How can I calculate the root mean square speed of carbon dioxide molecules at STP?

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Borys Shumyatskiy eNotes educator| Certified Educator

Hello!

As we know from the kinetic theory of gases, the mean kinetic energy of gas molecules is proportional to its absolute temperature `T.`  Because the kinetic energy of a molecule is `(m v^2)/2,` the root mean square is proportional to `sqrt(T).` The exact formula is

`v_(rms) = sqrt((3RT)/M),`

where `R approx 8.3 J/(mol * K)` is the ideal gas constant and `M` is the molar mass of a gas. `M` must be expressed in `(kg)/(mol).`

 

We know the relative atomic masses of  `O` and `C.` They are `16` and `12,` and we can compute the molar mass of `C O_2:` 

`12 + 2*16 = 44 (g/(mol)) = 4.4*10^(-2) ((kg)/(mol)).`

 

 

STP usually means `0` degrees Celsius, or about `273 K.` Now we can obtain the numerical result:

`v_(rms) = sqrt((3*8.3*273)/(4.4*10^(-2))) approx 393 (m/s).`

 

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