Question

Consider a helium ion He+.

a) Find the ratio between the ionization energy of the H atom and and the He^+ ion.

b) How does the Bohr radius, we defined for H atom, differ for He^+ ? Using the Bohr radius of He^+ and the representation of the ground state wave function for the H atom, write the ground state wave function for the He^+.

c) Find the average distance of a 1s electron from a nucleus in He+ ion. How does it compare with the Bohr radius you found in (b) for H atom and He^+ ion?

d) Find the most probable value of r characterizing the distance between a 1s electron and the nucleus in the He^+ ion. Compare your answer with (c) and discuss a physical meaning of obtained results.

e) Find the average distance of 2pz electrons from a nucleus in He+ ion. Compare your results with those for a 1s electron in (c). Are you surprised by the answer?

f) How your answer in (e) would be different, if you find the average distance of 2px electrons, instead of pz?

Answer 1

(a) Ionizatin energy is the energy rewuired to remove anelectron completely from the outermost orbit.Hence the electron will go from n1 = 1 to n2 = infinite.

formulae for the energy of nth orbit is En = - 13.6xZ² / n² eV

For H atom : When n1 = 1, E1 = - 13.6x1² / 1² eV = - 13.6 eV

When n2 = infinite, E_inf =  - 13.6x1² / infinite² eV = 0 eV

Hence inization energy of H, IE_H = 0 eV - (- 13.6 eV) = 13.6 eV

For He⁺ ion : atomic number, Z = 2

For He⁺ ion : When n1 = 1, E1 = - 13.6x2² / 1² eV = - 54.4 eV

When n2 = infinite, E_inf =  - 13.6x2² / infinite² eV = 0 eV

Hence inization energy of H, IE_He+ = 0 eV - (- 54.4 eV) = 54.4 eV

Hence the ratio is,  IE_H /  IE_He+  = 13.6 eV / 54.4 eV = 1/4 = 0.25 (answer)

(b): The formulae for the radius of nth orbit is r_n = 0.529 xn² / Z A⁰

For H, Z = 1 and for He⁺, Z = 2. Hence the radius of nth orbit of He+ is half of the radius of the nth orbit of H.

Here a0 = Bohr's radius = 0.529 A⁰