Effective Nuclear Charge and Electronegativity

Before we can use the framework of the electrostatic interpretation of bonding to explain different types of bonds (covalent/ionic/metallic/secondary), we need the concept of electronegativity. The electronegativity of an atom is a measure (and there are many different methods of calculation) of how strongly an atom will attract electrons towards itself. One way of thinking about this is to consider the force $F$ exerted on an electron in the outer shell. We can model this force by taking into account two things:

  1. The effective nuclear charge experienced by outer shell electrons, denoted $Z_{\text{eff}}$. The larger the charge, the stronger the force.
  2. The distance from the nucleus to outer shell electrons. The larger the distance, the weaker the force.

In the electrostatic interpretation, the simplest way calculate this force is to use Coulomb's law to describe that the force between two charged particles:

$$F=k\frac{q_1 q_2}{r^2}$$

where $k$ is the Coulomb constant, $q_1$ and $q_2$ are the charges of the two particles, and $r$ is the distance between the two particles.

If we approximate an electron as a point charge, the force of attraction between an electron and a nucleus would seem to be $F=-k\frac{e^2 Z}{r^2}$ where $e = 1.602 \times 10^{-19} \,\text{C}$ is the charge of an electron and $Z$ is the atomic number of the element which indicates the charge of the nucleus (i.e., the number of protons, each with charge $e$). However, it isn't so simple in in multi-electron atoms (i.e., every atom other than hydrogen) because any valence electron is also being repelled by the other electrons. To simplify, instead of keeping track of the attraction between the nucleus and the electron and the repulsion from all the other electrons, we can instead imagine that the other electrons are "shielding" the electron we are interested in from the full attraction of the nucleus. We then say that the electron experiences an effective nuclear charge, $Z_{\text{eff}}$ which is lower than the actual nuclear charge $Z$.

This is illustrated below in Figure 3.6.1. $Z_{\text{eff}}=Z-S$, where $S$ is the amount of shielding (sometimes also denoted $\sigma$). A rough approximation of $S$ for valence electrons is to simply count the number of core (non-valence) electrons. A better approximation of $S$ is given by Slater's rules which empirically accounts for partial charges in various orbitals. (We'll report Slater's numbers here, but we won't address its theory in this class.)

In Figure 3.6.1, we show Li has an electronic configuration of $1\text{s}^22\text{s}^1$. The $1\text{s}$ electrons are the core electrons. Our rough approximation, not accounting for shielding from the 2s shell, gives $Z_{\text{eff}} = Z-S = 3-2 = 1$. Slater's rules give a better approximation of $Z_{\text{eff}} = 1.28$ by considering electron contributions to shielding from different orbitals and sub-orbitals.

On the left a lithium atom is depicted with one valence electron in the 2s shell which is attracted to the nucleus with charge 3 and repelled by the two core electrons in the 1s shell. On the right, is a simplified depiction from the perspective of the valence electron in which the nuclear attraction and electron repulsion have partially cancelled to produce a single force equivalent to a lower effective nuclear charge $Z\_{\text{eff}}$ and no core electrons. Simply summing the nuclear charge $Z$ with the total charge of the core electrons is a simplification. A better approximation is $Z\_{\text{eff}} \approx 1.28$.

Figure 3.6.1 On the left a lithium atom is depicted with one valence electron in the 2s shell which is attracted to the nucleus with charge 3 and repelled by the two core electrons in the 1s shell. On the right, is a simplified depiction from the perspective of the valence electron in which the nuclear attraction and electron repulsion have partially cancelled to produce a single force equivalent to a lower effective nuclear charge $Z_{\text{eff}}$ and no core electrons. Simply summing the nuclear charge $Z$ with the total charge of the core electrons is a simplification. A better approximation is $Z_{\text{eff}} \approx 1.28$.

So, the effective force an electron, approximated as point charge, would feel is:

$$F_{\text{e}^-}=-k\frac{e^2 Z_{\text{eff}}}{r^2} \tag{3.6.1}$$

The force $F_{\text{e}^{-}}$ tells us essentially the magnitude of the attraction the electron feels from the nucleus — or, the strength of the attraction between a valence electron and the atom itself. The stronger this is, the more difficult it is to remove an electron from the atom. The weaker, the easier it is to pull an electron off. In short, this is a measure of the electronegativity! This measure of electronegativity was proposed by Allred and Rochow, and is well-correlated with the more common Pauling electrongativity.

From Eq. 3.6.1, we can observe that:

  1. The force of attraction increases proportionally with the effective nuclear charge $Z_{\text{eff}}$.
  2. The force of attraction decreases with distance since $r^2$ is in the denominator.

Let's use this interpretation to consider how the effective force on a valence electron, Eq. 3.6.1 influences trends in the periodic table such as atomic size and electronegativity.

Exercise 3.6.1: Effective Force: Periodic Trends in Atomic Size and Electronegativies
Not Currently Assigned

The effective force that electrons feel affects the size of atoms, because it determines the extent to which electron clouds will spread out from the nucleus. The effective force that the electrons feel is also a measure of the electronegativity, as it measures the force felt by the electron. The stronger the force, the higher the electronegativity.

Explain each of your predictions. Take about 5 minutes on this exercise.

  1. Based on the two considerations above, distance and $Z_{\text{eff}}$, predict how the size of atoms and electronegativity will change (increase or decrease) as you move from left to right across a given row of the periodic table ( e.g., Period 2).

  2. (Optional:) If you have time, consider how size/electronegativity changes as you you move from top to bottom in a given column of the periodic table (e.g., Group 14). This will be on your homework.