By Rosann Kozlowski
Jun 6, 2023 11:44 pm EST

AndreaObzerova/iStock/GettyImages

Effective nuclear charge is the net positive charge experienced by the valence electrons of a multi‑electron atom after accounting for the shielding effect of inner electrons. Across a period the charge increases, while down a group it generally decreases.

Effective Nuclear Charge Formula

For a single electron the relationship is expressed as:

\(Z_{eff} = Z – S\)

• Z – the atomic number (total protons in the nucleus)
• S – the shielding constant, representing the average electron density that reduces the nuclear attraction
• Z_eff – the effective nuclear charge felt by the electron

Calculating Effective Nuclear Charge

Determining Z and S requires a clear understanding of the electron configuration and the application of Slater’s Rules to quantify shielding. Below is a practical, step‑by‑step procedure.

Step 1: Identify the Atomic Number (Z)

Locate the element on a periodic table to obtain its atomic number. For example, sodium (Na) has an atomic number of 11.

Step 2: Write the Electron Configuration

Express the configuration using the standard grouping sequence: \\((1s) (2s, 2p) (3s, 3p) (3d) (4s, 4p) (4d), (4f), (5s, 5p), (5d), (5f) …\\

For sodium:

\(\text{Na: } (1s^2)(2s^2,2p^6)(3s^1)\)

Here the superscript indicates the number of electrons in each sub‑shell. The valence electron of interest is the single 3s electron.

Step 3: Assign Shielding Values Using Slater’s Rules

• Electrons to the right of the electron of interest contribute 0 to shielding.
• Electrons in the same group (same principal quantum number and subshell type) contribute 0.35 each.
• For s or p electrons of interest:
• Electrons one shell below contribute 0.85 each.
• Electrons two or more shells below contribute 1.00 each.
• For d or f electrons of interest, all other electrons contribute 1.00 each.

Applying this to sodium’s 3s electron:

• Right‑hand side: 0
• Same group (none): 0
• Shell 2 (8 electrons): 8 × 0.85 = 6.80
• Shell 1 (2 electrons): 2 × 1.00 = 2.00
• Shell 3 (none): 0

Sum of shielding constants: 8.80.

Step 4: Compute Z_eff

Insert the values into the formula:

\(Z_{eff} = 11 – 8.80 = 2.20\)

Thus, the effective nuclear charge acting on sodium’s valence electron is 2.20. This value can be used to predict ionization energies, atomic radii, and other electronic properties.

Applying the Method to Other Elements

The same procedure works for cations, anions, and any element where the electron configuration is known. Accurately recording the number of protons and electrons ensures a reliable estimate of the effective nuclear charge.