Write the electron configurations for each of the following elements and its \(3+\) ions: (a) La (b) Sm (c) Lu

Understanding the Aufbau principle is crucial when we want to determine the electron configuration of an atom. It's a guideline that helps us see how electrons populate an atom's orbitals. Aufbau is German for 'building up,' and thus the Aufbau principle is also called the 'build-up' principle.

The principle states that electrons fill atomic orbitals of the lowest available energy levels before occupying higher levels. In simple terms, electrons start filling up the 'seats' from the front of the room (lower energy levels) and work their way to the back (higher energy levels).

To properly fill the orbitals with electrons according to this principle, there's a specific order: 1s, 2s, 2p, 3s, 3p, and so forth. However, the order isn't always so straightforward due to some overlaps in energy levels, especially when we reach the d and f orbitals. Remembering this can be facilitated by using diagrams or mnemonics such as the 'diagonal rule' which can help visualise the order of orbital filling.

Closely associated with the Aufbau principle is the orbital filling order, which tells us the sequence in which subshells are filled with electrons. The general order (also mentioned in the Aufbau principle section) starts with 1s, the lowest energy orbital, and moves across different energy levels and subshells in a specific pattern. A popular tool for determining the order is the use of an energy level diagram, where arrows guide us through the proper sequence of filling.

Here's a simple break down of the order for the first few energy levels:

• 1s
• 2s
• 2p
• 3s
• 3p
• 4s
• 3d
• 4p
• 5s
• 4d

However, with more complex atoms, the order can get trickier, as is the case with transition metals and lanthanides, involving the 3d, 4d, 4f, and 5d orbitals. This is why the original step-by-step solutions for La, Sm, and Lu consider the specific order in which the 4f and 5d orbitals are filled.

The step-by-step solutions for ions of La, Sm, and Lu demonstrate the concept of ionic electron configuration. When an atom loses or gains electrons, it forms an ion. The electron configuration for these ions is not the same as their neutral atoms – they lose (or gain) electrons from their outermost orbitals.

For example, when discussing La^(3+), we start with La's neutral ground state configuration and then remove three electrons to reflect the positive charge, which indicates a loss of three electrons. It's essential to remove these electrons from the outermost orbitals first, following what's known as the 'reverse of the Aufbau principle'.

After removing the correct number of electrons, the electron configuration may simplify since some orbital sets become completely empty. The 'simplified' version of the electron configuration focuses on the filled orbitals that remain. This step is clearly illustrated in the solutions provided in the original exercise where the configurations for La^(3+), Sm^(3+), and Lu^(3+) ions are simpler than their neutral states.