Ionization Energy Calculator (Multi-Electron)
Estimate the ionization energy of an electron in a multi-electron atom using effective nuclear charge, IE = 13.6 eV × (Z_eff/n)².
⚛️ What is the Ionization Energy Calculator (Multi-Electron)?
This multi-electron ionization energy calculator estimates how much energy is needed to remove a specific electron from an atom or ion with more than one electron. Enter the atomic number, a screening constant, and the electron's shell number, and it returns the effective nuclear charge and the estimated ionization energy in both electronvolts and kilojoules per mole.
The exact hydrogen-atom formula IE = 13.6 eV/n² only applies to a single electron orbiting a bare nucleus. Every other atom or ion has additional electrons that partially screen the nucleus, reducing the charge any one electron actually feels. This calculator replaces the bare charge Z with an effective nuclear charge Z_eff = Z − σ, where σ is commonly estimated using Slater's rules based on which shells the other electrons occupy.
This model is used across introductory atomic physics and chemistry courses to explain trends like why ionization energy generally increases across a period (nuclear charge grows faster than screening) and decreases down a group (outer electrons sit in higher, more screened, more distant shells). It is also directly relevant to plasma physics, where an atom's ionization energy determines its ionization behavior in the Saha equation.
This calculator is useful for chemistry and physics students studying periodic trends, and for plasma physics students estimating ionization energies feeding into ionization-balance calculations.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 - Sodium (Na) 3s valence electron
Example 2 - Neon (Ne) 2p electron
Example 3 - Lithium (Li) 2s valence electron
❓ Frequently Asked Questions
🔗 Related Calculators
What is ionization energy in a multi-electron atom?
It is the energy needed to remove a specific electron from an atom or ion that has more than one electron. Unlike the exact hydrogen atom formula, multi-electron ionization energy must account for the other electrons partially screening (shielding) the electron from the full nuclear charge.
What is the formula used here?
IE = 13.6 eV × (Z_eff/n)², the same Bohr-model energy-level formula used for hydrogen, but with the actual nuclear charge Z replaced by an effective nuclear charge Z_eff = Z − σ that accounts for electron screening.
What is the screening constant σ?
σ is an estimate of how much the nuclear charge felt by one electron is reduced by the other electrons in the atom. It is commonly estimated using Slater's rules, which assign a screening contribution based on whether other electrons are in the same shell, the next-inner shell, or deeper core shells.
How do I estimate σ using Slater's rules?
Add up: about 0.35 for every other electron in the same group (same n, s/p type), about 0.85 for every electron in the shell just below (n−1), and about 1.00 for every electron two or more shells below (n−2 and lower). d and f electrons use slightly different group rules, but this covers the common s/p cases.
Why is Z_eff always less than Z?
Every other electron in the atom repels the electron being considered and partially cancels out some of the nucleus's attractive pull, so the electron being ionized never feels the atom's full nuclear charge. Z_eff = Z − σ captures this reduced, effective attraction.
Why does n appear in the denominator squared?
Electrons in outer shells (higher n) are both farther from the nucleus and more effectively screened, so their binding energy falls off rapidly, as 1/n², the same scaling that governs hydrogen's energy levels. This is why outer-shell (valence) electrons are almost always far easier to remove than inner-shell (core) electrons.
How accurate is this estimate compared to real measured ionization energies?
It is a useful order-of-magnitude estimate, typically within a factor of 1.5-2 of measured values for main-group elements, but it does not capture electron correlation, exchange effects, or fine shell structure that real spectroscopic measurements show. Treat results as instructive approximations, not precise reference values.
Can this calculator estimate successive ionization energies?
Yes, in principle, by using the appropriate Z (unchanged) and a recalculated σ that reflects one fewer electron in the atom after each ionization step, since removing an electron changes how much screening the remaining electrons provide to each other.
Why is ionization energy relevant to plasma physics?
In a plasma, an atom's or ion's ionization energy directly determines how easily it loses electrons under a given temperature, which is exactly the input the Saha equation needs to calculate the equilibrium ionization fraction of a gas. See the related Saha Equation Calculator.
What units does this calculator report?
Results are shown in electronvolts (eV), the standard unit for atomic-scale binding energies, and in kilojoules per mole (kJ/mol), the unit more commonly used in chemistry for comparing ionization energies across the periodic table.