Problem 159 - Introductory Chemistry Atoms First

Question

Write the ground-state electron configuration for each of the following atoms or ions. Which have a valence-shell octet? (a) \(\overline{\mathrm{Ar}}\) (b) \(\mathrm{Na}^{+}\) (c) \(\mathrm{C}^{2-}\) (d) \(\mathrm{O}^{2-}\) (e) \(\mathrm{Ca}^{2+}\)

Step-by-Step Solution

Answer

The ground-state electron configurations for the given atoms and ions are: (a) \(\overline{\mathrm{Ar}}\): \(\mathrm{1s^{2}\,2s^{2}\,2p^{6}\,3s^{2}\,3p^{6}}\) (b) \(\mathrm{Na}^{+}\): \(\mathrm{1s^{2}\,2s^{2}\,2p^{6}}\) (c) \(\mathrm{C}^{2-}\): \(\mathrm{1s^{2}\,2s^{2}\,2p^{4}}\) (d) \(\mathrm{O}^{2-}\): \(\mathrm{1s^{2}\,2s^{2}\,2p^{6}}\) (e) \(\mathrm{Ca}^{2+}\): \(\mathrm{1s^{2}\,2s^{2}\,2p^{6}\,3s^{2}\,3p^{6}}\) The atoms or ions having a valence-shell octet are: \(\overline{\mathrm{Ar}}\), \(\mathrm{Na}^{+}\), \(\mathrm{O}^{2-}\), and \(\mathrm{Ca}^{2+}\).

Step 1: Understanding Aufbau Principle, Hund's Rule and Pauli Exclusion Principle for electron distribution

To determine the ground-state electron configuration, it's important to remember these three principles: - Aufbau Principle: Electrons are placed in orbitals in order of increasing energy (lowest energy levels first). - Hund's Rule: Electrons occupy orbitals of the same energy level (degenerate orbitals) in a way that maximizes the total number of parallel spins. - Pauli Exclusion Principle: No more than 2 electrons can occupy the same orbital, and these 2 electrons must have opposite spins.

Step 2: Determine the ground-state electron configurations for each atom or ion

Now we will find the ground-state electron configurations for each atom/ion using the above principles. Here are the atomic numbers of each element corresponding to the protons, which equals the number of electrons in a neutral atom: - Argon (Ar): 18 - Sodium (Na): 11 - Carbon (C): 6 - Oxygen (O): 8 - Calcium (Ca): 20 For each element, we have to adjust the number of electrons according to the given ion charge. (a) \(\overline{\mathrm{Ar}}\) (neutral): 18 electrons Electron configuration: \(\mathrm{1s^{2}\,2s^{2}\,2p^{6}\,3s^{2}\,3p^{6}}\) (b) \(\mathrm{Na}^{+}\): 10 electrons (lost 1 electron) Electron configuration: \(\mathrm{1s^{2}\,2s^{2}\,2p^{6}}\) (c) \(\mathrm{C}^{2-}\): 8 electrons (gained 2 electrons) Electron configuration: \(\mathrm{1s^{2}\,2s^{2}\,2p^{4}}\) (d) \(\mathrm{O}^{2-}\): 10 electrons (gained 2 electrons) Electron configuration: \(\mathrm{1s^{2}\,2s^{2}\,2p^{6}}\) (e) \(\mathrm{Ca}^{2+}\): 18 electrons (lost 2 electrons) Electron configuration: \(\mathrm{1s^{2}\,2s^{2}\,2p^{6}\,3s^{2}\,3p^{6}}\)

Step 3: Identify which atoms or ions have a valence-shell octet

To determine which atoms or ions have a valence-shell octet, we need to check the outermost shell (highest principal quantum number) of each atom/ion: (a) \(\overline{\mathrm{Ar}}\): Valence-shell (n=3) has 8 electrons (3s² and 3p⁶), so it has an octet. (b) \(\mathrm{Na}^{+}\): Valence-shell (n=2) has 8 electrons (2s² and 2p⁶), so it has an octet. (c) \(\mathrm{C}^{2-}\): Valence-shell (n=2) has 6 electrons (2s² and 2p⁴), so it does not have an octet. (d) \(\mathrm{O}^{2-}\): Valence-shell (n=2) has 8 electrons (2s² and 2p⁶), so it has an octet. (e) \(\mathrm{Ca}^{2+}\): Valence-shell (n=3) has 8 electrons (3s² and 3p⁶), so it has an octet. Therefore, the atoms or ions having a valence-shell octet are: \(\overline{\mathrm{Ar}}\), \(\mathrm{Na}^{+}\), \(\mathrm{O}^{2-}\), and \(\mathrm{Ca}^{2+}\).

Key Concepts

Aufbau Principle

The Aufbau Principle is a fundamental guideline for determining the electron configuration of an atom. This principle states that electrons fill atomic orbitals in order of increasing energy levels. By following this principle, electrons will first occupy the lowest energy orbitals available before moving on to higher ones.

• The order of filling starts with the 1s orbital, the lowest energy level, followed by 2s, 2p, 3s, 3p, and so on.
• This sequence can be easily remembered through the use of the diagonal rule or an Aufbau diagram, which charts the filling order.

The Aufbau Principle ensures that the most stable electron configuration is achieved by minimizing the amount of energy in an atom. Failure to follow this principle can result in excited states, where electrons exist at higher energy levels than necessary.

valence-shell octet

The concept of a valence-shell octet revolves around the stability that comes when an atom's outermost electron shell is completely filled with eight electrons. This configuration is particularly stable because it mirrors the electron arrangement of noble gases, which are inherently non-reactive.

• Atoms strive to achieve a valence-shell octet by sharing, gaining, or losing electrons through chemical reactions.
• For many elements, such as carbon, oxygen, and sodium, reaching a valence-shell octet often involves forming ions or covalent bonds.

The octet rule is a guideline, which is central in predicting the chemical bond formation for many elements in the periodic table, especially those in groups 1-2 and 13-18. However, there are exceptions to this rule, particularly among transition metals and heavier elements.

Hund's Rule

Hund's Rule is another essential principle used when writing electron configurations. This rule states that electrons will fill degenerate orbitals (those of the same energy level) in a manner that maximizes multiplicity, meaning parallel spins are preferred.

• Electrons will occupy each orbital singly before any orbital is doubly filled with paired electrons.
• This arrangement helps to minimize electron repulsion, as electrons with parallel spins are less likely to overlap spatially.

By following Hund's Rule, atoms achieve a more stable electronic structure. This is especially important when dealing with p, d, and f orbitals, where multiple degenerate orbitals are present at the same energy level.

Pauli Exclusion Principle

The Pauli Exclusion Principle is a foundational concept in quantum mechanics and chemistry. According to this principle, no two electrons in an atom can have the same set of four quantum numbers. This essentially means that an orbital can contain a maximum of two electrons, and their spins must be paired (one with spin +1/2 and the other with spin -1/2).

• The limitation of two electrons per orbital ensures that the electrons maintain distinct characteristics, contributing to the unique electron configuration of each element.
• Paired opposite spins within the same orbital help to balance the magnetic fields generated by the electrons, stabilizing the atom.

The Pauli Exclusion Principle underpins the organization of the periodic table and affects how chemical bonds form between atoms, by dictating electron pairing and distribution within orbitals.