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Potentials of Atoms and Atomic Ions
The Bohr model of a hydrogen-like ion predicts that the total energy E is given by
where Z is the number of protons in the nucleus, 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
where e is the charge of the electron and rn is the orbit radius when the principal quantum number is n. The orbit radius is given by
h is Planck's constant divided by 2π and me is the mass of the electron.
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 8 in the third shell and 18 in each of the fourth and fifth shells.
The term hydrogen-like ion means that there is but one electron in a shell, the inner shells being 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.
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 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
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. As will be seen below 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 for the first eighteen elements from the CRC Handbook of Physics and Chemistry 82nd Edition (2001-2002) are shown below.
Z Element I II III IV V VI VII VIII 1 H 13.59844 2 He 24.58741 54.41778 3 Li 5.39172 75.64018 122.45429 4 Be 9.3227 18.21116 153.89661 217.71865 5 B 8.29803 25.15484 37.93064 259.37521 340.22580 6 C 11.2603 24.38332 47.8878 64.4939 392.087 489.99334 7 N 14.53414 29.6013 47.44924 77.4735 97.8902 552.0718 667.046 8 O 13.61806 35.1173 54.9355 77.41353 113.899 138.1197 739.29 871.4101 9 F 17.42282 34.97082 62.7084 87.1398 114.2428 157.1651 185.186 953.9112 10 Ne 21.5646 40.96328 63.45 97.12 126.21 157.93 207.2759 239.0989 11 Na 5.13908 47.2864 71.62 98.91 138.4 172.18 208.5 264.25 12 Mg 7.64624 15.03528 80.1437 109.2655 141.27 186.76 225.02 265.96 13 Al 5.98577 18.82856 28.44765 119.992 153.825 190.49 241.76 284.66 14 Si 8.15169 16.34585 33.49302 45.14181 166.767 205.27 246.5 303.54 15 P 10.48669 19.7694 30.2027 51.4439 65.0251 220.421 263.57 309.6 16 S 10.36001 23.3379 34.79 47.222 72.5945 88.053 280.948 328.75 17 Cl 12.96764 23.814 39.61 53.4652 67.8 97.03 114.1958 348.28 18 Ar 15.75962 27.62967 40.74 59.81 75.02 91009 124.323 143.46
Column I gives the energy in electron volts (eV) required to separate one electron from the neutral atom. Column II is the energy required to separate a second electron; i.e., to separate one more electron from the singly ionized element. And so forth.
The data can be rearranged and displayed in terms of the number of electrons in the shell, which is the more relevant quantity:
Number of Electrons in the Second Shell Element 8 7 6 5 4 3 2 1 O 13.61806 35.1173 54.9355 77.41353 113.899 138.1197 F 17.42282 34.97082 62.7084 87.1398 114.2428 157.1651 185.186 Ne 21.5646 40.96328 63.45 97.12 126.21 157.93 207.2759 239.0989 Na 47.2864 71.62 98.91 138.4 172.18 208.5 264.25 Mg 80.1437 109.2655 141.27 186.76 225.02 265.96 Al 119.992 153.825 190.49 241.76 284.66 Si 166.767 205.27 246.5 303.54
The number of electrons in a shell is the complement of the separation order. That is to say, the figure of 113.899 eV for oxygen in the V column is the energy required for the separation of an electron when there are two electron in the second shell. (There are also two electrons in the first shell.)
It was indicated above that the shielding by electrons in the same shell implies the ionization potential should decrease with the number of electrons in the shell. Here are the graphs for Oxygen, Fluorine and Neon.
The shielding effect of electrons in the same shell is thus borne out.
The regularity of the effect is revealed when the curves for the three elements are displayed in the same graph, as below.
According to the notion that an electron in the same shell shields a half unit of positive charge in the nucleus the effect from an increase in the positive charge of the nucleus should have twice the magnitude of the the effect on ionization energy of a unit increase in the number of electrons in a shell. For oxygen for only one electron in the second shell the ionization energy is 138.1197 eV, so the increase to two electrons in the second shell decreased the ionization energy by 24.2207 eV. On the other hand the increase in the ionization energy when the proton number is increased from 8 for oxygen to 9 for fluorine and number of electrons in the second shell is maintained as two is (157.1651-113.899)=43.2661 eV. The ratio of these two numbers is 0.560. It is not 0.500 but it is reasonably close. When the same computation is made for the fluorine ion with two electrons in the second shell the ratio is 0.559. For neon the same computation produces a figure of 0.559.
When the computation is carried out for different numbers of electrons in the second shell the ratio are quite different from 0.5. The ratio vary with the number of electrons in the shell but the values are very close for the different elements.
Number of Electrons in the Second Shell Element 8 7 6 5 4 3 2 O 1.006860003 0.718282345 0.697982257 0.990665034 0.559807794 F 0.745439979 0.973959924 0.709975706 0.693700058 0.982491439 0.55917886 Ne 0.754172725 0.733500518 0.949520587 0.704699612 0.690015227 0.975793949 0.558552044 Na 0.740584284 0.724920641 0.932247403 0.698511166 0.687358062 0.970240167 Mg 0.730816622 0.718241901 0.924217798 0.695636364 0.686452046 Al 0.723313736 0.712702887 0.915372255 0.694399482
This ratio, which could be described as the shielding factor for electrons, has the averages shown below:
Number of electrons in the shell 8 7 6 5 4 3 2 0.737221842 0.726961185 0.950362995 0.703584112 0.69110153 0.979797647 0.559179566
The graph of the data is as follows:
The values of the shielding factor suggest that when a second electron is added to the shell it is at nearly the same distance from the center of the atom as the first. When a third is added it is closer to the center than the first two and its shielding factor is one. For the fourth its distance is slightly less than the first two but greater than the third so its charge is spread over a range that puts 70 percent closer to the center than the first two electrons in the shell, thus shielding 0.7 of a positive charge. The same occurs for the fifth electron in the shell. The sixth electron is like the third in that its charge is closer to the center than the first two and it has a shielding factor of nearly one. The seventh and eight are like the fourth and fifth, being closer to the center than the first two but not so much closer that they have a shielding factor of one; instead their shielding factor is about 0.73.
(To be continued.)
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