What is the de Broglie wavelength of a nitrogen molecule in air at 300 K? Assume that the molecule is moving with the root mean square speed of molecules at this temperature. (Atomic mass of nitrogen = 14.0076 u)
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