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What Is the Effective Nuclear Charge?

By C.B. Fox
Updated May 21, 2024
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The effective nuclear charge is the attractive force of the protons in the nucleus of an atom on an electron after the repulsive force of the atom's electrons is factored out. The numerical value of this charge is found through the simple mathematical formula, Z(effective)=Z-S, where Z is the positive charge and S is the number of electrons in fully occupied orbitals. In neutral atoms, the effective nuclear charge is equal to the number of electrons in the outermost orbital, which are called valence electrons.

In atoms, protons all have a positive charge of one. An atom is distinguished from other atoms by the number of protons it has, though the number of neutrons may vary between different atoms of the same type, and some atoms of the same type may have ions with more or fewer electrons in orbit. The total positive charge of an atom is the number of its protons, which is also the atom's atomic number as shown on the periodic table of the elements. The first step in determining an atom's effective nuclear charge is to determine its total positive charge, which can be accomplished by looking up the atom's atomic number.

Electrons are drawn towards the nucleus of the atom and are found in orbitals that fill up in a predictable fashion. The first orbital can contain only two electrons. Subsequent orbitals each contain eight electrons when full. Under normal circumstances, and for the purposes of finding an atom's effective nuclear charge, electrons will occupy the closest orbital to the nucleus that they can.

Fully occupied orbitals counteract the same amount of positive charge as they contain negative charge. For example, an atom with 12 protons and 12 electrons, which is a neutral atom, will lose 2 positive charge from the fully occupied first orbital and 8 positive charge from the second. The other two electrons in the third orbital do not affect the atom's effective nuclear charge, which in this case would be 12 minus 10, or 2.

In most cases, the simplified equation is sufficient to discover an atom's effective nuclear charge. More complex versions of the equation take into account the small negative charge of the valence electrons, which is considered to be negligible for most purposes. Ions will also have an effective nuclear charge which deviates slightly from the standard equation, as the addition of an extra electron to the outer orbital will make it slightly less positive, and the loss of an electron will increase the atom's positive attraction.

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