The Bohr model of a hydrogen-like ion predicts that the total energy E of an electron is given by

E = −Z²R/n²

where Z is the net charge experienced by the electron, n is the principal quantum number and R is a constant equal to approximately 13.6 electron volts (eV). This formula is the result of the total energy being equal to

E = − Ze²/(2r_n)

where e is the charge of the electron and r_n is the orbit radius when the principal quantum number is n. The orbit radius is given by

r_n = n²h/(Zm_ee²)

where h is Planck's constant divided by 2π and m_e is the mass of the electron.

Shell Structure

Electrons in atoms are organized in shells whose capacities are equal to 2m², where m is an integer. Thus there can be at most 2 electrons in the first shell, 8 in the second shell and 18 in the third shell. Here only the second shell is being considered. For elements above boron all of the electrons have been removed except two.

Here are all the ionization potentials for such ions. The values are for the elements for which the data is available in the CRC Handbook of Physics and Chemistry 82nd Edition (2001-2002).

The Ionization Potential of the First Electron
as a Function of Proton Number

An ion with only one electron in a shell is equivalent to the hydrogen atom but having a positive charge of Z instead of one, where Z is the proton number #p of the nucleus less the amount of shielding by the electrons in the inner shells. The Bohr theory applies to such system. According to the Bohr theory the ionization potential should be

IE = Z²R/n²

R is constant and n, the quantum number, is equal to 1. Thus the ionization potential should be proportional to Z²; where Z=(#p−ε). For the first electron in the second shell it is usually presumed that the two electrons in the first shell shield exactly two units of charge. Here is the plot of the relationship.

This appears to be a quadratic relationship but shifted; i.e. something proportional to (#p−ε)². Thus equation is then

IE = R(#p−ε)²
which can be expressed as
IE = R(#p² − 2#p*ε + ε²)

where R is the Rydberg constant, 13.6 eV.

The appropriate regression equation would be

IE = c₀ + c₁#p + c₂(#p)²
in which
c₁<0

The regression results are

IE = 14.17457076 − 12.40435731#p + 3.479334381(#p)²
[3.1] [-19.7] [186.4]
R² = 0.999967446

The numbers in the square brackets are the t-ratios for the regression coefficients. For a regression coefficient to be statistically significantly different its magnitude must be greater than 2.0. As can be seen the regression coefficients for are highly significant.

The value of ε can be found as

ε = ½(−c₁/c₂) = 1.782576199

Thus the shielding of the first electron in the second shell by the two electrons in the first shell is not exactly 2. Instead it is 0.89 of that value. This could be due to the distributions of the charges of the two inner electrons, either their radial dispersion or their asymmetry.

When the same procedures are applied to the data for the second electron in the second shell the regression results are

IE = 19.29289271 − 16.01642327#p + 3.472960919(#p)²
[12.2] [-74.3] [542.8]
R² = 0.999995889

The shielding ε that these coefficients imply is

ε = ½(−c₁/c₂) = 2.305874388

Thus the additional shielding created by the second electron is (2.305874388−1.782576199)=0.523298189, notably close to the value of 0.5 that the simple theory of shielding by electrons in the same shell suggests.

(To be continued.)