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The atomic number of an element is a fundamental property that identifies the number of protons in the nucleus of an atom. This number is unique to each element and is critical in determining the element's place in the periodic table. More importantly for our discussion on electron configurations, the atomic number also tells us the number of electrons in a neutral atom, since the number of electrons equals the number of protons to balance the electric charge.

For example, in the exercise provided, Zirconium (Zr) has an atomic number of 40. This means a neutral atom of Zirconium will have 40 protons in its nucleus and, correspondingly, 40 electrons surrounding the nucleus. These electrons will be arranged in different orbitals following specific rules that minimize the energy of the atom. Proper understanding of the atomic number is crucial for writing the electron configurations of elements as it tells us where to start when filling electron orbitals.

Electron orbitals are regions around an atom's nucleus where electrons are most likely to be found. Each orbital can hold a maximum number of electrons: s-orbitals can hold up to 2 electrons, p-orbitals can hold up to 6, d-orbitals can hold up to 10, and f-orbitals can hold up to 14. The configuration of these electrons follows the aufbau principle, which states that electrons fill the lowest energy orbitals first before moving to higher-energy ones.

To illustrate, the ground state electron configuration for neutral Cobalt (Co), which has an atomic number of 27, is filled as follows: 1s虏 2s虏 2p鈦� 3s虏 3p鈦� 3d鈦� 4s虏. Here, you can see that electrons fill up orbitals from 1s to 4s, abiding by the energy levels and sublevel capacities. Helping students visualize the order of the orbitals can aid in understanding electron configurations.

When an atom loses electrons, it becomes a cation, a positively charged ion. The electron configuration of a cation is determined by removing electrons from the neutral atom's ground state configuration, starting with the highest energy orbital. This is in accordance with Hund's rule and the (n+l) rule, which dictate the order of filling and removing electrons.

For example, the technetium cation (Tc鲁鈦�), which loses three electrons, will have an electron configuration that starts with the neutral atom configuration and then removes the three electrons from the outermost orbitals. As illustrated in the solution, you remove electrons from the 4d orbital to arrive at the configuration for Tc鲁鈦�: 1s虏 2s虏 2p鈦� 3s虏 3p鈦� 4s虏 3d鹿鈦� 4p鈦� 4d虏. It is helpful for students to learn that for transition metals, electrons are generally removed from the s orbital before the d orbitals, despite s being filled after d at the ground state.