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, 8 in the third shell and 18 in each of the fourth and fifth shells. Here only the second shell is being considered.

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 = (R/n²)(#p−ε)²

Where R is constant known as the Rydberg constant and is equal to about 13.6 electron volts (eV), n is the principal quantum number which here is the same as the shell number. The quantity #p is the proton number of the nucleus and ε is the amount of shielding by electrons in inner shells or the same shell. 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. It is immediately discovered that this not the case. Here is the plot of the relationship for the first electron.

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

IE = (R/n²)(#p−ε)²
which can be expressed as
IE = (R/n²)(#p² − 2#p*ε + ε²)

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.

The regression results for all eight of the electrons in the second shell are

The graph of the shielding versus the number of electrons in the shell is:

The slope of the relationship is found by regressing the shielding S on the number of electrons #e. This gives

S = 0.991596406 + 0.715998632#e
[14.2] [51.6]
R² = 0.997755065

Thus, on average, each additional electron in the second shell shields about 0.716 units of positive charge of the nucleus.

The incremental (additional) shielding created by each additional electron is the difference between the shielding with that electron and what it is without that elecron. For example, for the second electron it 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. The values for the second through eighth electron are:

The graph of the relationship is

There is an upward trend with notably high values for the third and sixth electrons.

Conclusions

The values of the ionization potentials IE are accurately explained by a function of the form

IE = (R/n²)(#p−ε)²

where R is an empirical constant approximately equal to the Rydberg constant, n is the shell number, #p is the proton number of the nucleus, and ε is an empirical value which is a function of the number of electrons in the shell and the number which are in inner shells.

(To be continued.)