Which of the following lanthanoid is radioactive?

Promethium

The correct answer is option 4. Promethium.

Promethium (Pm) is the only naturally radioactive lanthanoid (lanthanide) element. Here's why:

1. Isotopes: Promethium does not have any stable isotopes. All its isotopes are radioactive, meaning they undergo radioactive decay by emitting radiation.

2. Promethium-145: The most stable isotope of promethium is promethium-145, but even this isotope is radioactive. It has a relatively short half-life of about 17.7 years. This means that over time, half of the promethium-145 atoms in a sample will decay into other elements.

3. Radioactive Decay: When promethium undergoes radioactive decay, it emits radiation, which can include alpha particles, beta particles, and gamma rays. This emission of radiation is characteristic of radioactive elements.

Because promethium lacks stable isotopes and undergoes radioactive decay, it is considered a naturally occurring radioactive element among the lanthanoids.

In contrast, the other lanthanoids like Thulium, Neodymium, and Terbium do not exhibit natural radioactivity because they have stable isotopes or no isotopes at all with significant radioactivity.

Therefore, among the lanthanoids, only Promethium is naturally radioactive.