Why does atomic radii decrease going from the bottom left to the upper right of the periodic table?

Nov 23, 2016

Two separate trends are occurring.

EFFECTIVE NUCLEAR CHARGE (LEFT/RIGHT)

Across a period, the number of protons and electrons both increase by $1$ for each element as we move from left to right. It appears as if no net attractive force is attained.

However, the added electron presents a bit of shielding; each electron can effectively block the others from the positive nucleus. That means an effectively-less negative charge interacts with the positively-charged nucleus, i.e. the net electron charge increases slower than the net nuclear charge does.

So, the nuclear attraction becomes larger from left to right, and the electrons get pulled in more easily.

We say that the effective nuclear charge increases from left to right on the periodic table for the same row.

We calculate effective nuclear charge as follows:

$\boldsymbol{{Z}_{e f f} = Z - S}$

where ${Z}_{e f f}$ is the effective nuclear charge, $Z$ is the atomic number, and $S$ is the shielding constant.

By definition:

• ${Z}_{e f f}$ is the average attraction of the electron(s) to the proton(s) after accounting for the rapid motion of electrons and the wall of core electrons in between the nucleus and the valence electron in question.
• $Z$ corresponds to the number of protons.
• $S$ is a value based on degree of "shielding" of the valence electrons from the positively-charged nucleus. For general chemistry, we approximate $S$ as the number of core electrons.

For example, comparing $\text{Mg}$ and $\text{F}$:

${Z}_{e f f , M g} \approx 12 - \left(\stackrel{\text{1s electrons")overbrace(2) + stackrel("2s electrons")overbrace(2) + stackrel("2p electrons}}{\overbrace{6}}\right) = 2$

${Z}_{e f f , F} \approx 9 - \left(\stackrel{\text{1s electrons}}{\overbrace{2}}\right) = 7$

and we have that ${Z}_{e f f , F} > {Z}_{e f f , M g}$, so the atomic radius of $\text{F}$ ($\text{42 pm}$) is smaller than that of $\text{Mg}$ ($\text{145 pm}$). The large difference is also due to having one more quantum level.

As a general rule then (besides for the transition metals), as ${Z}_{e f f}$ increases, atomic radii decrease. Thus, since ${Z}_{e f f}$ increases from left to right, atomic radii decrease from left to right.

QUANTUM LEVELS (UP/DOWN)

This is a simpler explanation. Each row of the periodic table corresponds to a new quantum level, denoted by the principal quantum number $n$.

Each quantum level is farther away from the nucleus than the previous, and is higher in energy. For instance, that's why the $3 s$ orbital is larger than the $2 s$ orbital---because $n = 3$ is higher in energy and farther away from the nucleus than $n = 2$.

Therefore, an element on the same column but a new row is automatically larger, because it has another quantum level.