Problem 128 - A Complete Resource Book in Chemistry for JEE Main

Question

The radius of \(\mathrm{La}^{3+}\) (atomic number \(=57\) ) is \(1.06 \AA\). Which one of the following given values will be closest to the radius of \(\mathrm{Lu}^{3+}\) (atomic number \(=71\) )? (a) \(0.85 \AA\) (b) \(1.06 \AA\) (c) \(1.40 \AA\) (d) \(1.60 \AA\)

Step-by-Step Solution

Answer

The radius of \(\mathrm{Lu}^{3+}\) is closest to \(0.85 \, \text{\AA}\).

Step 1: Understanding Lanthanide Contraction

The elements from La to Lu are part of the lanthanide series. A phenomenon occurs in this series called lanthanide contraction, which refers to the decrease in ionic radii of these elements as we move from La to Lu. This happens despite an increase in the atomic number because the additional electrons are entering an inner f-orbital which poorly shields the nuclear charge.

Step 2: Comparing La and Lu Ionic Radii

Given \(\mathrm{La}^{3+}\) has a radius of \(1.06 \, \text{\AA}\). Due to lanthanide contraction, the radius of \(\mathrm{Lu}^{3+}\) will be smaller. Among the given choices, we need to identify which is the smallest value less than \(1.06 \, \text{\AA}\).

Step 3: Selecting the Closest Value for Lu Radius

From the options, \(0.85 \, \text{\AA}\) is smaller than \(1.06 \, \text{\AA}\) and closest compared to other options which are all larger. This aligns with the expected decrease in ionic size due to lanthanide contraction.

Key Concepts

Ionic Radii

Ionic radii refer to the sizes of ions, which are atoms or molecules that have gained or lost one or more of their electrons. The size of an ion can significantly influence the chemical properties and behavior of elements in compounds.
Understanding ionic radii requires considering several factors.

• First, the charge of the ion: Cations (positive ions) tend to be smaller than their neutral atoms because losing electrons reduces electron-electron repulsion in the outer shells.
• Second, the electron configuration: Transitioning from neutral atoms to ions alters the electron configuration, affecting the radius.
• Third, the shielding effect: The reduction in effective nuclear charge felt by an electron due to the presence of inner electron shells also plays a critical role.

Understanding these components is essential to predict and compare the ionic radii of elements, as demonstrated in comparing La and Lu ions.

Lanthanide Series

The lanthanide series comprises fifteen elements with atomic numbers 57 to 71, from lanthanum (La) to lutetium (Lu). They are known as rare earth elements.
These elements are renowned for their similar properties, but they exhibit a unique phenomenon known as the lanthanide contraction. The lanthanide contraction is the gradual decrease in ionic radii and atomic sizes across the series from La to Lu.
This occurs because, as the atomic number increases, additional electrons are added to the 4f orbital. However, the 4f electrons do not shield the nuclear charge effectively. Their poor shielding means that the effective nuclear charge experienced by outer electrons increases, pulling these electrons closer to the nucleus and decreasing the radii.

• This contraction affects not only the chemistry of lanthanides but also their periodic properties and those of subsequent elements in the periodic table.
• It also plays a critical role in influencing the properties of metals and their compounds, making it a vital concept in inorganic chemistry.

Electrons and Orbitals

Electrons are negatively charged subatomic particles that orbit the nucleus of an atom. They are organized into orbitals, which are regions of space around the nucleus where electrons are most likely to be found.
These orbitals have different shapes (s, p, d, f), each capable of holding a specific number of electrons. The arrangement of electrons in these orbitals follows the principles set by quantum mechanics:

• The Pauli exclusion principle states that no two electrons can have the same set of four quantum numbers, essentially meaning each orbital can hold a maximum of two electrons with opposite spins.
• The Aufbau principle describes the "building up" process: electrons fill orbitals from lowest to highest energy.
• Hund’s rule explains that electrons will fill an unoccupied orbital before they pair up in occupied ones.

In the context of lanthanides, the electrons fill the 4f orbitals, but these do not efficiently shield the increasing nuclear charge. This results in the lanthanide contraction, mentioned earlier, which critically alters their ionic radii despite the increase in atomic number.