How To Calculate Effective Nuclear Charge

Effective nuclear charge refers to the charge felt by the outermost (valence) electrons of a multi-electron atom after the number of shielding electrons that surround the nucleus is taken into account. The trend on the periodic table is to increase across a period and increase down a group.

In a simple system, like a hydrogen-like ion where there is only one electron and a known number of protons in the nucleus, we can easily calculate the electric force between the negatively charged electrons and the positively charged nucleus using Coulomb's law. However, as atoms get more complicated, the actual nuclear charge felt by the outer electrons becomes vastly different from the charge/force felt by the inner electrons. The attraction of the nucleus decreases with distance (according to the laws of electromagnetism), so we must account for this distance. Additionally, all of the inner shells of core electrons create a shielding effect that reduces the effective charge and the attractive force felt by the outermost electrons.

While these values become extremely complicated as more electrons are added to a system, there are certain periodic trends we can take advantage of to estimate the effective charge. Finding the effective charge can then be used to calculate ionization energy, atomic radius, and other useful metrics in physics and fields like inorganic chemistry.

The formula for calculating the effective nuclear charge for a single electron is:

\(Z_{eff} = Z – S\)

Where

•‌Zeff‌ is the effective nuclear charge, also just called ‌Z‌ eff or ‌Z‌ effective

•‌Z‌ is the number of protons in the nucleus, the atomic number

•‌S‌ is the average amount of electron density between the nucleus and the electron

Calculating effective nuclear charge involves understanding the ‌Z‌ and ‌S‌ values. ‌Z‌ is atomic number, and ‌S‌ requires the use of Slater's Rules to determine an electron cloud shielding value between the nucleus and the electron under consideration.

Step 1: Find Atomic Number to Determine Z Value

‌Example problem‌: What is the effective nuclear charge for the valence electron in sodium?

Z is the number of protons in the nucleus of the atom, and this determines the positive charge of the nucleus. The number of protons in the nucleus of an atom is also known as the atomic number.

Using a periodic table of elements, locate the desired atomic number. In the example above, sodium, symbol Na, has atomic number 11.

Write the electron configuration of the element in the following order and groupings:

\((1s) (2s, 2p) (3s, 3p) (3d) (4s, 4p) (4d), (4f), (5s, 5p), (5d), (5f) \ldots\)

Recall that the numbers (1, 2, 3. . .) correspond to the principal quantum number or energy shell level of the electrons in the atom, and this designates how far away the electrons are from the nucleus. The letters (s, p, d, f) correspond to the given shape of an electron's orbital. For example, "s" is a spherical orbital shape, and "p" resembles a dumbbell shape figure 8.

For sodium, the electron configuration is

\(\text{Na: } (1s^2) (2s^2, 2p^2) (3s^2).\)

In the example above, the superscript number indicates the number of electrons in each shell. Sodium has 11 electrons: two electrons in the first energy level (1), eight electrons in the second energy level (2), and one electron in the third energy level. The electron in the 3s¹ orbital is the focus of the example.

The value S may be calculated using Slater's Rules, named after the scientist John C. Slater who developed them. These rules give shielding effect values to each electron to find a final shielding constant. Do ‌not‌ include a value of the electron of interest. Assign the following values:

1\. Any electrons to the right of the electron of interest contain no shielding value.

2\. Electrons in the same group as the electron of interest shields 0.35 nuclear charge units.

3\. For s or p electrons of interest: electrons with one less value of the principal quantum number

(energy level: 1, 2, 3. . .) are assigned 0.85 units of nuclear charge. Electrons found two or more energy levels lower shield by 1.00 unit.

4\. For d or f electrons of interest: all electrons shield by 1.00 unit.

‌For the example above, the answers for Na would be:‌

1\. 0; there are no electrons higher (or to the right in the electronic configuration)

2\. 0; there are no other electrons in the 3s orbital of Na.

3\. 8.8; Requires two calculations: first, there are eight electrons in the energy level 2 shell, two in the s shell, and six in the p; 8 × 0.85 = 6.8. Plus, since the 1s² electrons are two levels from the electron of interest: 2 × 1.

4\. 0; it is not a d or f shell electron on the outer subshell.

Add all the shielding charges calculated using Slater's Rules.

In the sample problem, the shielding values sum to

\((0 + 0 + 8.8 + 0) = 8.8\)

Place the values for Z and S into the effective nuclear charge formula:

\(Z_{eff} = Z – S \longrightarrow 11 – 8.8 \ \text{}\)
\(\text{For Na the } Z_{eff} = 2.2\)

The effective nuclear charge of the 3s¹ electron in the sodium atom is 2.2. This can be interpreted as the net positive charge exerted on the valence shell electron. Slater's formula can also be applied to anions, cations, and other variations of elements, as long as we correctly record the number of protons and the number of electrons.