# Energy of ionization

The potential of ionization or energy of ionization of a Atom or a Molécule is the energy necessary to tear off a electron to him. More generally, the nth energy of ionization is necessary energy to tear off the nth electron after the $n-1$ first electrons were torn off. In physical Chemistry, the concept of energy of ionization is the reverse of that of Electronic affinity, i.e. of its propensity to yield or on the contrary to retain an electron.

The reaction of ionization of the atom has is written:

$A_ \left\{\left(G\right)\right\} \left\{\ rightarrow\right\} A^+_ \left\{\left(G\right)\right\} + e^-$

## General information

The energy of ionization is expressed in eV or Joule or kilojoule/mole (kJ/mol). 1 electronvolt is very close to 100 kJ/mol. It is a size which is always positive, which means that it is always necessary to provide energy to an atom to tear off one to him (or several) electrons. The energy of ionization varies according to the atom or from the molecule considered, as well as its state of ionization.

One can ionize an atom having more than one electron in several stages. For example, a boron atom five electrons: two in an internal layer (1s2) and three in the layer of valence (2s2 and 2p1). The energy of ionization of order N is energy necessary successively to separate N electrons from the atom. The energy of the first ionization varies much according to the atoms. The energy of ionization increases along a line of the periodic table of the elements period then decreases abruptly when one passes to another line.

The torn off electron that one considers in the concept of energy of ionization comes from the layer of valence. But it can be made that an electron of the deep layers of the atom is torn off without the electrons of the surface layers being it beforehand; in this case the electrons reorganize then, giving place to a Rayonnement (Fluorescence X).

## Numerical values of energies of ionization

Generally, energies of ionization decrease along a column of the periodic Tableau of the elements and grow from left to right along a line of the table. The energy of ionization shows strong a anticorrelation with the atomic Rayon. Successive energies of ionization of a given element increase gradually, as one can see it on the table below. The increase is particularly strong when after the complete exhaustion of an atomic layer of Orbitale, one passes to a new layer. This comes owing to the fact that when all the electrons of orbital were extracted, the energy of following ionization will consist in extracting an electron from orbital nearer to the core, where the electrostatic force which binds the electron to the core is more intense.

In the table below, one gives some values for the third line of the periodic table.

The energy of ionization is a good indicator to determine how much electrons has an element given on its external layer. It is advisable to observe starting from how much successive ionizations occurs the significant jump corresponding to the passage of the external layer to the following layer. For example, if one needs 1.500 kJ/mol to tear off an electron and 5.000 kJ/mol to tear off the second, and then 6.000 kJ/mol for the third, that wants to say that the external layer has a single electron. It is thus a metal which will yield an electron easily. Once a stable byte was made up, it becomes much more difficult to tear off the following, but on the other hand, once this electron was withdrawn, the following will be slightly easier to tear off.

## Electrostatic interpretation and semi-traditional model

The atomic energy of ionization can be calculated starting from the electric Potentiel and of the Modèle of Bohr of an atom.

One considers an electron of load - E and a Ion with a load +ne , where N is the missing number of electrons to the ion. According to the model of Bohr, if the electron approached, there could remain related to the atom with a certain ray has . The electrostatic potential V at the distance has core is written:

$V = \ frac \left\{1\right\} \left\{4 \ pi \ epsilon_0\right\} \ frac \left\{\right\} \left\{does not have\right\} \, \!$

given that the zero of the potential are referred ad infinitum.

As the electron is negatively charged, it is attracted by this positive potential. (It is the potential of ionization ). Energy necessary to leave the well of potential and to leave the atom is given by:

$E = eV = \ frac \left\{1\right\} \left\{4 \ pi \ epsilon_0\right\} \ frac \left\{ne^2\right\} \left\{has\right\} \, \!$

It is the energy of ionization, which is equal to the potential of ionization if it is expressed in electronvolt S

At this stage of the traditional approach, the analysis is still incomplete since the distance has remains an unknown variable. It is then advisable to associate with each electron of a chemical element given a selected characteristic distance so that the expression of the potential of ionization is in agreement with experimental data.

A semi-traditional approach based on the assumption of Bohr extends the validity of the traditional model by quantifying the Quantité of movement. This approach is very well checked for the hydrogen atom which has one electron. The intensity of the Angular momentum for a circular orbit is:

$L = |\ vec \left(R\right) \ times \ vec \left(p\right)| = rmv = N \ hbar$

The total energy of the atom is the sum of its energies potential U and kinetics T , i.e.:

$E = T + U = \ frac \left\{p^2\right\} \left\{2m_e\right\} - \ frac \left\{ke^2\right\} \left\{R\right\} = \ frac \left\{m_e v^2\right\} \left\{2\right\} - \ frac \left\{ke^2\right\} \left\{R\right\}$
Speed can be eliminated from the term corresponding to the kinetic energy while posing that Coulomb attraction must be compensated with the Centrifugal force:

$T = \ frac \left\{ke^2\right\} \left\{2r\right\}$
What then makes it possible to express energy according to K , E , and R .

$E = - \ frac \left\{ke^2\right\} \left\{2r\right\}$
The quantification of the momentum expressed some lines higher according to the assumption of Bohr then makes it possible to write:

$\ frac \left\{n^2 \ hbar^2\right\} \left\{rm_e\right\} = ke^2$
From where one draws the relation between N and R :

$r \left(N\right) = \ frac \left\{n^2 \ hbar^2\right\} \left\{km_e e^2\right\}$
One calls the ray of Bohr a0 the ray for which N is equal to 1. One can then express the equation of energy by calling upon the ray of Bohr:

$E = - \ frac \left\{1\right\} \left\{n^2\right\} \ frac \left\{ke^2\right\} \left\{2a_0\right\} = - \ frac \left\{13.6eV\right\} \left\{n^2\right\}$
One can extend this model to other cores that of hydrogen by introducing the Atomic number into the equation.

$E = - \ frac \left\{Z^2\right\} \left\{n^2\right\} \ frac \left\{ke^2\right\} \left\{2a_0\right\} = - \ frac \left\{13.6 Z^2\right\} \left\{n^2\right\} eV$

## Simple but rigorous approach of Feynman

The disadvantage of the semi-traditional approach, which makes the implicit assumption of an electron in orbit around the core, with the centrifugal force which is opposed to the attraction force, following the example satellite in Orbite is that it is proven since the beginning of the 20th century that it is erroneous: an electron in orbit would not fail to radiate, and would crumble on the core according to a trajectory in spiral. Feynman showed that it was not necessary to make this assumption to estimate the ray of the hydrogen core:

By recalling that the total energy of the system core + electron is

$E = \ frac \left\{h^2\right\} \left\{2ma_0^2\right\} - \ frac \left\{e^2\right\} \left\{a_0\right\}$

" We do not know what a0 is worth, but we know that the atom will be arranged to make a kind of compromise so that its energy is as small as possible " , Feynman writing in its famous " Readings one Physics"

$\ frac \left\{of\right\} \left\{da_0\right\} = - \ frac \left\{h^2\right\} \left\{ma_0^3\right\} + \ frac \left\{e^2\right\} \left\{a_0^2\right\}$

By writing that the value of this derivative is null, one obtains the value of a0

$a_0= \ frac \left\{h^2\right\} \left\{me^2\right\} =$ 0.0528 nanometers

## The energy of ionization in quantum mechanics

The model of Bohr is not completely in conformity with the theory of the quantum Mécanique, described better by the Modèle of Schrödinger according to which the localization of the electron is not described in a deterministic way, but like a " nuage" localizations equipped with a certain probability of being more or less close to the core. This more rigorous approach is also more complicated, but one can give some tracks to approach it: The cloud corresponds to a Fonction of wave or, more precisely with a linear Combinaison of the determining of Slater, i.e., according to the Principe of exclusion of Pauli, of the antisymmetric products of the atomic Orbitale or the molecular Orbitale. This linear combination is a development in Interaction of configuration S of the electronic function of wave.

In the general case, to calculate N the ième energy of ionization, it is necessary to withdraw the energy of a system of $Z-n+1$ electrons of a system of $Z-n$ electrons. The calculation of these energies is not simple, but it is about a rather traditional problem of what one calls the computational Chimie, i.e. the study of chemistry by numerical digitalization. At first approximation, the energy of ionization can be deduced from the Théorème of Koopmans

## Notes and sources of the article

In its version of February 2007, this article must much with the article corresponding of the wikipedia in English language.
 Random links: Warrant | Ehud Tenebaum | The Dawn (film, 1927) | Gottfried Honegger | Billy Kidd | Federico_Confalonieri