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I'm a high school student and I'm learning about ionization energy and atomic radius of elements. I want to compare the ionization energy of lithium and magnesium.

Here is the information provided in my textbook:

In periodic table, the atomic radius of elements gradually decreases from left to right of a period and increases from up to down of a group.
If the atomic radius of an element is greater than the atomic radius of another element, then the ionization energy of the first element is less than the ionization energy of the second element.

So according to my book, magnesium has a greater atomic radius than lithium which means magnesium has less ionization energy than lithium. But when I checked on the internet, the result was completely opposite. Is there an explanation for this exceptionality?
Please share an explanation that is suitable for a high school student.

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    $\begingroup$ Note that I have used a brief, simple explanation for a high school student asking such a question. As the more complete answer could be much more complex, involving quantum chemistry. There is no high math used in the answer. $\endgroup$
    – Poutnik
    Commented Jul 31, 2021 at 11:46

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The book has omited the other factors.

The short answer is: In spite of the bigger atomic radius, the valence electrons of magnesium are attracted by the greater net electrostatic force compared to lithium. As this effect is stronger than the radius effect, the ionisation energy is greater.

The ionisation energy depends on the atomic radius AND the effective nucleus charge.

It can be expressed as:

$$E_\mathrm{ion}=\frac 12 \cdot \frac {e^2 \cdot Z_\mathrm{eff}} {4 \cdot \pi \cdot \epsilon_0 \cdot r_\mathrm{eff}}$$

where

  • $E_\mathrm{ion}$ is the electron ionisation energy
  • $Z_\mathrm{eff}$ is is effective nucleus proton number
  • $r_\mathrm{eff}$ is the radius of a hypothetical circular electron orbit to have the same ionisation energy for given effective nuclear charge.
  • The factor $\frac12$ is due the electron kinetic energy contribution to the total electron energy.

The effective nucleus charge means what charge should the nucleus have to cause the same electrostatic effect on the particular electron, if there were no other electrons in the atom. It's value is affected by the nucleus charge being screened/shielded away by other electrons and by mutual electron repulsion.

Therefore comparison of ionisation energies based solely on the radius is applicable only if the difference of effective nucleus charges, acting on valence electrons, is negligible.

For more, see e.g. Wikipedia pages Atomic radius and Slater's rules (for the effective nucleus charge).

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