Effective Nuclear Charge
Effective nuclear charge (Z_eff) is the net positive charge experienced by a valence electron after accounting for the shielding (screening) effect of inner electrons, which partially cancel the attraction from the nucleus. It is calculated as Z_eff = Z − S, where Z is the actual atomic number and S is the shielding constant. Effective nuclear charge increases across a period because additional protons are added while shielding remains approximately constant, explaining trends in atomic radius, ionization energy, and electron affinity.
Key Formula
Z_eff = Z − S
LaTeX: Z_{\text{eff}} = Z - S
| Symbol | Meaning | Unit |
|---|---|---|
| Z_eff | Effective nuclear charge | dimensionless (in units of elementary charge) |
| Z | Actual atomic number (total protons) | dimensionless |
| S | Shielding constant (sum of shielding contributions from inner electrons) | dimensionless |
Worked Example
Problem
Using Slater's rules, calculate the effective nuclear charge experienced by the 3s electron in sodium (Na, Z = 11). Electron configuration: 1s² 2s² 2p⁶ 3s¹.
Solution
Step 1 — Identify the electron in question: one 3s electron. Step 2 — Apply Slater's rules for the shielding constant S: Electrons in higher groups (none here): contribute 0. Electrons in the same (n = 3) group: 0 others in 3s → contribute 0 each. Electrons in (n−1) shell (2s, 2p): 8 electrons × 0.85 = 6.80 Electrons in (n−2) and lower shells (1s²): 2 electrons × 1.00 = 2.00 Step 3 — Total shielding constant: S = 0 + 6.80 + 2.00 = 8.80 Step 4 — Calculate Z_eff: Z_eff = Z − S = 11 − 8.80 = 2.20
Answer
Z_eff for the 3s electron in sodium ≈ 2.20
Effective Nuclear Charge for Valence Electrons Across Period 3
| Element | Z | Valence Shell | Shielding (S) | Z_eff |
|---|---|---|---|---|
| Na | 11 | 3s | 8.80 | 2.20 |
| Mg | 12 | 3s | 8.80 | 3.20 (approx) |
| Al | 13 | 3p | 9.00 (approx) | 4.00 (approx) |
| Si | 14 | 3p | 9.00 (approx) | 5.00 (approx) |
| Cl | 17 | 3p | 11.25 (approx) | 5.75 (approx) |
| Ar | 18 | 3p | 11.25 (approx) | 6.75 (approx) |
Interactive Tools
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Related Terms
Ionization Energy
Ionization energy (IE) is the minimum energy required to remove the most loosely bound electron from a gaseous atom or ion in its ground state, producing a positive ion. The first ionization energy (IE₁) removes the first electron; successive ionization energies increase because each removal leaves behind a more positively charged species that holds remaining electrons more tightly. Ionization energy increases across a period (due to greater effective nuclear charge) and decreases down a group (due to greater atomic radius and electron shielding), making it a key periodic trend.
Atomic Radius
Atomic radius is a measure of the size of an atom, typically defined as half the distance between the nuclei of two identical adjacent atoms in a covalent bond (covalent radius) or in a metallic lattice (metallic radius). Atomic radius decreases across a period (left to right) because increasing nuclear charge pulls electrons closer to the nucleus, while it increases down a group because additional electron shells increase the average distance of the outermost electrons from the nucleus. These periodic trends directly influence bond lengths, ionic sizes, and many physical properties.
Electron Affinity
Electron affinity (EA) is the energy change that occurs when a neutral gaseous atom gains one electron to form a negative ion (anion). A negative EA value indicates an exothermic process — the atom releases energy and the anion is more stable than the separated atom and electron — which is the case for most halogens. Electron affinity generally increases (becomes more negative) across a period and decreases down a group, though there are notable exceptions such as the anomalously low EA of fluorine compared to chlorine due to electron–electron repulsion in fluorine's compact 2p orbitals.
From Latin "effectivus" (productive, efficient) and "nucleus" (kernel). The concept of shielding and effective nuclear charge was quantified by John C. Slater in 1930 through his empirical shielding rules, and later refined by Clementi and Raimondi using quantum mechanical calculations in 1963.