# Ionization energy

The ionization energy (IE) of an atom is the energy required to strip it of an electron. More generally, the nth ionization energy of an atom is the energy required to strip it of an nth electron after the first [itex]n-1[itex] have already been removed. It is centrally significant in physical chemistry as a measure of the "reluctance" of an atom to surrender an electron, or the "strength" by which the electron is bound.

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## Values and trends

Generally speaking, ionization energies decrease down a group of the Periodic Table, and increase left-to-right across a period. Ionization energy exhibits a strong negative correlation with atomic radius. Successive ionization energies of any given element increase markedly. Particularly dramatic increases occur after any given block of atomic orbitals is exhausted. This is because, after all the electrons are removed from 1 orbital, the next ionisation energy involves removing an electron from a closer orbital to the nucleus. Electrons in the closer orbital experience greater forces of electrostatic attraction, and thus, require more energy to be removed.

Some values for elements of the third period are given in the following table:

Successive ionization energies in kJ/mol
Element First Second Third Fourth Fifth Sixth Seventh
Na 496 4,560
Mg 738 1,450 7,730
Al 577 1,816 2,744 11,600
Si 786 1,577 3,228 4,354 16,100
P 1,060 1,890 2,905 4,950 6,270 21,200
S 999 2,260 3,375 4,565 6,950 8,490 11,000
Cl 1,256 2,295 3,850 5,160 6,560 9,360 11,000
Ar 1,520 2,665 3,945 5,770 7,230 8,780 12,000

## Electrostatic explanation

Ionization energy can be predicted by a simple analysis using electrostatic potential and the Bohr model of the atom, as follows.

Consider an electron of charge -e, and an ion with charge +ne, where n is the number of electrons missing from the ion. According to the Bohr model, were the electron to approach and bind with the atom, it would come to rest at a certain radius a. The electrostatic potential at distance a from the ionic nucleus, referenced to a point infinitely far away, is:

[itex]V = \frac{1}{4\pi\epsilon_0} \frac{ne}{a} \,\![itex]

Since the electron is negatively charged, it is drawn to this positive potential. (The value of this potential is called the ionization potential). The energy required for it to "climb out" and leave the atom is:

[itex]E = eV = \frac{1}{4\pi\epsilon_0} \frac{ne^2}{a} \,\![itex]

This simple analysis is incomplete, as it leaves the distance a as an unknown. It can be made more rigorous by assigning to each electron of every chemical element a characteristic distance, chosen so that this relation agrees with experimental data.

## Quantum-mechanical explanation

According to the more sophisticated theory of quantum mechanics, the location of an electron is best described as a "cloud" of likely locations (specifically, an electron orbital) that ranges near and far from the nucleus. The energy can be calculated by integrating over this cloud. In the simplest possible case of the last ionization energy, it is given by

[itex]E = -\int \psi^* (\mathbf{r}) H \psi(\mathbf r ) d^3 \mathbf r[itex]

where H is the Hamiltonian and [itex]\psi[itex] is the ground state wavefunction, i.e. the eigenfunction of H with the lowest energy. In atomic units, H approximately given by

[itex]H = -\frac{Z}{\left|\mathbf{r}\right|} - \frac{1}{2} \nabla^2[itex]

where Z is the nuclear charge. With this Hamiltonian, the energy can easily be evaluated, and is in fact equal to the energy given by the Bohr model, with a characteristic distance [itex]a = a_0 / Z[itex], where [itex]a_0[itex] is the Bohr radius.

In general, calculating the nth ionization energy requires subtracting the energy of a [itex]Z-n+1[itex] electron system from a [itex]Z-n[itex] electron system. The first equation above is extended to integrate over the coordinates of every electron, and the second equation acquires an extra two terms for each electron and an extra electron repulsion term for each pair of electrons. Calculating these energies is not simple, but is a well-studied problem and is routinely done in computational chemistry.

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