﻿ Empirical Relationships for the Ionization Potentials of Atoms and Ions
San José State University

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 Empirical Relationships for the Ionization Potentials 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 the shell number. The value of Z is the number of protons in the nucleus #p less the shielding ε by the electrons in inner shells. Thus the ionization potential would be

#### IE = (R/n²)(#p−ε)² which can be put in the form IE = (R/n²)((#p)² − 2(#p)ε + ε²)

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 are however 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 turns out, even by electrons in the inner shells the shielding is less than the full value of their charge.

This partial shielding explains how there can be negative ions. In a negative ion such as O= there are outer electrons clinging to a structure with overall negative charge. That appears to be a puzzle. The oxygen nucleus has eight proton. There are two electrons in the first shell and six in the second shell of the oxygen atom. Overall that is electrostatically neutral. But for a seventh electron in the second shell the two electrons in the inner shell and other six electrons in the second shell shield only a portion of the positive charge of the nucleus. Thus there is a positive attraction for that seventh electron. And since the seventh electron shields only a fraction of a unit charge for the eighth there is a positive attraction for the eighth electron. However another electron would go into a third shell and the ten (2+8) other electrons would be inner shell electrons and could more than shield the eight positive charges of the nucleus. Thus there would be a repulsion of an eleventh electron.

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

The values given below are for the cases of the first shell.

Parameter Values and Statistical
Characteristics of the Empirical Ionization
Function for the Electrons in the First Shell
Electron
Number
εR
(eV)
ζCoefficient of
determination
10.179758108 13.892541.2371532250.999998221
20.777385892 13.85999 0.8685233860.999998302

The fact that ε is not zero for the first electron is an enigma. The amount that the second electron adds to the shielding is 0.59762782. Thus its shielding ratio is about 0.6.

For the electrons in the second shell the parameter values are:

Parameter Values and Statistical
Characteristics of the Empirical Ionization
Function for the Electrons in the Second Shell
Electron
Number
ε R
(eV)
ζCoefficient of
determination
1 1.782576199 13.917337521.237153314 0.999967446
2 2.305874388 13.891843680.868523386 0.999995889
3 3.261517659 13.93006743-0.263191344 0.999998748
4 3.806464455 13.72633367-1.576107221 0.999999107
5 4.472571375 13.68295704-2.105785389 0.999999898
6 5.315741296 13.69648352-3.414286539 0.999999584
7 6.012661928 13.71370719-3.955810442 0.999999288
8 6.751314712 13.83482438-4.346850852 0.999999695

The fact that the shielding for the first electron is 1.782576199 rather than 2.0 is notable. Instead it is about 89 percent of that value.

For the third shell the parameter values are:

Parameter Values and Statistical
Characteristics of the Empirical Ionization
Function for the Electrons in the Third Shell
Electron
Number
ε R
(eV)
ζCoefficient of
determination
1 8.23363938614.12814515 -3.8992706780.999999245
2 8.85005947814.20186513 -4.9041684240.999999245
3 9.56661352613.63683417 -10.214645550.999698201
4 11.0195510114.6263037 -3.0784633480.999573295
5 11.19255474514.10977023-8.1818980130.999999218
6 11.9601280613.99266234 -10.339578390.999998189
7 12.605321914.03815118 -11.51921010.999997968
8 13.5291813214.47872006 -10.190812970.999993592

For the first five electrons in the fourth shell the values are:

Parameter Values and Statistical
Characteristics of the Empirical Ionization
Function for the First Five Electrons in the Fourth Shell
Electron
Number
εζCoefficient of
determination
117.57065432 0.661428062 0.997488104
218.53150712 0.998053668 0.997193112
319.54117024 1.067763524 0.99726095
420.6161118 1.310662754 0.9969434
521.40504006 0.546564484 0.997849703

The value of the parameter R is not given for this case. The R values are found by multiplying the regression coefficient for (#p)² by n², where n is the principal quantum number for the shell. For shells one through three the value of n which seems appropriate is equal to the shell number. The principal quantum number could be based upon the maximum occupancy of the shell but this does not seem to fit for third shell where the occupancy is 8 and thus n would be 2. Instead a value of n=3 seems to fit. Thus for shells one through three the value of n is equal to the shell number. This is not the case for the fourth shell. For a value of n=4 the value R for the fourth shell is not close to the Rydberg constant. To get a value R equal to the Rydberg constant the regression coefficient for (#p)² would have to be multiplied by about 6.5 instead of 3²=9 or 4²=16. In other words, n would have to be approximately 2.55. A value of 2½ seems to be appropriate.

## Patterns to the Parameter Values

The shielding by electrons in the same shell should be a function of the number of electrons in the shell, #e, less one; i.e., #e−1. The plot of the shielding versus #e−1 is shown below.

The regression results are

#### IE = 0.991596406 + 0.715998632(#e-1) [14.2]       [51.7] R² = 0.997755065

This indicates that the shielding due to the two electrons in the inner shell of electrons is 0.992 units instead of 2.0. The additional shielding from each additional electron in the second shell is about 0.716 of a unit charge. The regression coefficients for the first through fourth shells are as follows:

 Shell εI γ R² 1 0.179758108 0.59762782 1.0 2 0.991596406 0.715998632 0.997755065 3 8.236353971 0.752364916 0.985388779 4 17.58222148 0.975337616 0.998039582

where εI stands for the shielding by the inner shell electrons and γ stands for the additional shielding per additional electron in the shell.

(To be continued.)

## Conclusions

• Electrons in the same shell shield each other of some of the positive charge of the nucleus. The shielding ratio is smaller for the first shells and approaches 1.0 for the fourth shell.
• The electrons in inner shells shield some of the positive charge of the nucleus but not in a one-for-one proportion.
• The Bohr model extends very well to the multiple electron case. The statistical fit is about the same or better than for the case of the hydrogen-like atoms or ions.
• The model explains the existence of negative ions.