The correct statement about 4f-series elements is:

Promethium is artificially synthesised 4f series element.

The correct answer is Option 3. Promethium is artificially synthesised 4f series element.

Let us delve into each statement about the 4f-series elements, focusing particularly on Promethium and the other lanthanides.

1. Paramagnetism falls to minimum in Nd (Neodymium)

Paramagnetism arises from unpaired electrons in the f-orbitals of the lanthanide elements. As you move across the 4f-series from Lanthanum to Lutetium, the number of unpaired f-electrons typically changes, leading to varying degrees of paramagnetism.

Neodymium (Nd) has a configuration of \([Xe] 4f^4\), meaning it has 4 unpaired electrons. Although Nd is indeed paramagnetic, it does not represent the point where paramagnetism is at a minimum. Instead, paramagnetism tends to increase across the series until it stabilizes or decreases at the end, particularly in the heavier lanthanides like Lutetium.

2. Lu³⁺ ions are bright green in aqueous solution

Lutetium (Lu) has a +3 oxidation state as Lu³⁺, which is often described as colorless or pale yellow when in solution. The coloration in lanthanide ions is typically due to f-f transitions. Lu³⁺ does not exhibit bright green coloration, making this statement incorrect.

3. Promethium is artificially synthesised 4f series element

Promethium (Pm) is unique among the lanthanides because it has no stable isotopes; all its isotopes are radioactive. It was first discovered in 1945 by researchers who synthesized it from uranium. Because of the lack of stable isotopes and its synthetic nature, Promethium is considered the only lanthanide that is artificially produced. 

Its synthesis typically involves the process of neutron capture by neodymium or other rare earth elements followed by beta decay. Promethium is used in certain applications, such as luminous paint and in atomic batteries.

4. Absorption bands of Lanthanoids are broad

The lanthanide series is known for its characteristic broad absorption bands due to the f-f electronic transitions. The broadness of these bands can be attributed to:

Spin-orbit coupling: This is significant in the f-orbitals, leading to a spread of energy levels.

Crystal field effects: When lanthanides are in a crystal lattice (like in solutions or solid-state), the local environment can affect the energy levels of the f-orbitals, resulting in broader transitions.

These broad absorption bands make lanthanides useful in various optical applications, such as phosphors in screens and lighting.

Summary

The statement about Promethium being an artificially synthesized 4f-series element is correct. Promethium’s unique properties, alongside its absence of stable isotopes, distinguish it from the other lanthanides, which mostly occur naturally. Understanding the behavior and characteristics of the lanthanide series is essential for applications in materials science, chemistry, and technology