(a) What electron subshell is being filled for the rare earth series of elements on the periodic table? (b) What electron subshell is being filled for the actinide series?

Firstly, let's review what electron subshells are. The electron configuration of an atom depends on the distribution of its electrons across various energy levels and subshells. Each energy level consists of several subshells (designated as s, p, d, and f), which can hold a specific maximum number of electrons: 2 for s, 6 for p, 10 for d, and 14 for f. Now let's study the periodic table, specifically the rare earth series and actinide series, to determine which electron subshell is being filled. The rare earth series consists of the lanthanides and are located in the sixth period. The actinide series consists of the actinides and are located in the seventh period. Both series belong to the f-block in the periodic table.

The rare earth series consists of the lanthanide elements, which have atomic numbers ranging from 57 (Lanthanum) to 71 (Lutetium). Excluding the first two elements (Lanthanum and Cerium), the remaining elements in the lanthanides have their electron configurations following the pattern [Xe] 4f^(n-54) 5d^1 6s^2, where n is the atomic number of the element. The f subshell is being filled in this series, specifically the 4f subshell.

The actinide series consists of the actinide elements, which have atomic numbers ranging from 89 (Actinium) to 103 (Lawrencium). Excluding the first two elements (Actinium and Thorium), the remaining elements in the actinides have their electron configurations following the pattern [Rn] 5f^(n-88) 6d^(0-1) 7s^2, where n is the atomic number of the element. The f subshell is being filled in this series, specifically the 5f subshell.

To answer the given exercise, the electron subshell being filled for the following series of elements is: (a) The rare earth series: 4f subshell (b) The actinide series: 5f subshell