Periodic Properties of Elements

Periodic properties of elements

Mirko Leccese

March 28, 2017

Abstract

The aim of these notes is to present a general overview of the periodic properties of elements, in order to rationalise their positions on Mendeleev’s table. Mainly, we will focus on atomic dimensions, electronegativity, ionization energies and electron affinities.

Contents

• Periodic classification of elements
• Effective nuclear charge
• Ionization energy
• Atomic and ionic radii
• Electron affinity
• Electronegativity
• Metals and Non-Metals
• Oxidation number
• Appendix A Radial Schrödinger Equation
• Appendix B Ionization energy in Bohr Model

Periodic classification of elements

The shape of the periodic table is related to the electronic structure of elements. Thus, for example, a block indicates the type of orbital, filled in sequence by electrons, according to the well-known rules. A period is any horizontal row and it corresponds to the complete filling of sub-shells; the number of a period corresponds to the principal quantum number of the level which is in phase of filling. For instance, period 2 corresponds to the shell and to the filling of sub-shells.

The number of a group or family (G), any vertical column, is related to the number of electrons filling the valence shell, that is the more external one. IUPAC recommends the use of the system 1-18 to enumerate groups. Within this convention, each element of G (alkali metals and alkaline earth metals) has a number of electrons in the valence shell equal to G. Considering that the valence shell of a d-block element is constituted by ns and (n-1)d orbitals, when we see, for instance, that Sc is the first element of the 3-group, it means that it has 3 valence electrons, two s electrons and one d electron.

Elements within a group have similar properties. Since they share an analogous valence electronic structure and chemical bonds involve valence electrons, they have similar chemistry. Instead, along a period there is a more accentuated variation of chemical properties.

Effective nuclear charge

For an atom with a single electron, like the hydrogen atom, n is the only quantum number connected to the energy of levels, that is, p, d, and f sub-levels have the same energy. We remind the expression of bound state energies for the hydrogen atom (see Appendix A, B), as obtained by the time-independent Schrödinger equation:

E_n = -13.6/n² eV, n = 1, 2, 3, ...

Thus, the energy of each orbital increases as the distance from the nucleus increases (higher n). In the case of polyelectronic atoms, the situation becomes quite complicated due to the interactions between electrons (the problem of electronic correlation). The first effect of this is the vanishing of degeneracy. Qualitatively, we could say that each electron moves within the attractive field of the nucleus and, at the same time, it experiences a mean nuclear charge generated by the other electrons. Electrons nearer the nucleus reduce the positive charge, such that electrons in external orbitals feel an effective nuclear charge we are going to indicate as Z^*. Then, we can define the effective nuclear charge as the real positive nuclear charge experienced by an electron.

Z^* = Z - σ

where σ is a constant called the screening Slater's constant. It takes into account two fundamental facts: i) the already cited screening effect produced by other electrons; ii) the different ability of electrons to "penetrate" internal shells. The former effect can be understood considering the wavefunction for a hydrogen atom in the form:

ψ_nml = R_nlY_lm

where R is the radial part, dependent on quantum numbers n and l, and Y are spherical harmonics, dependent on l and m. According to the statistical interpretation of quantum mechanics, the quantity:

|ψ(r, θ, φ)|²r²sinθdθdφdr

represents the probability to find the particle described by ψ within the elemental volume r²sinθdrdθdφ (in polar coordinates). Since we are interested in the probability to find the particle at a given distance to the nucleus, regardless of the direction (according to the previous definition of penetration), we can integrate over angular coordinates:

P(r) = ∫₀^π ∫₀^2π |Y_lm|² sinθ dθ dφ R_nl²r² dr

Since spherical harmonics are normalized in the sense that:

∫₀^π ∫₀^2π |Y_lm|² sinθ dθ dφ = 1

we get:

P(r)dr = R_nl²r² dr

The quantity P(r) = R_nl²r²dr is the radial distribution function, which multiplied by r² gives the probability to find the particle between r and r+dr. For an orbital with n = 1 and l = 0 (see Appendix A), it gives:

P(r) = 4Z³r²e^-2Zr/a₀

At r = 0, P(0) = 0, for a 1s-electron, the probability to find it near the nucleus vanishes. Some plots of radial distribution functions are given in Figure 2.

This "mean field approximation" is the theoretical basis of Hartree-Fock method. We remind that, within this method, the global wave function is assumed to be approximated by a single Slater's determinant. The total averaged potential acting on the electron in a_i arising from the N-1 electrons in the other spin-orbitals is given by:

V_HF(1) = ∑_b≠a ∫ dx₂ |χ_b(2)|²/r₁₂

that is, by summing over all the one-electron potential obtained by averaging the interaction of electron 1 and electron 2, weighted by the probability that electron 2 occupies the volume element at x. For further details of this topic see "Modern Quantum Chemistry", by A. Szabo, N. Ostlund.

As we can see, even if at r = 0, in general to s-orbitals is associated a greater radial probability near the nucleus than p-orbitals. We can conclude that s-electrons penetrate much more than p-electrons. Same considerations hold for d and f electrons. Then, the effect of penetration decreases as l increases, that is, it decreases with s > p > d > f.

Slater’s constant can be calculated according to these four following rules:

• All the electrons in orbitals whose principal quantum number is higher than the one for which we are calculating do not contribute;
• Each electron with the same principal quantum number contributes to σ with a factor 0.35. However, if the electron in question occupies a d or f-orbital, electrons in s or p orbitals with the same quantum number contribute with 1.00 each one;
• Each electron in orbitals with principal quantum number n - 1 contributes each one with a factor 0.85, except ones in d and f orbitals, which contribute with 1.00;
• All the electrons with lower n contribute with 1.00 each one.

These rules are also known as Slater's rules. To show their application we consider the case of Ca. Its complete ground state electron configuration is:

1s² 2s² 2p⁶ 3s² 3p⁶ 4s²

Then, for an electron in the 4s orbital, we get:

σ = 1 × 0.35 + 8 × 0.85 + 10 × 1 = 17.15

Since Z = 20, we find the following effective nuclear charge:

Z^* = Z - σ = 2.85

As expected, Z^* increases along a group and along a period (since Z itself increases much more than Slater's constant). Along a period, Z^* increases because the single electron added step by step is not completely screened by the nuclear charge "produced" by the other core electrons.