Problem 93 - Chemistry: An Atoms-Focused Approach

Question

What are the electron configurations of \(\mathrm{Li}^{+}, \mathrm{Ca}, \mathrm{F}^{-}, \mathrm{Mg}^{2+}\) and \(A 1^{3+} ?\)

Step-by-Step Solution

Answer

Answer: The electron configurations are: \(\mathrm{Li}^{+}: 1s^{2}\) \(\mathrm{Ca}: 1s^{2}2s^{2}2p^{6}3s^{2}3p^{6}4s^{2}\) \(\mathrm{F}^{-}: 1s^{2}2s^{2}2p^{6}\) \(\mathrm{Mg}^{2+}: 1s^{2}2s^{2}2p^{6}\) \(A 1^{3+}: 1s^{2}2s^{2}2p^{6}\)

Step 1: Determine the atomic number and number of electrons for each ion

We will use the periodic table to find the atomic number (Z) of each element: \(\mathrm{Li}\) has an atomic number of 3 \(\mathrm{Ca}\) has an atomic number of 20 \(\mathrm{F}\) has an atomic number of 9 \(\mathrm{Mg}\) has an atomic number of 12 \(\mathrm{A 1}\) has an atomic number of 13 (since it is Al, which is aluminum) Now, we will determine the number of electrons for each ion by adding or subtracting the charge from the atomic number: \(\mathrm{Li}^{+}\) has 2 electrons (3 - 1) \(\mathrm{Ca}\) has 20 electrons (20 - 0) \(\mathrm{F}^{-}\) has 10 electrons (9 + 1) \(\mathrm{Mg}^{2+}\) has 10 electrons (12 - 2) \(A 1^{3+}\) has 10 electrons (13 - 3)

Step 2: Use Aufbau principle, Hund's rule, and Pauli Exclusion Principle to find electron configurations

We will now use the rules to determine each electron configuration. For \(\mathrm{Li}^{+}\): 1s: 2 electrons Electron configuration: \(1s^{2}\) For \(\mathrm{Ca}\): 1s: 2 electrons 2s: 2 electrons 2p: 6 electrons 3s: 2 electrons 3p: 6 electrons 4s: 2 electrons Electron configuration: \(1s^{2}2s^{2}2p^{6}3s^{2}3p^{6}4s^{2}\) For \(\mathrm{F}^{-}\): 1s: 2 electrons 2s: 2 electrons 2p: 6 electrons Electron configuration: \(1s^{2}2s^{2}2p^{6}\) For \(\mathrm{Mg}^{2+}\): 1s: 2 electrons 2s: 2 electrons 2p: 6 electrons Electron configuration: \(1s^{2}2s^{2}2p^{6}\) For \(A 1^{3+}\): 1s: 2 electrons 2s: 2 electrons 2p: 6 electrons Electron configuration: \(1s^{2}2s^{2}2p^{6}\) So the electron configurations for the given ions are: \(\mathrm{Li}^{+}: 1s^{2}\) \(\mathrm{Ca}: 1s^{2}2s^{2}2p^{6}3s^{2}3p^{6}4s^{2}\) \(\mathrm{F}^{-}: 1s^{2}2s^{2}2p^{6}\) \(\mathrm{Mg}^{2+}: 1s^{2}2s^{2}2p^{6}\) \(A 1^{3+}: 1s^{2}2s^{2}2p^{6}\)

Key Concepts

Atomic Number

The atomic number of an element is fundamental to its identity. It is the number of protons found in the nucleus of an atom and is denoted by the symbol \(Z\). This number is essential because it defines the element and determines its position on the periodic table. For example, the atomic number of lithium (Li) is 3, which means it has three protons in its nucleus.

When determining electron configurations, knowing \(Z\) allows us to find out the neutral atom's number of electrons, as a neutral atom will have an equal number of protons and electrons. In the case of ions, the charge must be considered to calculate the number of electrons.

• Positive ions (cations) are formed by losing electrons, so the number of electrons is \(Z\) minus the positive charge.
• Negative ions (anions) gain electrons, so the number of electrons is \(Z\) plus the added charge.

Aufbau Principle

The Aufbau Principle is a fundamental rule for constructing electron configurations. In German, "Aufbau" means "building up" which is precisely what this principle describes.

According to this principle, electrons fill atomic orbitals from the lowest to highest energy levels to minimize the energy of the atom. It implies electrons will fill the \(1s\) orbital before \(2s\), then fill \(2p\), and so on up the energy ladder according to the standard sequence.

For example, when calculating the electron configuration for calcium (Ca), first the \(1s\) is filled, followed by \(2s\), then \(2p\), \(3s\), \(3p\), and finally \(4s\). This method ensures the atom is in its most stable (lowest energy) state. When constructing the electron configuration of an ion, the filled orbitals reflect this "building up" process.

Hund's Rule

Hund's Rule is a guiding principle that further specifies how electrons are to be distributed among orbitals of the same sublevel (equal energy orbitals).

The rule states that all orbitals in a given sublevel would be singly occupied before any orbital is doubly occupied. Moreover, all singly occupied orbitals will have electrons with the same spin direction. This minimizes electron repulsion, achieving more stability.

For example, when distributing electrons in the \(2p\) sublevel for \(F^{-}\), each of the three \(2p\) orbitals will first hold one electron before any start to pair, and they all will spin in the same direction. This rule is crucial in determining accurate electron configurations and is employed after considering the Aufbau Principle requirements.

Pauli Exclusion Principle

The Pauli Exclusion Principle is a foundational rule in quantum mechanics, asserting that no two electrons in an atom can have identical sets of quantum numbers. This effectively means that each orbital can hold a maximum of two electrons, which must have opposite spins.

For instance, in the \(1s\) orbital, after placing the first electron (spin-up), the second must be spin-down. This ensures each electron has a unique set of quantum numbers even though they're located in the same orbital.

• The principle is key in preventing more than two electrons from occupying the same orbital.
• It ensures that electrons are properly distributed across available orbitals within sublevels to maintain the individuality of each electron.

The Pauli Exclusion Principle complements the Aufbau Principle and Hund’s Rule in defining how electrons populate atomic orbitals, enforcing the unique configuration of every element and ion.