `~~ = 0.0320 ` Kg / mol and the molar mass `M_(N_2)` of diatomic nitrogen is `2 ( 14.0067 g // mol ) ~~ 0.0280 kg // mol `
Now we can substitute known data into Eq 20-27 for oxygen we find
` V_(rms ) = sqrt((2 ( 6.21 xx 10^(-21)J)(6.02 xx 10^(23) mol ^(-1)))/( 0.0320 kg // mol ))`
`=483 m//s `
Similarly ,for nitrogen we find
` V_(rms ) = 517 m//s `
the nitrogen we find
` V_(rms ) = 517 m//s`
The nitrogen molecules , with their smaller mass ( and the same average kinetic energy ) , have the greater rms speed .