Calculate the most probable radius for en electron in a 2p orbital in a Li2+ ion

Question

Calculate the most probable radius for en electron in a 2p orbital in a Li2+ ion. To answer this question, you need to know that the probability function of a non-symmetric orbital is best given by P(r) = r 2 2 radial, with the maximum probability determined at the point (value of r) that dP/dr = 0. For a 2p orbital, dP dr = 1 8 z 3 a 3 0 16zr3 a 2 0 8z 3 r 4 a 3 0 e 2zr/a0 Firstly, setting dP/dr = 0, derive an expression for determining r (the radius of maximum probability), and then determine the radius of maximum probability for an electron in a 2p orbital of a Li2+ ion.

Explanation / Answer

For a hydrogen-like ion such as Li2+, the energy in atomic units (also called hartrees) is given by € En = Z 2 2n 2 .

The ionization process can be described by the equation

€ Li2+ Li3+ + e.

The ionization energy is just the energy that it takes for the ionization process.

Thus, the ionization energy € IE can be defined as

€ IE = En = En =1 .

Substituting the hydrogen-like ion energy expression yields

€ IE = 32 2 × 2 32 2 ×12 % & ' ( ) * = 0 9 2 % & ' ( ) * IE = 4.5 a.u. (or hartrees).

Using the conversion factors,

1 hartree = 27.211 eV

and 1 hartree = € 4.3597 ×10–18 Joules,

we have

€ IE = 122.4 eV = 1.962 ×10-17 J.

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