A beam of electrons with a kinetic energy of 52.5 eV is used to bombard a sample containing He^+ ions in their ground state. The number of distinct spectral lines emitted in the resulting spectrum will be:

Step-by-step solution

The energy of an electron in the n^ th orbit of a hydrogen-like ion is given by: E_n = -13.6 Z^2 n^2 eV For He^+ , the atomic number Z = 2 . Thus: E_n = -13.6 4 n^2 eV = - 54.4 n^2 eV The energy of the ground state ( n=1 ) is: E₁ = -54.4 eV Let us calculate the energy required to excite the He^+ ion to higher states: For n=4 : E₄ = - 54.4 16 = -3.4 eV E_ 1 4 = -3.4 - (-54.4) = 51.0 eV For n=5 : E₅ = - 54.4 25 = -2.176 eV E_ 1 5 = -2.176 - (-54.4) = 52.224 eV For n=6 : E₆ = - 54.4 36 -1.511 eV E_ 1 6 = -1.511 - (-54