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Empirical Relationships for the
Ionization Potentials of of the First
Three Electrons in the Fifth Shell
of Atoms and Ions

The ionization energy IE, or as it is usually called the ionization potential, for an electron in an atom or ion is the amount of energy required to dislodge it. The Bohr model of a hydrogen-like atom indicates that the energy required to remove an electron, called the ionization potential, should follow the form

IE = RZ²/n²

where R is the Rydberg constant (approximately 13.6 electron Volts (eV), Z is the net charge experienced by the electron and n is the principal quantum number, effectively given by the maximum occupancy of the shell. The value of Z is the number of protons in the nucleus #p less the shielding ε by the other electrons. Thus the ionization potential would be

IE = (R/n²)(#p−ε)²

This is equivalent to a regression equation of the form

IE = c0 + c1(#p) + c2(#p)²

Such a form gives a very good fit to the data. The coefficient of determination goes as high as 0.9999998+. The value of ε is found as

ε = −½c1/c2

However, according to the equation, it also should be that c0/c2 is equal to ε² and thus equal to the square of the value found from c1 and c2. The regression coefficients are not constrained to achieve that equality. Thus effectively the form assumed for the relationship for ionization potential is

IE = (R/n²)[(#p−ε)² + ζ]

where R is an empirical value, rather than necessarily being the Rydberg constant, and ζ is a constant. The values of R found for the first three shells are notably close to the Rydberg constant.

The Bohr model is strictly for a hydrogen-like atom or ion; i.e., one in which there is a single electron in the outermost shell. However the regression equation fits very well the cases of multiple electrons in the outer shell. Thus the shielding ε is for the electrons in the inner shells and also in the same shell. But the shielding by electrons in the same shell is only a fraction of their charge. As it turned out, even by electrons in the inner shells the shielding is less than the full value of their charge. <

The Parameters Values for
the Empirical Relationships

There can be only two electrons in the first shell, eight in the second and third shells and eighteen in the fourth and fifth shells. The standard model presumes that ε is zero for the first shell, two for the second shell, ten for the third shell, eighteen for the fourth shell and thirty six for the fifth shell. The results indicate that there is only partial shielding by the electrons in the inner shells.

There are data only for the first three electrons in the fifth shell. The graphs of ionization energy versus the number of protons in the nucleus are given below.

The quadratic curvature is clearly perceptible in the case of the first electron but only barely perceptible in the case of the second electron. For the third electron the curvature appears to be nonexistent or cubic.

The values given below are for the cases of the first three electrons in the fifth shell.

Parameter Values and Statistical Characteristics of the Empirical Ionization Function for the First Three Electrons in the Fifth Shell
ζCoefficient of
135.1608298313.50056571 -0.696517560.999779761
229.658206299.002892857 -26.66090520.999998668
319.40061354.762987013 -236.15067520.997901056

The model is well borne out by the results for the first electron. There are 36 electrons in the first four shells and the shielding is nearly 36. The value of R is very close to value of the Rydberg constant, 13.6 eV. For the second electron, although the statistical fit is excellent, the values deviate from what the model predicts. The shielding should be greater for the second electron than for the first, but it is not. The value of R is quite different from 13.6 eV and the value of ζ is quite different from what was found in other cases. For the third electron the deviations from what the model predicts are even more drastic.

(To be continued.)

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