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.

Shielding by Electrons

The term hydrogen-like ion means that there is but one electron in a shell, the inner shells being completely filled. The value of Z in the above formulas can be considered to be the number of protons p in the nucleus less the number of electrons in inner shells. ε₀; i.e., Z=p−ε₀. Thus ε₀ positive charges of the nucleus are said to be shielded by the ε₀ electrons which are closer to the center of the atom than the electron being considered. On the other hand, there is no shielding by any electrons which are farther from the center of the atom.

Shielding by Electrons in the Same Shell

When a charge is distributed uniformally on a spherical surface it has the effect on another charge outside of the spherical equal to what that same charge would have concentrated at the center of the sphere. The effect on a charge within the sphere is zero. The effect on a charge located on the same sphere is equal to what half the charge would have located at the center of the sphere. Thus if the number of electrons in a shell is denoted as ε₁ then

Z = p − ε₀ − ½(ε₁-1)

Thus the energy required to remove an electron from a shell should decrease with the number of electrons in that shell. This is due to the shielding of some of the positive charge of the nucleus by electrons in the same shell.

The negatively charged ions are created when an atom acquires enough electrons to complete a shell. For example, the fluoring atom has nine protons and nine electrons. There are two electrons in the first shell and seven in the second shell. The capacity of the second shell is eight. The fluorine ion F^- has a net negative charge yet the electrons are some how clinging to it. Often chemistry students believe that the completion of the shell of eight involves some sort of attraction for the electron. The notion of shielding of electrons in the same shell provides a differents sort of justification for the F^-. The two electrons in the inner shell shield fully two protons. For any electron in the second shell there are seven other electrons in the same shell, each shielding a half unit of positive charge each. That make the charge experienced by each of the electrons in the second shell equal to (9-2-½(7))=3.5 positive charges. That is sufficient to hold each of the electrons in the second shell. The reality is more complicated but this computation explains how the electrons in the F^- ion could be clinging to a system with no net positive charge.

The Ionization Energies of the Elements from Neon to Sulfur

The ionization energies for the elements from neon to sulfur are plotted in the graph below. The data are from the CRC Handbook of Physics and Chemistry 82nd Edition (2001-2002).

The shielding effect of the electrons in the same shell is thus demonstrated. The effect is apporoximately, but no precisely, linear.

The regularity of the ionization energy is further demonstrated by plotting the ionization energy versus the charge in the nucleus, as below.

Again, the effect is apporoximately, but no precisely, linear. This suggests that a linear regression will capture the relationship between the ionization energies and the number of electrons in the shell and the number of protons in the nucleus.

Let #p be the number of protons in the nucleus and #e the number of electrons in the shell. A regression equation of the form

IE = c₀ + c₁#p + c₂#e

using the data for neon to chlorine gives the following results:

IE = -326.5349138 + 64.32181881#p − 49.60899613#e
[-12.5] [35.4] [-27.3]
R² = 0.9704

The numbers in square brackets are the t-ratios of the regression coefficients; i.e., the ratios of the coefficient to its standard deviation. There values indicate that the regression coefficients are highly significant statistically.

The ratio of the coefficient for #e to that of #p indicates that the electrons in the same shell shield about 0.77 of a charge.

The above equation presumes that the slopes of the relationship between ionization energies and the number of electrons in the shell are all the same. A previous graph indicates that the slope varies with the number of protons in the nucleus. Thus a regression equation of the form

IE = c₀ + c₁#p + c₂#e + c₃#p*#e

gives following improved results:

IE = -630.5502698 + 86.8414748#p + 17.94997185#e − 5.004368#p*#e
[-19.3] [20.0] [2.8] [-10.6]
R² = 0.9897

The previous regression equation can be expressed as

IE = -630.5502698 + 86.8414748#p + (17.94997185 − 5.004368#p)*#e

Thus the slope of the relationship between ionization energy and the number of electrons in the shell varies with #p, the number of protons in the nucleus. Therefore the shielding factor for an electron also varies with #p. For neon it is about 37 percent and icreases to 77 percent for chlorine.

(To be continued.)