Periodicity I: the atomic radius

The first periodic trend we'll learn is about the atomic radius. So what's atomic radius?

Atomic radius

Atoms are made of a nucleus at the center and one or more electrons moving around the nucleus. Electrons, however, do not travel like the Earth does. They are more like flashing at various locations. We can't determine exactly where the electrons are or where they will be. All we have is the probability of finding an electron at a particular position. Conventionally, we define a spherical surface that encloses the space where the probability of finding electrons inside is 90% as the boundary of the atom, and its radius as the atomic radius.

Bonding atomic radius

In practice, we would find out the atomic radius of an atom by measuring the distance between two nuclei of the same element in a molecule and defining half of the distance as the bonding atomic radius (\( r_b = \frac{d}{2} \)).

This would allow us to measure the locations of the nuclei instead of the electrons, which is a much easier task with modern technology. However, do take note of the overlap in the illustration above. When two atoms form a bonding (we'll talk more about bonding later), they are interacting with each other by sharing their valence electrons. Hence, the valence electron of one atom would "go into" the orbitals of the other atom, and consequently lead to an overlaping of the atoms, just like handshaking. As a result, the distance between the two nuclei is a little smaller than double the radius. However, it's still a good estimation of atomic radii.

Nonbonding atomic radius

In some cases, the atomic radii could be measured without forming the bonding. For example, for noble gases (elements in group 8A), the atoms do not form bondings with other atoms because they have a full valence shell, and are stable and inert to chemical reactions.

So we would measure their atomic radii by freezing the gases and reducing their volumes, until the atoms touch each other. We'll again measure the distance between the two adjacent nuclei and calculate the atomic radii accordingly. However, no overlap would be observed in such a scenario.

Nuclear charge and shielding effect

Before we discuss the periodic trends of atomic radius, there're two very important concepts that greatly affect the trends (and other trends too). The first one is the nuclear charge, Z.

The nucleus at the center of an atom possesses positive charges due to its protons. This would exert an attractive force on the electrons. This force of attraction is weaker when electrons are located far away from the nucleus, i.e. in an outer shell.

The second concept is the shielding effect, S. It is the repulsion that the valence electrons would experience because of the presence of the inner electrons. For example, in a sodium (Na) atom, electron configuration \( 1s^22s^22p^63s^1 \), its valence electron would see a total of 10 electrons sitting between itself and the nucleus. This leads to a weaker net attraction being experience by the valence electron. Hence we say the nucleus, or the nuclear charge, is being shielded by the inner electrons.

As a result, we define the effective nuclear charge, \( Z_{eff} \) being the net attraction force that the valence electrons would experience.

\[ Z_{eff} = Z - S \]

However, the formula above is only for definition and we would not do any calculation using it. The actual calculation of effective nuclear charge would be more complex. Nevertheless, the formula tells us that:

1. greater the number of protons (or atomic number), higher the attraction force;
2. greater the number of inner electrons, weaker the attraction force.

Periodic trends of atomic radius

Down the group

Now let's take a look at the periodic trend of atomic radius when we go down the group in a periodic table. We'll use alkali metals (group 1A) as the example.

As we have shared before, elements in the same main group would have the same number of their valence electrons, while the total numbers of energy levels would correspond to the period number. This could be shown clearly with the electron configurations of the alkali metals.

Element	Electron configuration	Number of energy levels
Li	\(1s^22s^1\)	2
Na	\(1s^22s^22p^63s^1\)	3
K	\(1s^22s^22p^63s^23p^64s^1\)	4
Rb	\(1s^22s^22p^63s^23p^64s^23d^{10}4p^65s^1\)	5

Just imagine yourself wearing more and more clothes, aren't you getting bigger and bigger in size? That's exactly what's happening to the atoms.

So we say, atomic radius increases down the group because of more energy levels.

Across the period

On the other hand, when we travel across the periods from the left to the right, we'll get atoms with the same number of energy levels despite their total numbers of electrons are increasing. So how does the atomic radius change? Is there any trend in such a case? Let's take a look at the nuclear charge and the shielding effect, and see if we could get an answer.

The elements across a period are in the sequence of increasing atomic number. Their nuclear charges are increasing from the left to the right of the period. This leads to greater attraction towards the valence electron.

Meanwhile, they all have the same number of inner electrons, and hence the same shielding effect. By considering the increasing nuclear charges and unchanged shielding effect, we could easily conclude that the effective nuclear charges increase across the period, and so do the attraction experienced by the valence electrons.

The increased attraction would pull the valence electrons closer to the nucleus, and hence a smaller radius. Since all elements are having the same number of energy levels, this little movement of valence electrons plays a significant role.

Ultimately, we say atomic radius decreases across the period because of 1) increasing effective nuclear charges, 2) greater attraction between nucleus and valence electrons, and 3) same number of energy levels.

So if we draw the atoms on a piece of paper, they may look like something similar to the illustration below.

In summary, when comparing atomic radii of two elements, consider two factors:

1. energy level;
2. nuclear charge or atomic number;

while the energy level would be the predominant factor.

Next, we'll apply the same principles to see if there is any periodic trend for ionic radius.

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