Question

Find the temperature at which the average thermal kinetic energy is equal to the energy needed to take a hydrogen atom from its ground state to n = 3 state. Hydrogen can now emit red light of wavelength 653.1 nm. Because of Maxwellian distribution of speeds, a hydrogen sample emits red light at temperatures much lower than that obtained from this problem. Assume that hydrogen molecules dissociate into atoms.

Answer 1

Given:

Wavelength of red light, λ  = 653.1 nm = 653.1 × 10  m

Kinetic energy of H₂ molecules (K) is given by

`K = 3/2 KT` .......(1)

Here, `K = 8.62 xx 10^-5   eV/K`

T = Temperature of H₂ molecules

Energy released (E) when atom goes from

ground state to n = 3 is given by

`E = 13.6 (1/n_1^2 - 1/n_2^2)`

For ground state, n₁ = 1

Also, n₂ = 3  

`therefore E = 13.6 (1/1 - 1/9)`

= `13.6 (8/9)` .............(2)

 kinetic energy of H₂ molecules = Energy released when hydrogen atom goes from ground state to n = 3 state

`therefore 3/2 xx 8.62 xx 10^-5xxT = (13.6xx8)/9`

`rArr T = (13.6xx8xx2)/(9xx3xx8.62xx10^-5)` 

= 9.4 × 10⁴  K