The Shielding Effect and Effective Nuclear Charge

The Shielding Effect

Electrons in an atom can shield each other from the pull of the nucleus. This effect, called the shielding effect, describes the decrease in attraction between an electron and the nucleus in any atom with more than one electron shell. The more electron shells there are, the greater the shielding effect experienced by the outermost electrons.

In hydrogen-like atoms, which have just one electron, the net force on the electron is as large as the electric attraction from the nucleus. However, when more electrons are involved, each electron (in the n-shell) feels not only the electromagnetic attraction from the positive nucleus but also repulsion forces from other electrons in shells from 1 to n-1. This causes the net electrostatic force on electrons in outer shells to be significantly smaller in magnitude. Therefore, these electrons are not as strongly bound as electrons closer to the nucleus.

The shielding effect explains why valence shell electrons are more easily removed from the atom. The nucleus can pull the valence shell in tighter when the attraction is strong and less tight when the attraction is weakened. The more shielding that occurs, the further the valence shell can spread out. As a result, atoms will be larger.

Example:

Why is cesium bigger than elemental sodium?

Solution:

The element sodium has the electron configuration 1s²2s²2p⁶3s¹. The outer energy level is n = 3 and there is one valence electron. The attraction between this lone valence electron and the nucleus with 11 protons is shielded by the other 10 core electrons.

The electron configuration for cesium is 1s²2s²2p⁶3s²3p⁶4s²3d¹⁰4p⁶5s²4d¹⁰5p⁶6s¹. While there are more protons in a cesium atom, there are also many more electrons shielding the outer electron from the nucleus. The outermost electron, 6s¹, therefore, is held very loosely. Because of shielding, the nucleus has less control over this 6s¹ electron than it does over a 3s¹ electron.

Effective Nuclear Charge

The magnitude of the shielding effect is difficult to calculate precisely. As an approximation, we can estimate the effective nuclear charge on each electron.

Effective nuclear charge diagram

Diagram of the concept of effective nuclear charge based on electron shielding.

The effective nuclear charge (often symbolized as Z_eff or Z*) is the net positive charge experienced by an electron in a multi-electron atom. The term "effective" is used because the shielding effect of negatively charged electrons prevents higher orbital electrons from experiencing the full nuclear charge.

The effective nuclear charge on an electron is given by the following equation:

Z_eff = Z - S

where Z is the number of protons in the nucleus (atomic number), and S is the number of electrons between the nucleus and the electron in question (the number of nonvalence electrons).

Example:

Consider a neutral neon atom (Ne), a sodium cation (Na⁺), and a fluorine anion (F^-). What is the effective nuclear charge for each?

Solution:

Start by figuring out the number of nonvalence electrons, which can be determined from the electron configuration.

Ne has 10 electrons. The electron configuration is 1s²2s² 2p⁶. The valence shell is shell 2 and contains 8 valence electrons. Thus the number of nonvalence electrons is 2 (10 total electrons - 8 valence). The atomic number for neon is 10, therefore:

Z_eff(Ne) = 10 - 2 = 8+

Flourine has 9 electrons but F^- has gained an electron and thus has 10. The electron configuration is the same as for neon and the number of nonvalence electrons is 2. The atomic number for F^- is 9, therefore:

Z_eff(F^-) = 9 - 2 = 7+

Sodium has 11 electrons but the Na⁺ ion has lost an electron and thus has 10. Once again, the electron configuration is the same as in the previous examples and the number of nonvalence electrons is 2 (by losing one electron, the valence shell becomes the n=2 shell). The atomic number for Na⁺ is 11, therefore:

Z_eff(Na⁺) = 11 - 2 = 9+

In each of the above examples (Ne, F^-, Na⁺) an atom has 10 electrons but the effective nuclear charge varies because each has a different atomic number. The sodium cation has the largest effective nuclear charge, which results in electrons being held the tightest, and therefore Na⁺ has the smallest atomic radius.