This carbon isotope is radioactive and very rare

The correct option is Carbon – 14.

To understand why Carbon-14 is the correct answer, we can look at the characteristics of the different isotopes of carbon: Carbon-12, Carbon-13, and Carbon-14.
An isotope is defined by the number of neutrons in its nucleus, while the number of protons remains constant (which is 6 for carbon).

1. Carbon-12: This is the most common isotope of carbon, making up about 98.9% of all natural carbon on Earth. It has 6 protons and 6 neutrons, and it is completely stable (non-radioactive).

2. Carbon-13: This is another stable isotope of carbon, accounting for about 1.1% of natural carbon. It has 6 protons and 7 neutrons and is also non-radioactive.

3. Carbon-14: This isotope has 6 protons and 8 neutrons. Unlike Carbon-12 and Carbon-13, Carbon-14 is unstable and undergoes radioactive decay (specifically beta decay) with a half-life of approximately 5,730 years. Furthermore, it is extremely rare, occurring in trace amounts of about 1 part per trillion (1 in 10¹²) carbon atoms in the atmosphere.

Therefore, Carbon-14 is both radioactive and very rare, matching the description in the question.