Question

Explain the following variations in atomic or ionic radii: (a) \(\Gamma>\mathrm{I}>\mathrm{I}^{+},(\mathrm{b}) \mathrm{Ca}^{2+}>\mathrm{Mg}^{2+}>\mathrm{Be}^{2+}\) (c) \(\mathrm{Fe}>\mathrm{Fe}^{2+}>\mathrm{Fe}^{3+}\)

Short answer

In all three cases provided, the variations in atomic or ionic radii can be explained by the differences in effective nuclear charge and electron-electron repulsion. In example (a), Γ>I>I⁺, due to increased electron-electron repulsion and an additional electron shell in xenon compared to iodine. In example (b), Ca²⁺>Mg²⁺>Be²⁺ as the atomic number increases, resulting in more protons in the nucleus and more energy levels, causing an increase in ionic radius. In example (c), Fe>Fe²⁺>Fe³⁺, the decrease in radius is attributed to the increase in effective nuclear charge experienced by the remaining electrons as they are pulled closer to the nucleus when the electrons are lost.

Step-by-step solution

Understanding Atomic Radius

Atomic radius is the distance between the nucleus and the outermost shell of an atom. The atomic radius depends on the number of protons in the nucleus, number of energy levels (shells), and the effective nuclear charge experienced by the outermost electrons.

Understanding Ionic Radius

Ionic radius is the measure of an ion's size, which varies depending on whether the ion lost or gained electrons. Cations, formed by losing electrons, typically have a smaller radius compared to the parent atom. Conversely, anions, formed by gaining electrons, usually have a larger radius.

(a) Explaining \(\Gamma>\mathrm{I}>\mathrm{I}^{+}\)

Here, we're comparing the radii of a neutral iodine atom (I) and its positive ion (I+). The neutral iodine atom has 53 protons and 53 electrons, whereas the iodine cation (I+) has 53 protons but only 52 electrons due to the loss of one electron. The effective nuclear charge increases in I+ (more protons per electron) causing the remaining electrons to be pulled closer to the nucleus, resulting in the smaller radius for I+. Gamma (Γ) represents the noble gas xenon, with 54 protons and 54 electrons. Due to additional electron-electron repulsion in xenon and the presence of a new electron shell compared to iodine, its atomic radius is larger than that of iodine, so Γ > I > I+.

(b) Explaining \(\mathrm{Ca}^{2+}>\mathrm{Mg}^{2+}>\mathrm{Be}^{2+}\)

In this example, we're comparing the radii of three doubly positive ions: Ca²⁺, Mg²⁺, and Be²⁺. As we move from Be to Mg and then to Ca in the periodic table, the atomic number increases, meaning that there are more protons in the nucleus and more energy levels. Consequently, the ionic radius increases. So, Ca²⁺ has the largest ionic radius, followed by Mg²⁺ and then Be²⁺, resulting in Ca²⁺ > Mg²⁺ > Be²⁺.

(c) Explaining \(\mathrm{Fe}>\mathrm{Fe}^{2+}>\mathrm{Fe}^{3+}\)

Here, we're comparing the radii of neutral iron (Fe), its divalent cation (Fe²⁺), and its trivalent cation (Fe³⁺). The neutral iron atom has 26 protons and 26 electrons, while its ions have lost 2 and 3 electrons, respectively. The positive charge on the nucleus remains the same for all cases, but as the number of electrons decreases, the effective nuclear charge experienced by the remaining electrons increases, and they are pulled closer to the nucleus, resulting in a smaller radius for the ions. Consequently, Fe³⁺ has the smallest radius due to the highest effective nuclear charge, followed by Fe²⁺ and then neutral Fe, resulting in Fe > Fe²⁺ > Fe³⁺.

Key concepts

Atomic Radius

Atomic radius, simply put, is the distance from the nucleus to the outermost layer of electrons in an atom.

It's like the size of a balloon that encloses the atom's electrons. When an atom has more layers of electrons, usually as a result of having a higher atomic number, it’s like adding more rubber to the balloon—increasing its size. The further away these layers are, the larger the atomic radius. But it’s not just about layering; the atomic radius also depends on the level of 'hug' the electrons feel from the nucleus, which is known as the effective nuclear charge.

• If the 'hug' is firm (a high effective nuclear charge), electrons are pulled closer, and the radius decreases.
• If the 'hug' is gentle (a low effective nuclear charge), the radius increases.

This hug gets tighter across a period in the periodic table because the number of protons increases, enhancing the attraction, and therefore, generally leading to a decrease in atomic radius.

Ionic Radius

When an atom gains or loses an electron, it becomes an ion, and this changes its size, a bit like how a sponge swells up with water or shrinks when it dries out.

The ionic radius is the measure of an ion's 'balloon' when electrons have been added or removed. Cations—ions with a positive charge—have smaller radii than their parent atoms because they've lost one or more electrons and the electrons that are left feel a stronger ‘hug’ from the nucleus. Conversely, anions—negatively charged ions—grow in size because they gain extra electrons, which adds to electron repulsion and increases the overall size of the electron cloud. It’s a game of balance: if electrons leave, the ion shrinks; if electrons join, the ion grows.

Periodic Table Trends

The periodic table is like a map for chemists, showing trends in atomic and ionic sizes which can be quite predictable.

Horizontal Trends

As you move from left to right across a period (rows on the periodic table), both atomic and ionic radii tend to decrease. This is because protons are added to the nucleus, pulling electrons closer and creating a smaller 'balloon'.

Vertical Trends

The trend is different when you move down a group (columns on the periodic table). The radii usually increase because each step down introduces an additional layer of electrons or a new 'balloon layer', making the atom or ion bigger even though the effective nuclear charge might also increase.

Effective Nuclear Charge

Imagine a track team running while tied together. That’s like electrons orbiting a nucleus with the effective nuclear charge (ENC) being how tightly the runners are bound to each other.

The ENC is the net positive charge experienced by an electron in a multi-electron atom. It's not just about the total number of protons in the nucleus, because each electron is also repelled by others in the cloud—like runners wanting their own space. This phenomenon is called electron shielding.

• In an atom with a high ENC, electrons are pulled more strongly towards the nucleus, which can lead to a smaller atomic or ionic radius.
• However, when there are lots of electrons (such as in heavier atoms), they shield each other from the full effect of the nuclear charge, so the ENC felt by outer electrons is less, leading to a larger radius.

Ultimately, the ENC helps explain why atoms and ions have the sizes they do and why these sizes change in different circumstances.