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Effective nuclear charge

The effective nuclear charge (represented as Z eff and in some cases as Z * ) is the net nuclear charge that an electron experiences when it is in a polyelectronic atom (that is, it has more than one electron).

More precisely, it is the electric charge that the nucleus of a hypothetical atom would have capable of attracting its only electron with the same force with which the nucleus of the real atom attracts that same electron in the presence of all other electrons.

It is a corrected nuclear charge that takes into account the effect of the presence of other electrons in a polyelectronic atom. This reduced charge explains why the outermost electrons in a polyelectronic atom are more loosely bound to the nucleus than the inner electrons.

Effective nuclear charge is a concept of great importance in chemistry, as it allows you to understand the periodic trend of many properties such as atomic radius, ionic radius, electronegativity, ionization energies, and more.

The effective nuclear charge arises from two phenomena:

  • The shielding effect of electrons in polyelectronic atoms.
  • Electrostatic repulsion between electrons due to all having the same electrical charge.

The shielding effect consists of a kind of shield formed by the internal electrons of an atom that covers the nucleus. This causes the outermost electrons to “feel” less attraction to the nucleus than they would if the other electrons were not present.

For example, the nuclear charge of the sodium atom is +11 (its atomic number is Z = 11), but, the only valence electron it has, it actually feels the attractive force of a charge of only +2.2 .

In other words, the shielding of the other 10 inner electrons causes sodium’s valence electron to feel a nuclear attractive force only one-fifth of what it should be.

In addition to the shielding effect, the repulsion between electrons (which have the same electrical charge) also helps to counteract the ability of the nucleus to attract external electrons. That is, this repulsion also contributes to reducing the effective nuclear charge.

Important characteristics of effective nuclear charge

It should be noted that the shielding effect responsible for reducing the effective nuclear charge only affects the electrons that are at the same energy level or in the upper layers, but not the innermost electrons. Also, the effect is not the same for electrons found in s and p atomic orbitals as those found in d and f orbitals.

Periodic trend of effective nuclear charge

Over a period

Electrons at the same energy level are less screening than those at lower energy levels.

Because of this, the shielding effect does not increase considerably as we move over a period, but the actual nuclear charge does. For this reason, the effective nuclear charge increases from left to right on the periodic table.

Throughout a group

On the other hand, when moving from one period to another in the same group (that is, moving down through a group), entire layers of highly shielding internal electrons are added. This causes the effective nuclear charge to decrease from top to bottom or, in other words , to increase from bottom to top .

Formula of effective nuclear charge

The effective nuclear charge can be calculated using a very simple semi-empirical equation that takes into account the true value of the nuclear charge (given by the atomic number, Z) and a term called the shielding constant. The latter encompasses in one single the effects of the presence of the other electrons.

The equation is given by:

where Z is the atomic number and σ (Greek letter sigma) represents the shielding constant, which depends on the electronic configuration.

Slater’s Rules

The shielding constant can be estimated from a system known as Slater’s rules. These rules allow us to calculate the shielding constant of an electron by adding the contributions of the other electrons to this shielding constant. These rules can be summarized like this:

  • Any electron at the same energy level (level n 0 ) contributes an amount of 0.35 to the shielding constant, unless they are both at level 1, in which case it contributes 0.35.
  • Each electron that is in the immediately previous level (in level n 0 -1) in a sop orbital, contributes 0.85; On the other hand, if it is in a dof orbital it contributes 1.
  • All other electrons at lower energy levels (n 0 -2, n 0 -3, etc.) contribute 1 to the shielding constant.

Example of effective nuclear charge calculation

Valence electron of the sodium atom

The electron configuration of the sodium atom is 1s 2 2s 2 2p 6 3s 1 . In other words, if we want to calculate the effective nuclear charge felt by the valence electron (the 3s 1 electron ), we must add the contributions of the other 10 electrons following Slater’s rules.

Since we are calculating the shielding constant of the electron 3s 1 (n 0 = 3) and it is only in the valence shell, there are no other electrons at the same energy level.

The level immediately above is n 0 -1 = 2, where there are 8 electrons in sop orbitals that contribute 0.85 each, and there are no electrons in do f orbitals.

Finally, the only level lower than 2 is n = 1, in which there are only 2 electrons. All of this is summarized in the following table:

As can be seen, the 10 internal electrons of sodium provide a shielding constant of 8.8, so the effective nuclear charge felt by the 3s 1 electron is:

Valence electrons of arsenic

The electron configuration of arsenic is 1s 2 2s 2 2p 6 3s 2 3p 6 3d 10 4s 2 4p 3 . The valence shell is shell 4 (n 0 = 4) that has 5 electrons: (4s 2 4p 3 ). In this case, each of these 5 electrons will feel the effect of the other 4 that are in its same shell and that of the other 28 internal electrons as shown in the table:

Therefore, the effective nuclear charge felt by the valence electrons of arsenic is:

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