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 first shell is being considered. For elements above helium all of the electrons have been removed except two.

Here are all the ionization potentials for such ions.

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

An ion with only one electron is equivalent to the hydrogen atom but having a positive charge of Z instead of one. 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 the proton number squared. Here is the plot of the relationship.

It certainly looks like a quadratic relationship. To test this proposition more precisely the logarithms of the ionization potential and the proton numbers are computed. The plot of these quantitites is shown below.

The regression equation of the logartithm of ionization potential on the logarithm of the proton number is

ln(IE) = 2.604026184 + 2.004055992ln(#p)
[2029.1] [4123.2]
R² = 0.999998471

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 coefficient for ln(#p) is highly significant. And the value appears to be essentially 2, as the Bohr theory predicts. However this proposition is tested by comparing the difference between the regression coefficient and 2 with the standard deviation of the regression coefficient. Although the difference is small, 0.004056, the standard deviation of the regression coefficient is even smaller, 0.000486039, the ratio is 8.3. Common sense however says that the Bohr theory is verified for the first electron in the the first shell.

The Z in the formula is the net charge experienced by the electron, the number of protons #p in the nucleus less any shielding ε. The Bohr theory 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.

Thus the appropriate regression equation would be

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

Since in this case there seems to be no reason for there to be any shielding for the first electron the value of ε should be essentially zero.

The results of the regression are:

IE = 17.63611461 − 4.994593419#p + 13.89254281(#p)²
[5.1] [-9.7] [858.9]
R² = 0.999998221

The coefficient of #p is negative but statistically different from zero. The coefficient of (#p)² is notably close to 13.6. The value of ε can be found as

ε = ½(−c₁/c₂) = 0.179758072

Somehow some small portion of the charge of the nucleus is being shielded.

The Bohr theory works marvelously well for an atom or ion with one electron but not nearly so well for multiple electrons. One line of approach is to treat the atom or ion with two electrons as a many-body problem. This approach does not come up with much in the way of definite results. An alternate line of approach is to say that electrons in the same shell shield some fraction of a positive charge in the nucleus. This shielding ratio would be based upon the proportion of the charge of an electron which is closer to the nucleus than the center of an electron. The value of one half would be plausible. However the shielding ratio could be affected by any deviation from spherical symmetry.

The Shielding Ratios of the Electrons in the First Shell
as a Function of Proton Number of the Nucleus

The shielding ratio ρ for the electrons in the first shell can be computed as

ρ = 1 − E₂(#p)/E₁(#p)

where E₁ and E₂ are the ionization potentials for the first and second electrons, respectively.

The values for the elements for which the data is available in the CRC Handbook of Physics and Chemistry 82nd Edition (2001-2002) are given below.

The graph of these data displays a very regular pattern.

This appears to be a relationship of the form

ρ = α(#p)^-β

The value of the parameters α and β can be found by plotting the logarithms of ρ and #p, as below.

A regression of ln(ρ) on ln(#p) gives:

ln(ρ) = 0.093012504 − 0.956463724ln(#p)
[16.2] [-439.2]
R² = 0.999865247

The results indicate that the shielding ratio is predictable on the basis of the proton number, but the shielding of positive charge in the nucleus by the electrons in the first shell is primarily important for small nuclides.

The Ionization Potential for the Second
Electronas a Function of the Proton Number

Previously a regression equation of the form

IE = c₀ + c₁#p + c₂(#p)²

was applied to the data for the first electron and it was found that there was a shielding of 0.179758072 units of charge. When the data for the ionization potentials of the second electron is used the regression results are

IE = 20.41371337 − 21.54911763#p + 13.85998758(#p)²
[6.2] [-44.6] [912.9]
R² = 0.999998302

Again the coefficient of #p is negative and the coefficient of (#p)² is notably close to the Rydberg constant of 13.6. The value of the shielding that is found from the coefficients is 0.777385892. The difference between this value and the value for the first electron is 0.59762782, indicating that the second electron shields about 0.6 of a unit charge of the nucleus. This is notably close to the value of 0.5 suggested by the simplest formulation of the notion of shielding by electrons in the same shell.

(To be continued.)