Effective Nuclear Charge Calculator

Effective nuclear charge (Z_eff) is the net positive charge felt by an electron from the nucleus, after accounting for shielding by other electrons. It explains atomic size, ionization energy, and electron affinity trends across the periodic table. Z_eff is always less than the actual atomic number due to electron-electron repulsion.

Use Slater's rules: Z_eff = Z - S, where S is the shielding constant. For valence s/p electrons: same-group s/p electrons shield 0.35 each (0.30 for 1s); all electrons in n-1 and lower shells shield 1.00 each; d/f electrons: all lower d/f and s electrons shield 1.00. Example: Na (Z=11), valence 3s electron: S = (10 × 1.00) - 0.35 = 9.65? Wait, use exact rules: S = 8 × 1.00 (1s, 2s, 2p) + 2 × 0.85 (3s electrons) = 9.7? Actually for Na: S = (10 inner e⁻ × 1.00) = 10, then Z_eff = 11 - 10 = 1 for 3s electron.

Z_eff explains periodic trends. Going across a period, Z increases but shielding stays similar, so Z_eff increases→ smaller atoms, higher ionization energy. Down a group, n increases so electron is further from nucleus, outweighing increased Z→ larger atoms, lower ionization energy. Z_eff also determines chemical reactivity and bond lengths.

Slater's rules estimate electron shielding: (1) Write electron configuration in groups: (1s) (2s,2p) (3s,3p) (3d) (4s,4p) (4d) (4f) (5s,5p)... (2) Electrons in same group: 0.35 each (0.30 for 1s); (3) Electrons in n-1 shells: 1.00 each for s/p electrons; (4) Electrons in n-2 or lower: 1.00 each.

Z_eff increases left to right across a period: more protons without proportionally more shielding. This pulls electrons closer, decreasing atomic radius and increasing ionization energy. Fluorine has highest Z_eff in period 2, explaining its high electronegativity. Noble gases are exceptions where adding p electrons causes slight radius increase.