Relationship between the Atomic Structure and Electrochemistry. 1. Electric Force, Standard Reduction Potential E°, and Standard Reaction Gibbs Free Energy ΔG°.

The relationship among the standard reaction Gibbs free energy ΔG°, the standard reduction potential E°, and the atomic structure parameters of radius, nuclear charge, and isoelectronic orbitals nl is accomplished through the attraction electric force F elec. In relationship with E°, it was necessary to define two new reference scales: E 0 ° with a final state of E° in the element, which allowed to have a parabolic trend of ΔG° versus F elec, and E °,0 whose final state is the ion with a more negative charge (e.g., -1, -2, -3). The relationship with ΔG° is related to the concept of chemical stability, and the relationship with E °,0 is more related to the concept of electronegativity. In relationship with ΔG°, it was necessary to predict the values of possible new cations and noncommon cations in order to find a better trend of ΔG° versus F elec, whose stability is analyzed by Frost diagrams of the isoelectronic series. This dependence of ΔG° on F elec is split into two terms. The first term indicates the behavior of the minimum of ΔG° for each isoelectronic orbital nl, while the second term deals with the parabolic trend of this orbital. For the minima of the configuration np6, a hysteresis behavior of the minima of ΔG° is found: an exponential behavior from periods 1 and 2 and a sigmoidal behavior from periods 5 and 4 to interpolate period 3. It is also found that the proximity of unfilled np or (n + 1)s orbitals induces instability of the ion in configurations ns2/nd2/4f2 and nd10/nd8(n + 1)s2, respectively. On the contrary, the stability of the orbitals np6 does not depend on the neighboring empty (n + 1)s0 orbitals. Both phenomena can be explained by the stability of the configuration of noble gas np6 and the nd10(n + 1)s2 configuration. We have also found that it is possible to increase the reduction potential E °,0 (macroscopic electronegativity), although the electric force F elec decreases because the orbital overlap influences the electronegativity.

Introduction

At present, the analysis of the electrochemical phenomena is currently made through the standard reduction potential E°(Z/Z), which is a macroscopic average variable used when an element is reduced from an oxidation state n′ to an oxidation state n through the reception of (n – n′) electrons. Thus, for instance, in cerium, in aqueous acidic solution at pH = 0

We can define the standard reduction potential relative to zero from the oxidation state Z to the oxidation state Z0 named E0°(Z/Z0) or simply E0°. Thus, the standard reaction Gibbs free energy of the reduction potential E0° with respect to Z0, ΔG0,Red°(Z/Z0), or simply ΔG0,Red° is also evaluated in order to associate the macroscopic thermodynamic parameters to electrochemistrywhere F is the Faraday constant which is equivalent to 96.485 kJ/mol/V or eV/V and n is the number of electrons transferred. Note that n can be negative.

We also know that the standard reaction Gibbs free energy of oxidation ΔG0,Ox° is the negative of ΔG0,Red°. Thus

ΔG0,Ox°/F has been selected because the stability of the cations is commonly plotted through the Gibbs free energy in the Frost diagrams, and its minimum indicates the highest cation stability. We also consider that the Frost diagrams with ΔG0,Ox°/F provide a deep insight into the electrochemical behavior of the elements. For example, ΔG0,Ox°/F indicates the importance of the metals in superconductivity alloys[1] and hence (1) the great stability of La3+, Ba2+, and Y3+ (with a concomitant high negative value of ΔG0,Ox°/F) and the highest negative reduction potential E0° (the highest negative slope of ΔG0,Ox°/F), as shown in Figures S1 and S2). (2) The lower proclivity of Ag and Au, and noble metals Pd and Pt to be used as superconductor materials are correlated to the high value and positive slope of ΔG0,Ox°/F, indicating a striking tendency to reduction (Figure S2). (3) The topological similitude in the electrochemical behavior of Cu, Tl, and Hg (Figures S3 and S4), in particular, Hg+ with Hg22+.[2] On the other hand, there is a parabolic trend of the energy E on the formal charge Q. In this case, the Gibbs free energy of oxidation ΔG0,Ox° for the isoelectronic series 4s2/3d2 is shown in Figure S5 of the Supporting Information.[3] Importantly, the isoelectronic series only indicates the same number of electrons in the element or ion not the electronic configuration. Thus, the virtual isoelectronic configurations ns2 can take the electronic configuration ns2 or the electronic configuration (n – 1)d2 and in the 6th period the electronic configurations 6s2 or 4f2. The same happens in the virtual isoelectronic configuration 4f146s2 that can take the electronic configuration 4f146s2 or the electronic configuration 4f145d2. The criterion for selecting a specific electronic configuration is its lower energy in relation to the other configurations. In this context, the orbital energies of the one-electron configurations ns1 and (n – 1)d1 taken from Mann et al.[4] and reported in Table S2 are used as reference for the selection of the electronic configuration of the ion or element in an isoelectronic series. Besides, we consider that the energy of the orbital 4f is similar to the orbital 5d with 4f ≲ 5d < 6s2 in the transition elements, as explained by Huheey et al.[3a] and Mann et al.[4] Thus, the selection of the electronic configuration affects the value of the effective nuclear charge Z*, and jumps are possible. Nevertheless, Felec, in general, has continuity in the isoelectronic series because the trend of radius counterbalances the jumps of Z*, as observed below. In the case of the isoelectronic configuration nd10, for example, in 3d10, the sequence of Ga3+, Zn2+, and Cu+ (all with configurations 3d10) follows Ni0, but this element has a configuration [Ar] 3d84s2 (in reality, [Ar] 3d8.4694s1.531).[4] The other elements of group G10, Pd and Pt, also have the same characteristic with a tendency to have the d10 electronic configuration (Pd: [Kr] 4d9.6674s0.333 and Pt: [[Xe] 5d9.2096s0.791).[4] In this case, we have evaluated the configuration nd8(n + 1)s2 for Z* for Ni, Pd, and Pt. The confirmation of this assignation was corroborated by the trend of the other cations in the electric force Felec, as observed below. In order to systematize the notation in an isoelectronic series, the representative configuration is written first and the next possible electronic configurations are written separated by a slash, for example, ns2/(n – 1)d2/4f2, where the representative configuration is ns2. Besides, the Gibbs free energy of oxidation ΔG0,Ox is not affected by the selection of the electronic configuration because it considers the experimental oxidation state and not the electronic configuration. In this context, the reduction potential E0° can be defined as the derivate of the standard reduction potential ΔG0,Ox° or at least an average of the increment or decrement of ΔG0,Ox° per electronwhere NA is Avogadro’s constant. Thus, it is expected that the reduction potential E0° has a lower curved trend than ΔG0,Ox°, as shown in Figure S5 of the Supporting Information. Even, a departure from the linear behavior indicates a more asymmetric behavior of the parabolic behavior of ΔG0,Ox°. Besides, as observed in Figure S6, there is almost a linear trend of decrease of E0° from the anions to the element until the monovalent cation. This linear trend indicates that the average of the reduction potential per interchanged electron decreases until the monovalent cation. The following linear trend of increase of E0° with an increase of oxidation of the isoelectronic series indicates an increase of the positive reduction potential per electron interchanged. On the other hand, the absolute electronegativity of an element χ may be defined as the variation of the energy E with the change of the formal charge Q on an atom or ion, that is, the electronic chemical potential μ or Lagrange multiplier with the constraint of N electrons at constant external one-potential V.[3a,5]where δ indicates the functional derivative[5c] and ∂E indicates a positive increase of the energy with the increase of ∂N electrons, and we must consider the discreteness of the number of electrons N in an isolated molecule but not in solution.[5e−5g]

If we relate the energy E to the standard Gibbs free energy of oxidation ΔG0,Ox° and eqs and 5, we observe that the reduction potential E0° is linked to the concept of electronegativity χ and has an approximately linear behavior with respect to Z in an isoelectronic series, as shown in Figure S6, for the isoelectronic configuration 4s2/3d2. Thus, E0° is the normalization of the value of ΔG0,Ox° per electron and can be considered as a macroscopic electronegativity. Besides, the separation of the linear trend indicates the perturbation of the atomic structural factors in the electronegativity, the lowest value of E0° in the monocation indicates a lower electronegativity, and the increase of E0° in the higher oxidation states indicates a higher electronegativity, as shown in Figure S6.

We have also defined a reduction potential that starts from the cation and ends at the element (Z0) for the cations and that starts from the element and ends at the anion, named E°,0. The relation of the values for cations and anions for this reduction potential E°,0 is more related to the ranking of electronegativity: higher for the elements which are prone to be converted to anions and lower for the elements which tend to form cations, as shown in Figure S9, for the isoelectronic configuration 3p6: (1) a more positive value of E°,0 specifies an increment of the macroscopic electronegativity, that is, an increase of the macroscopic power to attract the electrons = more easiness of being reduced. (2) A more negative value of E°,0 indicates less macroscopic electronegativity, that is, more capacity to release the electrons = more easiness of being oxidized. On the other hand, a negative value ΔG0,Ox° for a cation indicates that it is stable, and a positive value for a cation indicates that it is unstable. Then, ΔG0,Ox° determines the macroscopic stability of the cations, and E°,0 determines the macroscopic power to attract or release the electrons. We must note that the values for E0° are equal to E°,0 for cations and opposite in sign to E°,0 for anions. Thus, there is an abrupt change of the Gibbs free energy of the reaction associated with E°,0 named ΔG°,0,Ox, as shown in Figure S10, as compared with ΔG0,Ox° of Figure S11, for example. Accordingly, we have used the trend of ΔG0,Ox° because it gives a more continuous trend of the values in the isoelectronic configuration ns2/(n – 1)d2/4f2 and np6. Indeed, the Frost diagrams of the first 80 elements of the periodic table, without counting the noble gases, give better continuity of ΔG0,Ox°/F than that of ΔG°,0,Ox/F with the exception of N, As, Se, Sb, Au, Bi, and Po.

In general, the expectation is that an increment of the absolute electronegativity increases the reduction potential E0° (=E°,0 in the case of cations, see above). For example, the Pauling electronegativity values of sodium and silver are χP,Na = 0.93 and χP,Ag = 1.93, respectively, and the corresponding values of the standard reduction potential in aqueous acidic solution at pH = 0 are E0°(Na+/Na0) = −2.71 V and E0°(Ag+/Ag0) = 0.80 V. However, this rule does not apply in other cases, for example, the absolute electronegativity of Li is higher than that of Na, but the reduction potential E0° is lower: χP,Li = 0.98 and E0°(Li+/Li0) = −3.04 V. These discrepancies could be explained based on the atomic structure. In this case, the more negative standard reduction potential E0°, which indicates a lower macroscopic power to attract electrons, can also be explained by the elevated higher hydration of Li+, which has a lower radius (128 pm) than Na+ (166 pm), and then the attraction potential becomes higher. There are cases where this rule is not valid, which can be observed when the radii in Figure S12 and their standard reduction potential E0° in Figure S13 are compared: in the case of the isoelectronic configuration 1s2, the radii of C4+ (CO2) and B3+ are lower than those in Be2+ and Li+, as shown in Figure S12. However, B3+ and C4+ have a higher standard reduction potential E0° than Be2+ and Li+, as shown in Figure S13. Then, we consider that the reduction potential E0° depends not only on the hydration radius but also on the physicochemical factors such as the orbital overlap, the nephelauxetic effect, the width of the band gap, the low value of energy of the highest occupied molecular orbital orbitals, the ligand field stabilization energy, the penetration of the orbitals, and the lattice energy. These factors are explained in an article that has been submitted for publication in order to propose a new more applicable scale of electronegativity.[6]

In principle, the electronegativity can be defined for elements or ions.[3,7] Besides, the electronegativity for the elements according to Alfred and Rochow (in Pauling units) χP,AR is directly proportional to the electric force Felec.[3a] Then, we can make the relationship of the electronegativity to the charge of the nucleus and the radius of the element through the evaluation of Felecwhere

Z* is the effective nuclear charge given by the rules of Slater[3a] with the supposition that the additional electron of the other atom also participates in the screening of the nuclear charge.[3a]Rcov is the radius of the covalent bond of the elements. The values of Z* for all the elements, cations, and anions of the isoelectronic series and lanthanides are given in Figure S14A–C of the Supporting Information. On the other hand, the definition of the Mulliken electronegativity in the scale of Pauling χP,M is[3a]where EIv is the ionization energy of the electron in the valence orbital and AEv is the electronic affinity of the valence shell. EIv indicates how much the electron is attracted to the nucleus and AEv indicates how much the electron is held by the atom.[8] Thus, if we match eqs and 8, we can conclude that the evaluation of Felec corresponds to the effect of both EIv and AEv, that is, the attraction and holding of the valence electrons, respectively. Indeed, it is also possible through Felec to consider the electronegativity of cations and anions because the Slater rules also consider the nuclear effective charge Z* for cations and anions. Thus, the electronegativity of Os8+ is χP,AR,Os8+ = 48 and that of Cl5+ is χP,AR,Cl5+ = 37 in the scale of Pauling when we transform the corresponding values of Felec (5p6, Figure S15A) in χP,AR (3s2, Figure S16A) through eq . We consider that the electronegativity values obtained by the approximation of Allred and Rochow are overestimated because a linear relationship between the electric force Felec and electronegativity is considered. Sanderson has explained that when an ion is bound to another ion, the electrons flow from the more electronegative ion to the less electronegative ion until their electronegativities are equalized. Indeed, the parabolic trend of the cations was shifted to a lower value of ΔG. Nevertheless,[9] Thus, the calculated electronegativity for ions in the Alfred Rochow approximation would be the value of the atom or ion when it is beginning to accept or donate electrons.

In this work, we use the periodicity of the standard reaction Gibbs free energy of oxidation ΔG0,Ox° in the isoelectronic series of the orbitals nl (e.g., nd10/nd8(n + 1)s2). Thus, the isoelectronic configurations allow to have a continuity in the values of ΔG0,Ox° (as shown in Figure S5) and Felec (as shown in Figure S15A–C) between the periods of the periodic table. This continuity in the periodicity is broken in other physical properties such as radius or electronegativity and even Z* (Figure S14A–C) as a function of the nuclear charge Z. Thus, in the configuration 3p6, the values of ΔG0,Ox°, Felec for the anion S2–, the noble gas Ar, the alkaline earth cation Ca2+, and the transition metal cation Mn7+ are related. Similarly, for the electric force Felec, the anion Au– is related to the cation At5+ in the isoelectronic configuration 5d106s2. Thus, we analyze the relationship between anions, elements, and cations. Besides, we have found that the kind of isoelectronic orbital nl determines the relationship between ΔG0,Ox° and Felec, and two fittings are necessary: the fitting of the trend of the minima of the curve of ΔG0,Ox° and the fitting from the minimum of the curve of ΔG0,Ox° for a specific isoelectronic orbital nl. In order to split the cations with higher incertitude in the electronic configuration because they have not been found experimentally and cations which have been found experimentally, whose electronic configuration can be determined experimentally, we have named the first cations as possible new cations and the second ones as noncommon cations. These cations can have a filled or unfilled electronic configuration. Besides, the methodology used to evaluate the parameters of this work is given after the Conclusions section.

Results and Discussion

Relation of ΔG0,Ox° with Z

Through the periodicity of the isoelectronic series of the nl orbitals for the electronic configurations ns2/(n – 1)d2/4f2, np6, nd10/nd8(n + 1)s2, nd10(n + 1)s2, 4f14, and 4f146s2/4f146d2 (see above), we have found the physicochemical properties of possible new cations (Table ) and of noncommon cations (Table ) not reported in the literature. In the case of the values of ΔG0,Ox°, we have used the periodicity of the values of columns and rows for ΔG0,Ox° and E0° of the isoelectronic series and the trend of the respective Frost diagram for an element.

Table 1

Parameters of Possible New Cations Predicted by the Periodicity of ΔG0,Ox°, Felec, and Radius

orbital	cation	Z	Z*	ΔG0,Ox°/F/V	E0°/V	Felec/μN	R/pm
2s2	B+	5	2.11	–1.53	–1.53	110	67
4d2	Sc+	21	6.71	–2.47	–2.47	87	134
5d2	Y+	39	16.01	–3.06	–3.06	203	135
6s2	La+	57	19.66	–3.50	–3.50	81	160
4f2	Ce2+	58	20.66	–5.80	–2.90	291	128
4f14	Tm+	69	28.5	–3.34	–3.34	300	148
4f146s2	Lu+	71	8.67	–1.110	–1.11	151	115

Table 2

Parameters of Noncommon Cations Predicted by the Periodicity of ΔG0,Ox°, Felec, and Radius

orbital	cation	Z	Z*	ΔG0,Ox°/F/V	E0°/V	Felec/μN	R/pm
3s2	Al+	13	3.72	–1.3	–1.3	106	90
5d2	Zr2+	40	14.17	–4.20	–2.10	334	93
4f146s2	Hf2+	72	9.02	–1.56	–0.78	208	100

Concerning all the isoelectronic configurations, we have found a parabolic functionality for ΔG0,Ox° at pH = 0 of the isoelectronic nl orbitals versus Z for a certain range of Z, as shown in Figure . This parabolic behavior has also been found for the ionization of the electronic affinity curve.[3a,3b,10] For the isoelectronic ns2/(n – 1)d2/4f2 orbitals, a parabolic trend of ΔG0,Ox° for their minima with the increment of Z is observed, as shown in Figure . Indeed, there are two parabolas for the minima of the electronic configuration ns2/(n – 1)d2/4f2, which are joined in the extremes by Ti2+ (Z = 22, 3d2) as seen in Figure . Importantly, the value of Ti2+ (ΔG0,Ox°/F = −3.26 V) was found by the trend of the Frost and reduction potential E0° diagrams for the isoelectronic configuration 4s2/3d2 and also for the different oxidation states of Ti at pH = 0 at standard conditions, as shown in Figures S5–S8, respectively. This value indicates the higher stability of this ion. On the other hand, the proximity of the empty np0 orbital to the occupied ns2 orbitals for n = 2, 3 produces a low increase of the stability for their minima with an increase of Z: B+ (Z = 5, 2s2, ΔG0,Ox°/F = −1.526 V) and Si2+ (Z = 14, 3s2, ΔG0,Ox°/F = −1.62 V). Then, after Si2+, there is a higher increase of stability for Ti2+ (Ti2+, Z = 22, 4d2, ΔG0,Ox°/F = −3.26 V). This cation has unfilled s and d orbitals. Thus, the lower value of ΔG0,Ox° for Ti2+ with a higher curvature in the first parabola indicates that the stability of the cation at the minima in 2s2 and 3s2 and 4d2 increases with an increment of Z when there is a change of a proximal p0 orbital to unfilled s and d orbitals. It is also observed that the curvature of the second parabola of the minima of the configurations nd2 or 4f2 in n = 4, 5, 6 is lower with higher Z than that of the first parabola, and then, the increment of stability is also lower. On the other hand, for the filled isoelectronic np6 orbitals, the minima of the parabolas of ΔG0,Ox° with the increment of Z has an exponential decay: Be2+ (Z = 4, 1s2) > Al3+ (Z = 13, 2p6) > Sc3+ (Z = 27, 3p6) > Y3+ (Z = 39, 4p6) > La3+ (Z = 57, 5p6), as shown in Figure . We have included the cation Be2+ because the sequence of 4p6, 3p6, 2p6 orbitals followed the 1s2 orbital which is the ns2 orbital more similar to the np6 orbitals than to the other ns2 orbitals. As observed, the trend of the np6 isoelectronic configurations reaches Be2+. Also, the exponential decay indicates an increment of the stability of the cations of the minima of the filled isoelectronic np6 orbitals with an increment of the period because of an increment of the radius, which allows more stability of cations (less attraction of electrons), as it is expected.

Figure 1

Periodicity of ΔG0,Ox° vs Z of the isoelectronic series ns2 [representative, rather ns2/(n– 1)d2/4f2] and np6 configurations for anions, elements, and cations in aqueous solution at pH = 0. The values of ΔG0,Ox° for the five possible new cations (cyan) and two noncommon cations (black) are shown in circles.

On the other hand, in the particular case of S2–, although the oxide was supplanted by a hydride in H2S in the isoelectronic configuration 3p6, the trend was also present, as observed in Figure . Thus, there is a trend of the anion S2– with the other anions, such as Si4–, P3–, and Cl–, which form hydrides. Indeed, the parabolic trend of the cations was shifted to a lower value of ΔG0,Ox°. Nevertheless, we have used the anion S2– and the other anions with a negative oxidation state in the derivation of the equations of the parabolic trend. Besides, it is also observed in Figure that the value of the Gibbs free energy of oxidation ΔG0,Ox° of Ar obtained in the gas phase has a sequential trend with anions and cations because of the physical state of Ar in the gas phase. This value of ΔG0,Ox° for Ar was obtained by the trend of the thermochemical data,[11] and, as explained, it is more congruent with the isoelectronic trend of 3p6 than that of ΔG0,Ox° = 0 for an element.

Figure 2

Periodicity of the Gibbs free energy of oxidation ΔG0,Ox° vs Z of the isoelectronic configuration 3p6 in aqueous solution at pH = 0. The pink square indicates the value of ΔG0,Ox° predicted for Ar0 in the gaseous state through thermodynamic data.

On the other hand, for the isoelectronic nd10/nd8(n + 1)s2 orbitals, the trend of the minimum of ΔG0,Ox° with the increment of Z was a slightly concave parabola with a positive slope, as shown in Figure : Ga3+ (Z = 31, 3d10) > In3+ (Z = 49, 4d10) > Au+ (Z = 79, 5d10). This indicates an increase in the tendency to be reduced from the cations. If we compare the value of ΔG0,Ox° of Figures and 3, we observe that the values of the configuration nd10/nd8(n + 1)s2 are more positive, indicating that these ions are more prone to be reduced with the increment of Z. We consider that the nd10/nd8(n + 1)s2 orbitals tend to get the nd10(n + 1)s2 configuration, resulting in their instability. Thus, the proximity of the partial empty (n + 1)s orbitals allows an increase of the possibility of reduction of these cations with the increment of the period. An example of this trend is the reduction of Ag+ (4d10) and Au+ (5d10). The relativistic expansion and contraction of orbitals d and s, respectively, produce a superior band gap between the d and s orbitals for Ag+ than for Au+, as reported by Mann.[3,4,12] For this reason, Au+ (5d10) is more easily reduced than Ag+ (4d10). The decrement of the band gap in Au+ is also aided by its relativistic spin–orbit splitting of d orbitals and the indirect relativistic shell expansion of these orbitals.[12]

Figure 3

Periodicity of ΔG0,Ox° vs Z of the isoelectronic series nd10/nd8(n + 1)s2, nd10(n + 1)s2, 4f14, and 4f146s2/4f146d2 for anions, elements, and cations in aqueous solution at pH = 0. The values of ΔG0,Ox° for the two possible new cations (cyan) and two noncommon cations (black) are shown in circles.

For the filled isoelectronic nd10(n + 1)s2 orbitals, the trend of the minimum of ΔG0,Ox° with the increment of Z was almost linear, but in this case, the slope was slightly negative and almost 0, as shown in Figure : Ga+ (Z = 32, 3d104s2) > Sn2+ (Z = 50, 4d105s2) > Tl+ (Z = 81, 5d106s2). The difficulty of the reduction (acceptation of electrons) of these cations at the minimum can be assigned to the nd10(n + 1)s2 electronic configuration, not as stable as the configuration ns2np6 when their corresponding values of ΔG0,Ox° are compared, as shown in Figures and 3. Indeed, the filled 4d105s2 and 5d106s2 orbitals suffer a relativistic contraction of the 5s2 and 6s2 orbitals.[4,13] This contraction is less in 5s2 than in 6s2 (comparing the energy of 5s2 in Cd to the energy of 6s2 in Hg).[4] For example, in the case of Sn2+ (4d105s2) and Pb2+ (5d106s2), the lower band gap is between the s and p orbitals in Sn2+ than in Pb2+, as shown in Figure 11 of ref (13). Therefore, the lower band gap in Sn2+ increases the possibility of the reduction of Sn2+ in comparison to that of Pb2+. The increase of electronegativity (possibility of reduction) with a decrease of the band gap is due to the inverse variation of the hardness η (half of the band gap) to the electronegativity[5c,6]

In the case of noble gases, we have interpolated the values of ΔG0,Ox°, and the predicted values are (/kJ mol–1 K–1) as follows: 132 (He), 0.238 (Ne), −218 (Ar), −193 (Kr), −145 (Xe), −77 (Rn). The trend is shown in Figures and 4. The negative values of ΔG0,Ox° indicate that they are stable for oxidation. On the other hand, the trend of the filled isoelectronic np6 orbitals for elements and anions, labeled with X0, X–, and X2– as shown in Figure , is an increment of the value of ΔG0,Ox° with the period n. This increment of ΔG0,Ox°, with a concomitant increase of the stability of elements and anions to rest in states of reduction, probably happens because they can release less electrons with the increment of the radius. This trend is opposite to the stability of the cations for the minima of configuration np6, which increases with the increment of the period n, as seen above (lower value of ΔG0,Ox°). On the other hand, higher stability is observed for Pt in the isoelectronic configuration 5d86s2, more exactly [Xe] 5d9.2096s0.791,[4] as shown in Figure , that is, Pt is more prone to remain reduced = not to be oxidized = more stable to the oxidation. The opposite happens with Pd (4d85s2, more exactly [Kr] 4d9.6674s0.333),[4] as also shown in Figure . A middle value is found in Ni (3d10, more exactly [Ar] 3d8.4694s1.531).[4] The more stability of Pt (5d86s2) is produced by the difficulty of accepting electrons in the 6s orbital as a consequence of the relativistic contraction of the orbital 6s and expansion of the orbital 5d.[3,4,13,14] We consider that the proximity of the partially filled d and s orbitals avoids the acceptation of the electrons in the d orbital of the configuration [Xe] 5d9.2096s0.791 of Pt. Similarly, the orbital configuration [Kr] 4d9.6675s0.333 in Pd is less prone to be reduced because of a lesser contraction of the orbital 5s2 with a subsequent higher band gap. Also, the higher stability of the orbital d in the electronic configuration [Ar] 3d8.4694s1.531 of Ni, due to its lower energy than the orbital s in comparison to Pd and Pt,[4] counterbalances the higher band gap between these orbitals, and then ΔG0,Ox is intermediate between Pd and Pt. Besides, it is also observed in Figure that the stability remains constant with ΔG0,Ox° = 0 (no tendency to be oxidized and no tendency to be reduced) in the case of elements X0 of configuration ns2/(n – 1)d2/4f2, nd10(n + 1)s2, 4f14, and 4f146s2/4f146d2, with no tendency to be reduced in the noble gases. In the case of the anions of the configuration ns2 (with the trend given by the line X–), the higher value of ΔG0,Ox° indicates that these anions are more stable than the anions of the other isoelectronic configurations.

Figure 4

Periodicity of ΔG0,Ox° vs Z for elements and anions in aqueous solution at pH = 0. The charge is defined with n in X. The dashed lines show the trend between the different filled isoelectronic nl orbitals. The predicted values of ΔG0,Ox° for the noble gases are shown in big circles (cyan).

Relation of E0° and E°,0 with Z

The evaluation of the trends of periodicity for E0° with Z was based on the isoelectronic orbitals nl: ns2/(n – 1)d2/4f2, np6, nd10/nd8(n + 1)s2, nd10(n + 1)s2, 4f14, and 4f146s2/4f146d2 as it was made in ΔG0,Ox°. Figure shows the values of E0° for these orbitals: (a) the dashed line indicates the trend of the minima for the np6 orbital, which corresponds to the point of change from a decrease of E0° with an increase of the oxidation state to an increase of E0° (abrupt change of the slope). It is also remarkable that the values of the mínima are almost constant. As also observed, the filled isoelectronic configuration np6 includes the isoelectronic configuration 1s2 (see Li+ in Figure ), and then, all the minima lay in the same dashed curve. This is also observed in the previous section for ΔG0,Ox° versus Z where the minimum of 1s2 is in Be2+, as shown in Figure , but in this case, Li+ is the minimum of the isoelectronic 1s2 orbital for E0° versus Z. (b) The dotted–dotted–dashed line indicates the values of ΔG0,Ox° = 0 for the configuration ns2, which corresponds to the elements Be, Mg, Ca, Sr, and Ba (X0). (c) The dotted line indicates the values of the anions of the alkali metals (X–). All these values allowed to have a continuity in the trend of the ns2/(n – 1)d2/4f2 electronic configuration and then to predict the values of the cations B+, Al+, Sc+, Y+, and La+. We also used the periodicity of ΔG0,Ox° and the trend of the configurations ns2/(n – 1)d2/4f22 for n = 4, 5, 6.

Figure 5

Periodicity of E0° vs Z of the isoelectronic configurations ns2/(n – 1)d2/nf2 and np6 in aqueous solution at pH = 0. The circles in cyan are the possible new cations (5), and the circles in black (2) are the values of the noncommon cations.

The reduction scale E°,0, as explained above, is more related to the concept of electronegativity. Besides, a more positive value of E°,0 increases the power to form the anion and a more negative value increases the power to be oxidized, and a value of zero indicates no tendency to be reduced or to be oxidized. Thus, it is observed in Figure that the noble gases (gray squares) have a higher reduction potential E°,0, especially Ne, and La+ (new cation) has the highest tendency to be oxidized (to La3+), with a lower value of E°,0. We consider that it would be interesting to react Ne and La+, seeking the possibility to form a compound, in which the configuration goes from the configuration 3p6 to the 4s1. Besides, it is also observed that the cations X+ (red dashed line below the X– line) of the configuration np6 and anions X– of the configuration ns2, which correspond to the same elements of Li, Na, K, Rb, and Cs, have similar negative values of E. A little less negative the anions, as seen in Figure . Thus, the cations X+ of the isoelectronic configuration ns2/(n – 1)d2/4f2 have a less macroscopic electronegativity than the anions X–.

Figure 6

Trend E°,0 vs Z of the isoelectronic configurations ns2/(n – 1)d2/nf2 and np6 in the aqueous solution at pH = 0. The circles in cyan are the possible new cations (5), and the circles in black (2) are the values of the noncommon cations. The noble gases are marked by gray squares.

We have applied the same methodology of first using the trend of the reduction potential E0° and then to find the relation of electronegativity by the reduction potential for the configurations nd10/nd8(n + 1)s2, nd10(n + 1)s2, 4f14, and 4f146s2/4f145d2. Figures S17 and S18 are plotted in the Supporting Information. Figure S18 shows that the reduction potential E°,0 of the elements Ni, Pd, and Pt is sequentially more negative (less electronegativity, less power to attract electrons), indicating that Pt is more susceptible to form hydrides. Besides, as observed in Figure and explained above, the sequence of stability, that is, an increase in the negative value of ΔG0,Ox°, is Pt < Ni < Pd. On the other hand, Er and Au+ have a more positive E°,0, as shown in Figure S18, with an added possibility to be reduced. In the case of gold, it is known that it tends to rest as an element.

Relation of ΔG0,Ox° with Felec

One of the objectives of this work is to find a relation between the parameters of the atomic structure of the atom (effective nuclear charge Z*, radius, and type of orbital). For this reason, we have evaluated Felec following eq . We have found, as expected, an exponential variation of Felec with Z in almost all the orbitals, as shown in Figure S15 of the Supporting Information. The next step was to relate ΔG0,Ox° with Felec. The last one is, in general, proportional to the electronegativity, as seen above through eq . We have found that this relationship was dependent on the isoelectronic nl orbitals. With this constraint, we have evaluated ΔG0,Ox° in two forms. First, we have evaluated the relationship of the minima of ΔG0,Ox° versus Felec for each isoelectronic nl orbital (eq , the first term on the right). Then, we have found a general variation of isoelectronic nl orbitals since the minimum (eq , the second term on the right). Second, we have considered the tabulated value of ΔG0,Ox° for the cation, which has the minimum of ΔG0,Ox° in a specific isoelectronic orbital (e.g., 3s2) (eq , the first term on the right), and then applied the general variation of the nl orbitals with respect to the minima (eq , the second term on the right): we must note that an isoelectronic series can include several electronic configurations such as ns2/(n – 1)d2/nf2.where ΔG0,Ox°(Felec[Zmin,]) = ΔG1° is dependent on the electric force Felec of the cations at the minimum of ΔG0,Ox°(Z) for each isoelectronic orbital nl; ΔG0,Ox°(Felec,Zmin,) = ΔG2° is the parabolic dependence of ΔG0,Ox° to the electric force Felec of an isoelectronic orbital nl with the vertex in Zmin; and ΔG1° is the tabulated ΔG0,Ox° for Zmin. It must be emphasized that the first term on the right of eqs and 11 gives the value of ΔG0,Ox° for Zmin, and the second term gives the change of ΔG0,Ox° on Felec with respect to Zmin.

In order to obtain the first term of the right side of eq , we have interpolated the values of ΔG1° with Felec for Zmin of the isoelectronic orbitals; however, the isoelectronic configuration for ns2/(n – 1)d2/4f2 does not allow establishing a trend, as shown in Figure a, as a consequence of the changes of configuration among the periods. In the isoelectronic series ns2/(n – 1)d2/nf2, there is an increment of ΔG1° with a decrement of Felec in B+, Si2+, and Ti2+ (a decrement of stability of the oxidation state), which is shown in the pink square of Figure a. We consider that the volume of the empty np0 and unfilled (n – 1)d orbitals increases the value of ΔG1° in B+, Si2+, and Ti2+ with a concomitant decrease of Felec, respectively. In other words, the possibility of reduction is increased because the new charge can be dispersed in the empty np0 and unfilled (n – 1)d orbitals. On the contrary, the cations Be2+ and Zr2+ and Pr3+ have a lower value ΔG1° with a higher Felec, as shown in the blue square of Figure a (an increase of the stability of the oxidation state). This behavior is opposite to the relation of Felec to electronegativity: an increment of Felec increases the capacity of reduction of the cation. In the case of Zr2+ and Pr3+ with configurations [Kr] 4d2 and [Xe] 4f2, respectively, the less proclivity of dispersion of the new charge on the energetic degenerate orbitals d and f produces the high stability of these ions. We consider that the similar behavior of Be2+ is due to its electronic configuration of the noble gas of Be2+, [He], that gives more stability to the cation. On the other hand, with respect to the configuration np6: (1) if a cation with an isoelectronic configuration np6 belongs to the 2nd and 3rd periods of the periodic table, an exponential decay of ΔG1° with Felec is found from Be2+ to Al3+, as shown in Figure b, as expected from the relation of Felec versus electronegativity. (2) If a cation with an isoelectronic configuration np6 belongs to the 3rd and 4th periods, an exponential decay versus Felec is predicted from Al3+ to Sc3+ or a sigmoidal growth versus Felec from La3+ to Al3+, including Sc3+. This indicates a zone of transition by the appearance of the orbital 4d and hysteresis in the value of ΔG1° versus Felec, as shown in Figure b. (3) If a cation with an isoelectronic configuration np6 belongs to the 4th, 5th, and 6th periods of the periodic table, with the (n – 1)d and 4f orbitals, the change of ΔG1° is sigmoidal with Felec with a remarkable similarity of ΔG1° between Y3+ and La3 and almost a vertical change from these cations to Sc3+, as shown in Figure b. (4) A regular behavior of the minima of the filled electronic configuration np6 is observed: an increase of the possibility of reduction with an increase of the electric force, different from the behavior of the configuration ns2/(n – 1)d2/4f2.

Figure 7

Interpolation of the data of ΔG1° with Felec for the cations of the isoelectronic orbitals (a) ns2/(n – 1)d2/4f2 and (b) np6.

In the case of nd10/nd8(n + 1)s2 orbitals, the dependence of ΔG1° for the minima of the parabolas versus Felec is almost linear, with a negative slope, as shown in Figure S19 of Supporting Information, Section S-5, indicating that the reduction of the cations is diminished (better stability) with a higher Felec (higher macroscopic power of attraction of electrons and higher macroscopic electronegativity). This trend is opposite to the trend related to Z for nd10/nd8(n + 1)s2. This decrease of ΔG1° in the orbital nd10/nd8(n + 1)s2 with the increase of Felec contradicts the asseveration that when Felec augments, its electronegativity increases, and its stability goes up. We consider that the relativistic effects in In3+ and Au1+ produce an increase of their values of ΔG1° (an increment of the reduction power, an increment of the macroscopic electronegativity, and an increment of the instability as cations), although these cations have a less value of Felec than Ga3+. On the other hand, the values of ΔG1° for the cations of the minima of the nd10(n + 1)s2 configuration (Ga+, Sn2+, Tl+) are similar [when we compare them to the scale of the nd10/nd8(n + 1)s2 configuration], as shown in Figure S20 of the Supporting Information. This happens although the electric force Felec varies. This is indicative that the proximity of filled d10 and p0 empty orbitals produces stability to oxidation and reduction of these cations because their ΔG1° values are almost 0 (a little more stable to be reduced because of ΔG1° < 0). Indeed, Ga+ is more prone to be oxidized, as shown in Figure S19. As a note, the higher electric force Felec in the cation of Sn is due to its higher oxidation state (2+) than those of Ga (1+) and Tl (1+).

In order to obtain the second term of the right side of eq , ΔG2°, we have normalized the nuclear charge Z to (Z – Zmin) for each isoelectronic nl orbital. In principle, we have found an exponential growth trend of Felec versus (Z – Zmin) and a parabolic trend of ΔG2° versus (Z – Zmin), for example, for nd10/nd8(n + 1)s2, as shown in Figures S15B, 1, and 8. The other nl isoelectronic orbitals are shown in Supporting Information, Section S-7 in Figures S21–S25. We have also fitted the cations, elements, and anions, and this gives a parabolic behavior in ΔG2°. The anions in an acidic aqueous environment pH = 0 are more stabilized, and then the value of ΔG2° becomes more negative, as shown in Figure for the isoelectronic configuration np6. In particular, we observe a lower value of Felec in the configuration 5d10/5d86s2 with respect to the exponential growth curve fitting, as shown in Figure a. The decrement of Felec in the configuration 5d10/5d86s2 can be associated with the expansion of the radius of d orbitals because of relativistic effects,[13] as shown in eq . It is also possible that this expansion of the radius is due to the delocalization of the charge in the next 6s orbital because of the tendency to fulfill the configuration 5d106s2. Also, the electric force Felec of the nd10/nd8(n + 1)s2 isoelectronic series decreases from n = 3 to n = 5 in relation to Z – Zmin, as shown in Figure a. In the case of n = 3 and 5, this is due to the high oxidation state of Zmin in n = 3 (Ga3+) than that of n = 5 (Au+).a Besides, a lower value of ΔG2° versus Z – Zmin than the overall trend is observed, as it happens in relation to the electric force Felec, for the configuration 5d10/5d86s2 in Figure b, for example, in Bi5+ (Z5+, Z = 83, Z – Zmin = 3, ΔG2°/F = 0.98 V, 5d10). This indicates that the cations in the configuration 5d10/5d86s2 are, with respect to their minimum of ΔG2°, more stable with the addition of electrons in order to occupy the 6s orbital (lesser possibility of reduction) than the other periods (nl = 3d10/3d84s2, 4d10/4d85s2) because the relativistic contraction of the orbital 6s and expansion of the orbital 5d decrease the capability of reduction. Probably, the high dispersion of the new charge in orbitals s and d does not allow this reduction. As a consequence, the minima of the configuration 3d10/3d84s2 (Zmin at Ga3+) and configuration 4d10/4d85s2 (Zmin at In3+) have a different oxidation state than the configuration 5d10/5d86s2 (Zmin at Au+). This indicates additional stability at a low oxidation state in gold for the configuration 5d10. There is also an abrupt increment of ΔG2° in platinum Pt from its minimum (Z0, Z = 78, Z – Zmin = −1, ΔG2°/F = 3.7 V, 5d86s2), as shown in Figure b, that is, an abrupt decrement of possibility to remain oxidized in comparison to the isoelectronic configurations 3d10/3d84s2 and 4d10/4d85s, as explained above.

Figure 8

Behavior of (a) Felec and (b) ΔG2° vs Z – Zmin for the isoelectronic nd10/nd8(n + 1)s2 orbitals.

On the basis of the relationships of Felec versus (Z – Zmin) and ΔG2° versus (Z – Zmin), we have found the relationship of ΔG2° versus Felec. Thus, the parabolic average behavior of ΔG2° versus Z in the isoelectronic configurations ns2/(n − 1)d2/4f2 and np6 in Figures 21b and 22b, respectively, is contracted in their relationship of ΔG2° versus Felec in Figure . This contraction is influenced by the atomic structure represented by Felec. We also found a higher value of ΔG2° of the anions at lower Felec, that is, the further possibility to remain reduced, probably due to the overlapping of the orbitals with a lower ionic interaction. We must also emphasize that the general trends of the isoelectronic ns2/(n – 1)d2/4f2 and np2 orbitals hide the particular behavior of a specific nl configuration, as also shown in Figure .

Figure 9

Behavior of ΔG2° vs Felec for the isoelectronic electronic configurations (a) ns2/(n – 1)d2/4f2 and (b) np6 orbitals. The continuous line is the average.

Figure shows all the average values of ΔG2°. We can say that Figure gives the reference value at the minimum of ΔG°(ΔG1°), which results from the increment of the effective nuclear charge and the variation of radius represented by Felec, taking into account the behavior of the overall nl orbitals (in the different periods). On the other hand, Figures and 9 give the change in the value of ΔG0,Ox° with the electric force Felec, (ΔG2°), in an nl isoelectronic configuration. Thus, we have found that there is an increment of ΔG2° when Felec is almost 0 in the following sequence: ns2/(n – 1)d2/4f2 > np6 > 4f14 > nd10/nd8(n + 1)s2 > 4f146s2/4f145d2 ≈ nd10(n + 1)s2, as shown in Figure (this trend is of the average of ΔG2°). This behavior is due to the fact that the ns2/(n – 1)d2//4f, np6, 4f146s2/4f145d2 and nd10(n + 1)s2 orbitals at Felec near 0 are more prone to be reduced by the filling of the empty np0 or nd0 or nonempty 4d and 4f, ns0, 4f146s2nd0, and nd10(n +1)s2np0 orbitals, respectively. On the other hand, with the increment of Felec, ΔG2° goes through a minimum in all of these orbitals, as shown in Figure . After the minimum, there is an increment of ΔG2° in the sequence 4f146s2/4f145d2 > nd10(n + 1)s2 > np6 ≈ nd10/nd8(n + 1)s2 > ns2/(n – 1)d2/4f2 > 4f14 (in reference to their minimum of ΔG2°). This sequence indicates the highest possibility of the cations of the isoelectronic configuration 4f146s2/4f145d2 to be reduced in the average and the lowest possibility of the cations of the isoelectronic configuration 4f14 in the average to be reduced after their minima of ΔG2° versus Felec.

Figure 10

Behavior of ΔG2° vs Felec for the isoelectronic nl orbitals.

Thus, we can evaluate ΔG0,Ox°(Z) by using Figures and 10 (eq ) for ns2/(n – 1)d2/4f2 and np6. The other method also uses Figure , but the tabulated values for the minima of the isoelectronic nl orbitals are employed.

Frost Diagram of the Possible New Cations

The analysis of the stability of the possible new cations can be evaluated through the Frost diagrams of ΔG0,Ox°/F(Z) = nE0° (eq ) versus the electric charge Q, as shown in Figure . It is seen that the cations that comproportionate (they are more stable) are as follows: Ce2+ > Tm+ ≈ La+ ≈ Y+ > Sc+ > B+ and the ion that disproportionate is Lu+ (less stable). Thus, the more stable cation is Ce2+.

Figure 11

Frost diagram of the predicted cations in aqueous solution at pH = 0.

We must also take into account that internal and external factors can influence the stability of the ions and the macroscopic electronegativity, such as the orbital shielding, the polarizability of the electronic cloud, the hydration of the ions, the similarity of the band gap, the relativistic effects with contraction of the s orbital and the expansion of the d orbital, the more penetration of the s orbitals, which increases the electronegativity, the orbital overlapping, the electronegativity alternation of the elements, the formation of multiple bonds, the bond length, the silicon rule, π backbonding, the lattice energy with the Madelung constant, the Racah’s constant, the spin of the orbital, the inaccuracy of the metal bond energy, and so forth. Thus, the above factors form a pool for future technological and scientific research.

Conclusions

Commonly, it is said that qualitatively the first elements of the transition series in a period can be easily oxidized, which decreases in the half of the period in the d5 configuration, and the reactivity is lower in elements with almost a filled configuration d10.[3a] Concerning the oxidation states, a common trend is that the low ones tend to be reducing and the higher ones to be oxidizing, such as happens to Mn, Co, and Ni in Figure S3. Thus, Frost diagrams are used for the analysis of the reduction potential of the different oxidation states for the same element.[15] It is also explained in the literature that high oxidation states are more common in the half of a period of the transition elements near to the electronic configuration (n – 1)d5ns2 (e.g., V, Cr, Mn in the 4th period).[16] Following our methodology, we have found quantitative curves of chemical behavior given by the stability (ΔG0,Ox°) and the possibility of reduction (E°,0) for the isoelectronic series, with the last concept related to the electronegativity. Thus, we analyze the stability of ions with Frost diagrams of ΔG0,Ox°/F versus the oxidation states of isoelectronic series with respect to Z. The parabolic trends indicate, for example, a higher stability to the oxidation of the anions and elements in the isoelectronic configuration ns2/(n – 1)d2/4f2 than in the configuration np6. In particular, the configuration ns2/(n – 1)d2/4f2 has a higher tendency to reduce in lower and higher oxidation states than the configuration 4p6 considering the minima of their isoelectronic parabola. Thus, we have analyzed the effect of structural factors such as the radius, the effective nuclear charge, and the kind of orbitals joined all in the electronic force Felec to predict the stability and reduction potential of the elements based on the Frost diagrams. Besides, the periodicity of ΔG0,Ox° and E° versus Z for the isoelectronic nl orbitals can also help to predict the respective values of possible new cations and noncommon cations and noble gases. We have determined that the dependence of ΔG0,Ox° versus Z (or Felec) indicates that the form of the orbital, its penetration, the proximity of other unfilled orbitals of low energy, and radial probability density influence the value of the reduction potential. Then, we have found that the cations tend to be or to go into the configuration of noble gas ns2np6 or an nd10(n + 1)s2 configuration, which are wells of stability in periods 4–6, as indicated by the first ionization energy.[17] We have also found that averaging the change of ΔG0,Ox° per electron, that is, the analysis of the reduction potentials E0° and E°,0 versus Z, is more related to the concept of electronegativity, and a nonlinear trend indicates a variation of the effect of structural factors on the electronegativity, as explained in the paragraph related to eq .

Methodology

In the evaluation of Felec versus ΔG0,Ox° for cations and anions and elements, we have used the following:

The radius of Shannon and Prewitt in octahedral coordination.[3a,15,18] When this was not possible, the tetrahedral coordination was employed, which has a lower ionic radius, and then Felec increased. For the elements, we have used the values of their covalent radius given in the Cambridge Structural Database.[19] In the case of anions, we have also obtained the anionic radii values from Pauling.[20]

Tables for evaluating the nuclear effective charge Z*. In order to have a more exact evaluation of the nuclear effective charge Z* up to Z = 86, the diagrams of Clementi and Raymondi were employed, which are considered more rigorous than the Slater evaluation.[3a,21] They proposed their evaluations for cations and neutral atoms, but their results can also be applied to anions, as can be concluded when one evaluates the Slater formulae and the results of Clementi and Raymondi’s equations and diagrams in the first three periods of the periodic table. However, the ratio of Z* for Clementi and Raymondi in relation to the values of Slater in higher periods reached up to 3. It has been remarked that the values of electronegativity from the Alfred and Rochow (Equation ) and the Pauling scale are more used by chemists and physicists.[7,22] The Slater equations give a lesser value of Z*, but the general behavior of Felec versus ΔG0,Ox° of both procedures is similar. In the case of elements with nuclear charge Z > 36, Clementi and Raymondi’s diagrams were the only available for these authors. We also corrected the deshielding by the decrement of electrons in the same orbital, for example, orbital d (see Supporting Information, Section S-3 for more information and the corresponding shield constants for each orbital). On the other hand, we consider that the values of the radius of elements and ions and nuclear effective charge Z* are well approximated in comparison to modern evaluation methods, although the derivation was from the 1960s.

Experimental values of the standard reduction potentials E0° in acidic aqueous solutions (pH = 0) from the tables of Bratsch, Bard, Atkins, and Pourbaix.[2,11,15,23] The values of the cations are mainly for the oxides, a middle strength field ligand, while we do not specify another thing. For the anions, we used the values of the hydrides. We can say that the periodicity of ΔG0,Ox° gives the trend of stability of the ions, elements, and anions in an isoelectronic series. On the other hand, the periodicity of E0° and E°,0 gives a trend more related to the concept of macroscopic electronegativity, which is influenced in this case by the hydration of the ions or anions, the nephelauxetic effect, and overlap of the orbitals, for example.