Air is approximately 21% #O_2# and 78% #N_2# by mass. What is the root-mean-square speed of each gas at 273 K?

1 Answer
Dec 28, 2017

Answer:

The #rms# speed for oxygen is 461 m/s and for nitrogen, 493 m/s

Explanation:

The relation between molecular kinetic energy and temperature is given by the equation

#1/2 mv^2 = 3/2 kT#

where #k# is the Boltzmann constant,

Solving this for #v# gives us the root mean square speed

#v_(rms) = sqrt ((3kT)/m) = sqrt((3RT)/M)#

where the final relation is given in terms of the gas constant #R# and the molar mass of a gas. (It basically amounts to multiplying top and bottom of the middle relation by the Avogadro constant #(6.02xx10^(23))#

Inserting values gives

#v_(rms) = sqrt((3(8.314)(273))/M) = sqrt(6809/M)#

Using #M=0.032# kg we get, for oxygen, #v_(rms)=461 m/s#

and using 0.028 kg for nitrogen #v_(rms)=493 m/s#

Note that the molar mass had to be changed to kg per mole in order to be consistent with the other units in the equation.