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Ionization Energy and Charge
Shielding of Electrons
in Atoms and Ions (Part II)

This is a continuation of a study on the ionization energy for electrons in different positions within atoms and ions. Ionization energy IE, or as it is usually called the ionization potential, for an electron is the amount of energy required to dislodge it. The previous study was limited to only five cases for each electron position. The more extensive database is displayed in the appendix.

Ionization Energies

The Bohr model of a hydrogen-like atom or ion indicates that the energy required to remove an electron, called the ionization potential, should follow the formula

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.

Here are the graphs of the ionization potential of the three innermost electrons.

Clearly the relationships are very regular and quadratic. The graph for the first electron exhibits an unexplained jump in ionization energy after ten protons. The data for none of the other electron positions exhibit such a jump.

Let p denote the number of protons in the nucleus of the atom or ion. The value of Z in the above Bohr formula is the number of protons in the nucleus p less the shielding ε by the electrons in inner shells or in the same shell. Thus the ionization energy would be

IE = (R/n²)(p−ε)²
which can be put in the form
IE = (R/n²)(p² − 2pε + ε²)

There may be other phenomena that affect the ionization energy besides the charge of the nucleus and the shielding. For example there may be energy associated with the spin pairing of electrons. Thus the formula should be

IE = (R/n²)p² − 2(R/n²)εp + ζ

where the ζ term includes the ε² term and any other factors affecting ionization energy.

Charge Shielding

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 also fits very well the cases of multiple electrons in the outer shells if charge shielding is taken into account. That shielding is by the electrons in the inner shells and may also be by electrons in the same shell. But the shielding by electrons in the same shell is likely only for a fraction of their charge. As it turns out, shielding even for electrons in the inner shells the shielding is less than the full value of their charges.

Here is the rationale for the shieldings. If inner shell electrons execute trajectories that take them over a spherical shell it is as though their charges are smeared over a spherical shell and their effect on outer shell electrons is the same as though the charges of the inner shell electrons are concentrated at the center of the atom or ion and thus cancel out an equal number of positive charges.

The electrons in a lower subshell may be entirely interior to an outer subshell. Thus the analysis should be in terms of shielding by electrons in inner shells and subshells versus shielding by electrons in the same subshell.

The shielding by electrons in the same subshell is a bit more complicated. The effect of a charge distributed over a spherical shell on an electron entirely within that spherical shell is zero. If the electron is entirely outside of the spherical shell the effect is the same as if the charge were concentrated at the center of the spherical shell. But if the center of the electron is located exactly on the shell then roughly half of the electron is inside of the spherical shell and is unaffected by its charge. Thus an electron is shielded by an amount approximately equal to one half of the charges in the same shell. That is the rough theory. It needs to be tested empirically.

This partial shielding by electrons in the same shell 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 protons. 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.

Empirical Analysis

The extended Bohr formula for ionization energy suggests a regression equation of the form

IE = c2p² + c1p + c0

Such a form does give a very good fit to the data. The value of ε can be found from the regression coefficients as

ε = −½c1/c2

For the second electron the quadratic regression that fits the data is

IE = 13.72673p² − 18.61822p + 8.80897
       [1607.6]     [-96.8]     [9.5]

Note that the Rydberg constant is 13.60569 eV.

The coefficient of determination for this equation is 0.999999673 and the standard error of the estimate is 0.9945 eV.

The figures in square brackets are the t-ratios for the regression coefficients 1above them; i.e., the ratios of the regression coefficients to their standard deviations. `A t-ratio of less than 2 would indicate that at the 95 percent degree of confidence the perceived relationship between the independent variable for the coefficient and the dependent variable is not real; i.e., is just due to chance. The t-ratios for the above regression equation are astronomically high.

The estimate of ε which comes from this equation

ε = −½(-18.61822)/13.72673 = 0.67817 electron charges.

Thus the shielding by another electron in the same shell is real and is approximately 0.5 electron charges.

The s-subshell of the second shell can contain at most two electrons. Therefore the third and fourth electrons are in that subshell. The "expected" shieldings for the third and fourth electrons are 2.0 and 2.5, respectively. This expectation is in the nature of "naively expected."

For the third electron (the first in the second shell) the regression equation is

IE = 3.43641p² − 11.32175p + 8.60728
       [2472.7]     [-347.3]     [51.5]

The coefficient of determination for this equation is 0.999999872 and the standard error of the estimate is 0.14129 eV.

The estimate of ε which comes from this equation

ε = −½(-11.32175)/3.436413 = 1.64732 electron charges.

Full shielding by the two electrons in the first shell would give ε=2.0.

The coefficient of p² is R/n² where n for the second shell is 2. Thus the R value for this case is 2²(3.436413)=13.7456517, close to the Rydberg constant of 13.60569 eV.

The fifth through tenth electrons are in the p-subshell of the second shell. Therefore their "expected" shieldings are 4.0, 4.5, 5.0, 5.5, 6.0 and 6.5, respectively. The shielding values found for the fifth through eighth electrons are 3.16858, 3.82658, 4.52719 and 5.31449. Thus there is a close match for the eighth electron. The measured shielding for the ninth electron is 5.99813, reasonably close to 6.0.

The measured shielding for the tenth electron is 6.81818, surprisingly exceeding the "expected" value 6.5.

Now it is worthwhile to apply the above methodology to the case of the first electron. There is no shielding in this case so ε should be zero. But it should not be applied to interval that includes the jump.

The regression equation for the first electron for p=1 through p=11 is

IE = 13.64589p² − 0.27454p + 0.36410
       [3434.8]     [-5.6]     [2.8]

The coefficient of determination for this equation is 0.999999965 and the standard error of the estimate is 0.11637 eV.

The estimate of ε which comes from this equation is 0.01006 charges. This is essentially zero, thus confirming the methodology.

The eleventh electron is in the third shell. There are ten electrons in the first two shells so the shielding would be expected to be 10.0. Its measured value is 8.17159, about 0.8 of its "expected" value.

The measured value of the shielding for the twelfth electron is 8.17159, about 0.8 of its "expected" value. 8.78703 charges, 0.61544 charges higher than the value for the eleventh electron.

The table of the above values and the values for subsequent electron positions is:

Shielding of Electrons in Different Positions in Atoms and Ions
Electron
Position
Inner
Electrons
Same
Shell
Electrons
Shielding"Expected"
Shielding
1000.0100.000
2010.6780.500
3201.6472.00
4212.2212.500
5403.1694.00
6413.8274.500
7424.5275.000
8435.3145.500
9445.9986.000
10456.8186.500
111008.1715910.000
121018.7870310.500
1310210.1325611.000
1410310.6186411.500
1510411.2619412.000
1610512.0162112.5
1710612.6834713.0
1810713.3301613.500
1918016.4575618.000
2018117.3761818.500

Before proceeding further it is necessary to note a feature of the electron structure of atoms beyond Argon (p=18). The 19th electron goes into the s-subshell of the fourth electronic shell even though there is additional capacity in the d-subshell of the third shell. It does this because of energy considerations.

The 20th electron also goes into the s-subshell of the fourth shell but subsequent electrons go into the d-subshell of the third shell. This creates a problem of the identification of the data for the 21st through the 30th electron. Apparently this is for the outermost electron. For the cases of 21 through 28 there are two electrons in the s-subshell of the fourth shell. Cu (p=29) is the anomaly it has only one electron in the s-subshell of the fourth shell. Zn (p=30) again has two electrons in the s-subshell of the fourth shell.

Shielding of Electrons in Different Positions in Atoms and Ions
Electron
Position
Inner
Electrons
Same
Shell
Electrons
Shielding"Expected"
Shielding
2119118.3330519.500
2220119.3266220.500
2321120.2348821.500
2422121.9943322.500
2523123.0111323.500
2624124.5748624.500
2725125.6736725.500
2826123.4936026.500
2928024.0509228.000
3028125.1722728.500

Shielding of Electrons in Different Positions in Atoms and Ions
Electron
Position
Inner
Electrons
Same
Shell
Electrons
Shielding"Expected"
Shielding
3130026.503530.000
3230127.2486630.500
3330228.4611431.000
3430329.0187931.500
3530429.9861032.000
3630529.9374032.500
3736034.9774236.000
3836137.4255736.500
3937138.6934337.500
4039039.1044839.000

Average Shielding Ratios

The "expected" shieldings of an electron based on the number of electrons in inner subshells and shells and the number of electrons in the same subshell are not precise estimates of the measured shieldings but are rough approximations. It might be the ratios used of 1.0 for the inner electrons and 0.5 for the same subshell electrons are not the best ratios for predicting the shieldings.

There are forty observations in the above empirical analysis which can be used to obtain the best ratios. Let Inner and Same be the number of the shielding electrons. Then the regression equation presumed is

ε = rI(Inner) + rS(Same)

There is no intercept constant for the equation because if there are no shielding electrons there is no shielding.

The regression results are:

ε = 0.94935(Inner) + 0.47607(Same)
       [92.1]     [5.6]

Lo and behold! The ratio values of 1.0 and 0.5 are very close to the best values for predicting the shielding of an electron. The coefficient of determination for the regression equation is 0.99685 and the standard error of the estimate is 1,20 eV. Thus about 99.7 percent of the variation in the shielding ε is explained by the variation in the numbers of shielding electrons.

If a constant is included in the regression the t-ratio for its estimate is −1.2, indicating the constant is not significantly different from zero at the 95 percent level of confidence.

(To be continued.)

Conclusions

The energy required to dislodge an electron from a position in an atom is determined by the net positive charge it experiences from the nucleus. The net experienced charge is the charge of the nucleus less the amount that it is shielded from by the electrons in the inner shells and subshells and also by the electrons in the same subshell. The shielding by the electrons in the same subshell is a fraction of their charge, roughly one half. The shielding by electrons in inner shells or shells is generally less than one for one, but roughly 0.95.

Fractional shielding accounts for the existence of negative ions.


Appendix

This data was compiled from a database published by the Physics Department of Ohio State University.

Ionization Energies (eV)
  Electron Number
Elem. Number of
Protons
1 2 3 4 5 6 7 8 9 10
H 1 13.598
He 2 54.416 24.587
Li 3 122.451 76.638 5.392
Be 4 217.713 153.893 18.211 9
B 5 340.217 259.368 37.93 25.154 8.298
C 6 489.981 392.077 64.492 47.887 24.383 11.26
N 7 667.029 552.057 97.888 77.472 47.448 29.601 14.534
O 8 871.387 739.315 138.116 113.896 77.412 54.934 35.116 13.618
F 9 1103.089 953.886 185.182 157.161 114.24 87.138 62.707 34.97 17.422
Ne 10 1362.164 1195.797 239.09 207.27 157.93 126.21 97.11 63.45 40.962 21.564
Na 11 1648.659 1465.091 299.87 264.18 208.47 172.15 128.39 98.91 71.64 47.286
Mg 12 2304.08 1761.802 367.53 327.95 265.9 224.94 186.5 141.26 109.24 80.143
Al 13 2673.108 2085.983 442.07 398.57 330.21 284.59 241.43 190.47 153.71 119.99
Si 14 3069.762 2437.676 523.5 476.06 401.43 351.1 303.17 246.52 205.05 166.77
P 15 3494.099 2816.943 611.85 560.41 479.57 424.5 371.73 309.41 263.22 230.43
S 16 3946.193 3223.836 707.14 651.63 564.65 504.78 447.09 379.1 328.23 280.93
Cl 17 4426.114 3658.425 809.39 749.74 656.69 591.97 529.26 455.62 400.05 348.28
Ar 18 4933.931 4120.778 918 854.75 755.73 686.09 618.24 538.95 478.68 422.44
K 19 5469.738 4610.955 1034 968 861.77 787.13 714.02 629.09 564.13 503.44
Ca 20 5129.045 1157 1087 974 895.12 816.61 726.03 656.39 591.25
Sc 21 926 829.79 755.47 685.89
Ti 22 940.36 861.33 787.33
V 23 974.02 895.58
Cr 24 1010.64 1010.64
Mn 25 1136.2 1136.2
Fe 26 1266.1 1266.1
Co 27 1403 1403
Ni 28 1547
Cu 29 1698
Zn 30 1856

Ionization Energies (eV)
  Electron Number
Elem. Number of
Protons
11 12 13 14 15 16 17 18
Na 11 5.139
Mg 12 15.035 7.646
Al 13 28.447 18.828 5.986
Si 14 45.141 33.492 16.345 8.151
P 15 65.023 51.37 30.18 19.725 10.486
S 16 88.049 72.68 47.3 34.83 23.33 10.36
Cl 17 114.193 98.03 67.8 53.46 39.61 23.81 12.967
Ar 18 143.456 124.319 91.007 75.02 59.81 40.74 27.629 15.759
K 19 175.814 154.86 117.56 100 82.66 60.91 45.72 31.625
Ca 20 211.27 188.54 147.24 127.7 108.78 84.41 67.1 50.908
Sc 21 249.832 225.32 180.02 158.7 138 111.1 91.66 73.47
Ti 22 291.497 265.23 215.91 193.2 168.5 140.8 119.36 99.22
V 23 336.267 308.25 255.04 230.5 205.8 173.7 150.17 128.12
Cr 24 384.3 355 298 270.8 244.4 209.3 184.7 161.1
Mn 25 435.3 404 343.6 314.4 286 248.3 221.8 196.46
Fe 26 489.5 457 392.2 361 330.8 290.4 262.1 235.04
Co 27 546.8 512 444 411 379 336 305 276
Ni 28 607.2 571 499 464 430 384 352 321.2
Cu 29 671 633 557 520 484 435 401 368.8
Zn 30 738 698 619 579 542 490 454 419.7

Ionization Energies (eV)
  Electron Number
Elem. Number of
Protons
19 20 21 22 23 24 25 26 27 28 29 30
K 19 4.341
Ca 20 11.871 6.113
Sc 21 24.76 12.8 6.54
Ti 22 43.266 27.491 13.58 6.82
V 23 65.23 46.707 29.31 14.65 6.74
Cr 24 90.56 69.3 49.1 30.96 16.5 6.766
Mn 25 119.27 95 72.4 51.2 33.667 15.64 7.435
Fe 26 151.06 125 99 75 54.8 30.651 16.18 7.87
Co 27 186.13 157 129 102 79.5 51.3 33.5 17.06 7.86
Ni 28 224.5 193 162 133 108 75.5 54.9 35.17 18.168 7.635
Cu 29 266 232 199 166 139 103 79.9 55.2 36.83 20.292 7.726
Zn 30 310.8 274 238 203 174 134 108 82.6 59.4 39.722 17.964 9.394
Ga 31 64 30.71 20.51
Ge 32 93.5 45.71 34.22
As 33 127.6 62.63 50.13
Se 34 155.4 81.7 68.3
Br 35 192.8 103 88.6
Kr 36 230.39 126 111
Rb 37 277.1 150 136
Sr 38 324.1 177 162
Y 39 374 206 191

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