Effective Nuclear Charge

At the heart of all of the other periodic table trends is effective nuclear charge (Z_eff). Z_eff refers to the actual felt pull that the outermost electrons of an element “feel”. Three major factors control Z_eff including the number of protons in the nucleus, the number of total shells an atom possesses, and lastly how many core electrons an atom has.

Coulomb’s Law Again

At a more basic level all of these factors are explained by Coulomb’s law.

[latexpage]
\[F=K\frac{q_1q_2}{r^2}\]
If we label q1 as the charge of the nucleus and q2 as the charge of one of the outermost electrons. We can easily see how the number of protons in the nucleus will increase the force. If we then replace distance (r) with the outermost shell number (n) we can see how having more shells rapidly decreases the force of the nucleus on the outermost electrons.
\[F=K\frac{q_{nuclues}q_{electron}}{n^2}\]

We can think of this by analogy as well. Imagine that the protons represent the physical strength of the nucleus. So an element with more protons is like a competitive weightlifter while an atom with fewer protons is like me, fairly wimpy when it comes to lifting heavy things.

Regardless of your strength the further you have to reach the harder it is to pull on that object. So electrons that are further away from the nucleus are harder to hold onto as well.

Lastly, anything that gets in the way of grabbing an object makes it more difficult to pull on as well. Imagine if you had to reach through a vat of honey to grab an object versus grabbing it out of the air. Other electrons inside of the valence electrons act like honey diminishing the attractive force that the nucleus can exert on them.

Calculating Z_eff

We can calculate the Z_eff by subtracting the number of core electrons from the number of protons in an element. Where the value represents the charge that the outermost electrons actually experience.

[latexpage]
\[
Z_{eff}=Protons – Core\; Electrons
\]

The core electrons are all electrons excluding the valence electrons. Typically the easiest way to calculate the core electrons is to subtract the valence electrons from the total number of electrons.

As an example let’s look at O (oxygen) and O^2-. Oxygen’s highest shell number is 2 and it has both a 2s and 2p subshell. The s subshell is fully filled with two electrons and the 2p subshell is partially filled with 4 electrons. In total oxygen has 6 valence electrons in the n=2 shell. We don’t have to go subshell by subshell though. You can simply count the position of an element from left to right making sure to skip the d-block.

Therefore oxygen with an atomic number of 8 has 8 proton and when neutral 8 electrons. Since 6 of the 8 are valence electrons 2 are core electrons.

[latexpage]
\[
Z_{eff}=8 – 2 = +6
\]

The only difference between O^2- and O is an additional 2 electrons. Both of these electrons are added to oxygen’s open 2p subshell so they are valence electrons too. This means that both O^2- and O will have a Z_eff of +6. This makes sense because we didn’t mess with any of the three factors that influence Zeff: number of protons, number of shells, or number of core electrons.

The Trend

With all of this in mind let’s look at the periodic table trend for Z_eff then onto the rest of the periodic trends.

More protons = higher Z_eff = bigger muscles. So elements near the right hand side of the periodic table will have higher Z_eff values.

Less shells = higher Z_eff = less reaching. So elements near the top of the periodic table will have higher Zeff values

In summary Z_eff increases up and to the right in the periodic table.