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

1 Periodic classification of elements 2

2 Effective Nuclear Charge 2

3 Ionization Energy 5

4 Atomic and ionic radii 6

5 Electron affinity 7

6 Electronegativity 7

7 Metals and Non-Metals 8

8 Oxidation number 8

Appendix A Radial Schröedinger Equation 9

Appendix B Ionization energy in Bohr Model 12

1

1 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-know

s p

rules. A period is any horizontal row and it corresponds to the complete filling of and sub-shells;

n

the number of period corresponds to the principal quantum number of the level which is in phase

n = 2 2s

of filling. For instance, period 2 corresponds to the shell and to the filling of sub-shells

2p.

and In stead, 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

s-block

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 valence shell equal to

−

G 10 p-block G d-block.

equal to in the case of and to for Considering that valence shell of a

−

d-block ns (n 1)d

element is constituted by and orbitals, when we see, for instance, that Sc is

4s 3d

the first element of 3-group, it means that it has 3 valence electrons, two electrons and one

electron. Figure 1: General structure of periodic table

Elements within a group have similar properties. Since they share an analogous valence elec-

tronic structure and chemical bonds involves valence electrons, they have a similar chemistry. In

stead, along a period there’s a more accentuated variation of chemical properties.

2 Effective Nuclear Charge n

For an atom with an single electron, like the hydrogen atom, is the only quantum number

s, p, d f

connected to the energy of levels, that is and 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 Schröedinger equation,

E 1

E = n = 1, 2, 3, . . . (1)

n 2

n

−13.6

E =

with eV. Thus, the energy of each orbital increases as the distance from the nucleus

1 n).

increases ( higher 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

2

1

generated by the other electrons . Electrons nearer the nucleus reduces the positive charge, such

?

Z

that electrons in external orbitals feels an effective nuclear charge we are going to indicate as .

Then, we can define the effective nuclear charge as the real positive nuclear charge experienced by

?

Z

an electron. can be expressed as follows, ? −

Z = Z σ (4)

σ

where is a constant called screening Slater’s costant. It takes in 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 an hydrogen atom in the form, ψ = R Y (5)

nml nl lm

R n l, Y

where is the radial part, dependent to quantum numbers and and are spherical harmonics,

l m.

dependent to and According to statistical interpretation of quantum mechanics, the quantity,

2 2

|ψ (r, θ, φ)| r sin θdθdφdr (6)

nml ψ

represents the probability to find the particle described by within the elemental volume

2

r sin θdrdθdφ ( in polar coordinates). Since we are interested in the probability to find the particle

at a given distance to nucleus, regardless the direction (according to the previous definition of

penetration), we can integrate over angular coordinates,

π 2π

Z Z 2 2 2

|Y |

P (r) = dθ sin θ dφR r (7)

ml

nl

0 0

Since spherical harmonics are normalized in the sense that,

π 2π

Z Z 2

|

dφdθ sin θ|Y = 1 (8)

ml

0 0

we get, 2 2

P (r)dr = R r dr (9)

nl

2

2 r dr

P (r) = R is the radial distribution function, which multiplied by gives the

the quantity nl r r + dr. n = 1 l = 0

probability to find the particle between and For an orbital with and (see

Appendix A), it gives, 3

Z −2Zr/a

2

r e

P (r) = 4 (10)

0

a

0

r = 0 P (0) = 0, 1s-electron

At we get that is for an the probability to find it near the nucleus

vanishes. Some plots of radial distribution functions are given in Figure 2.

1 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, namely in second

quantization formalism, † †

† |0i |χ i

a . . . a = χ . . . χ

a (2)

i k l

i k l

χ χ

where is a spin-orbital. The total averaged potential acting on the electron in arising from the N-1 electrons

a

i

in the other spin-orbitals is given by, 2

|χ

Z (2)|

b

X

HF

V (1) = dx (3)

2

a r

12

b6 =

a

6

b = a

that is, by summing over all the one-electron potential obtained by averaging the interaction of electron 1

2

|χ

dx (2)| dx x

and electron 2, weighted by the probability that electron 2 occupies the volume element at . For

2 2 2

b

further details of this topic see "Modern Quantum Chemistry", by A.Szabo, N.Ostlund.

3

Figure 2: Plots of radial distribution functions for some hydrogenoid orbitals

r = 0, P (0) = 0, s-orbital

As we can see, even if at in general to is associated a greater radial

p-orbitals. s-electrons

probability near the nucleus than We can conclude that penetrates much

p-electrons d f

more than do. Same considerations hold for and electrons. Then, the effect of

s > p > d > f l.

penetration decreases as , that is it decreases with

σ

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

n

i) All the electrons in orbitals whose principal quantum number is higher than the one for

σ

which we are calculating do not contribute; σ

ii) Each electron with the same principal quantum number contributes to with a factor 0.35.

d f s p

However, if electron in question occupies a or -orbital, electrons in or orbitals with

the same quantum number contributes with 1.00 each one; −

n 1

iii) Each electron in orbitals with principal quantum number contribute each one with a

d f

factor 0.85, except ones in and orbitals, which contributes with 1.00;

n

iv) All the electrons with lower contributes 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,

2 2 6 2 6 2

1s 2s 2p 3s 3p 4s (11)

4s

Then, for an electron in orbital, we get,

· · ·

σ = 1 0.35 + 8 0.85 + 10 1 = 17.15 (12)

Ca,4s

Z = 20

Since, we find the following effective nucleare charge,

? −

Z = Z σ = 2.85 (13)

Ca,4s

?

Z Z

As expected, increases along a group and along a period (since itself increases much more

?

Z increases because the single electron added step by

than Slater’s constant). Along a period,

step is not completely screened by the nucleare charge "produced" by the other core electrons.

Of course n