Despite the similarities in the chemical reactivity of elements in the lanthanide series, their abundances in Earth's crust vary by two orders of magnitude. This graph shows the relative abundance as a function of atomic number. Which of the following statements best explains the sawtooth variation across the series? (a) The elements with an odd atomic number lie above the belt of stability. (b) The elements with an odd atomic number lie below the belt of stability. (c) The elements with an even atomic number have a magic number of protons. (d) Pairs of protons have a special stability.

The belt of stability is an area on a graph that represents where stable nuclei exist. It represents combinations of protons and neutrons for which the nuclei are stable. Stable nuclei generally have more neutrons than protons.

In nuclear physics, magic numbers are the number of protons or neutrons (separately) that give rise to stable nuclei. They are the closed-shell configurations, meaning the corresponding energy levels are fully occupied, making it more stable.

Pairs of protons in a nucleus are often more stable due to the strong nuclear force. They can be part of a spin-coupled pair, which is more stable than single particles within the atomic nucleus.

(a) Elements with an odd atomic number lie above the belt of stability. - The elements with an odd atomic number can lie either above or below the belt of stability, so this statement is not necessarily correct. (b) Elements with an odd atomic number lie below the belt of stability. - Same as for option (a), so it is also not correct. (c) Elements with an even atomic number have a magic number of protons. - Having an even atomic number doesn't automatically mean there is a magic number of protons. Magic numbers are specific for particular values of atomic numbers, which may not cover the entire even atomic numbers. Therefore, this statement is not correct. (d) Pairs of protons have a special stability. - This statement is true, and the stable nature of pairings of protons and neutrons can explain the variation in the relative abundance of elements in the Lanthanide series.

Based on the analysis of each statement, we can conclude that the correct statement is (d) Pairs of protons have a special stability.