Answer :

We have a relation to find kinetic energy of a gas molecule at any Temperature which is



Where EK is the kinetic energy of a particle at temperature T and K is boltzmann constant


K = 1.38 × 10-23 JK-1


And we know Kinetic energy of a particle is given by the relation



Where vrms is the root meansquare speed of the particle and m is the mass of the particle


So equating both equations we have



On solving we get the root meansquare speed of the particle as



Where Vrms is the root mean square speed of the particle of mass m at temperature T and K is boltzmann constant


K = 1.38 × 10-23 JK-1


Here the temperature is


T = 300K


We are given the particle is a nitrogen molecule (N2) mass of the Nitrogen atom is


mN = 14.0076 u


so mass of N2 molecule is


m = 2 × 14.0076 u = 28.0152 u


converting it to Kg


we know 1 u = 1.66 × 10-27 Kg


so mass of Nitrogen molecule is


m = 28.0152 × 1.66 × 10-27Kg = 46.50 × 10-27Kg


= 4.650 × 10-26Kg


We know de-Broglie wavelength of a particle is given by relation



where 𝜆 is the de de-Broglie wavelength of a particle having mass m and moving with velocity v , here velocity is the root mean square speed of the particle, h is Planck’s constant


h = 6.63 × 10-34 Js


putting the value root mean square speed



in above equation we get



Solving we get the de Broglie wavelength is



Putting value of m,T,K and h in above equation we get




Or 𝜆 = 0.028 × 10-9m = 0.028 nm


So, the de Broglie wavelength of Nitrogen Molecule is 0.028 nm


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