The energy needed to detach the electron of a hydrogen like ion in ground state is a system 4 rydberg (a) what is the wavelength of the radiation emitted when the electron jumps from the first excited state to the ground state? (b) What is the radius of the orbit for this atom?

Explanation

(a)In energy units , 1 rydberg `= 13.6eV` The energy needed to detacis electron is `4 xx 13.6eV` .The energy in the ground state is therefore, `E_(1) = -4 xx 13.6eV` The energy of first excited state `(n= 2) is E_(2) = (E_(1))/(4) = - 13.6eV` The energy difference is `E_(2) - E_(1) = 3 xx 13.6eV = 40.8eV` The wavelength of theradiation emitted is
`lambda = (hc)/(DeltaE)`
`= (1242eV nm)/(40.8eV) = 3.04nm`
(b) The energy of a hydrogen like in ground state is `E= Z^(2)E_(0) where Z = atomic number and E_(0) = -13.6eV Thus Z = 2` the ratio of the first orbit is `a_(0))/(z) where a_90) = 53pm` Thus,`r = (58pm)/(2)= 26.5 pm`