What is the de-Broglie wavelength of nitrogen molecule in air at 300K ? Assume that the molecule is moving with the root mean square speed fo molecules at this temperature. (Atomic mass of nitrogen `=14.0076U`) Plank's constant=`6.63xx10^(-34)Js` , Boltzmann constant `=1.38xx10^(-23)JK^(-1)`

Explanation

Here, temperature T=300K, Atomic
mass of nitrogen =14.0076U , mass of nitrogen
molecule `m=2xx14.0076=28.0152U`
`=28.0152xx1,66xx10^(-27)kg`
Let `v_(rms)` be the not mean square speed of the
nitrogen molecule at temperatureT(=300K), then
KE of nitrogen `=3/2 kT`
or `1/2mv_(rms)^2=3/2kT`
or `v_(rms)=sqrt((3kT)/m)`
De-Broglie wavelength of nitrogen molecule
`lambda=h/(mv_(rms))=h/(msqrt(3kT//m))=h/(sqrt(3mkT))`
`=(6.63xx10^(-34))/(sqrt(3xx(28.0512xx1.66xx10^(-27))xx(1.38xx10^(-23))xx300))`
`0.028xx10^-8m=0.028nm`