Problem 34 - General Chemistry: Principles and Modern Applications

Question

Which of the following species has the greatest number of unpaired electrons (a) \(\mathrm{Ge} ;\) (b) \(\mathrm{Cl} ;\) (c) \(\mathrm{Cr}^{3+}\) (d) Br -?

Step-by-Step Solution

Answer

The species with the greatest number of unpaired electrons is \(Cr^{3+}\) as it has 3 unpaired electrons.

Step 1: Identify Atomic Numbers

The atomic numbers for each species are as follows: Ge (32), Cl (17), Cr (24), Br (35). These atomic numbers are found on the periodic table.

Step 2: Determine the electron configuration

Write down the electron configurations according to the Aufbau principle, Hund's rule, and Pauli's exclusion principle.\n For Ge: \[1s^2 2s^2 2p^6 3s^2 3p^6 3d^{10} 4s^2 4p^2\]\n For Cl: \[1s^2 2s^2 2p^6 3s^2 3p^5\]\n For Cr3+: Chromium has a special electron configuration. Its ground-state electron configuration is: \[1s^2 2s^2 2p^6 3s^2 3p^6 4s^1 3d^5\] however in case of Cr3+, three electrons are lost, two from 4s orbital and one from 3d, hence the configuration is: \[1s^2 2s^2 2p^6 3s^2 3p^6 3d^3\]\n For Br-: When Bromine gains an electron, become Br-. Its electron configuration is: \[1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^{10} 4p^6\]

Step 3: Identify Number of Unpaired Electrons

Now, count the number of unpaired electrons in the last shell for each species. Ge has 2, Cl has 1, Cr3+ has 3 and Br- has 0 unpaired electron.

Key Concepts

Unpaired Electrons

Unpaired electrons are electrons that occupy an atomic or molecular orbital on their own, without a partner with opposite spin. These electrons play a significant role in chemical reactivity and magnetism. When examining the electron configurations of elements or ions, look for partially filled orbitals to identify unpaired electrons.
In the given exercise, each species: Ge, Cl, Cr^{3+}, and Br^- has specific unpaired electrons. Ge has two unpaired electrons in its 4p orbital, Cl possesses one unpaired electron in the 3p orbital, Cr^{3+} has three unpaired electrons residing within the 3d orbital, and Br^- has no unpaired electrons as its orbitals are fully paired. Understanding the counting of unpaired electrons helps in predicting an element's magnetic properties, with more unpaired electrons leading to stronger magnetic effects, a concept known as paramagnetism.

Aufbau Principle

The Aufbau Principle is a fundamental guide for constructing the electron configuration of atoms in ground state. Its name is derived from the German word 'Aufbau,' which means 'building up.' According to this principle, electrons occupy orbitals from lowest to highest energy.
Here's how it works:

• Begin filling orbitals from 1s, then 2s, 2p, 3s, and continue onwards, following the increasing order of energy levels.
• Each orbital can hold a maximum of two electrons with opposite spins.

For instance, in the complete electron configuration of Ge mentioned in the exercise, each orbital fills according to energy hierarchy starting from 1s up to the 4p orbital. It ensures that at each step, the lowest energy arrangement for the electrons is achieved, which is crucial for describing an atom's ground state.

Hund's Rule

Hund's Rule states that for electrons filling degenerate orbitals, which are orbitals of equal energy, electrons will fill the orbitals singly as much as possible, pairing up only when necessary. This minimizes electron-electron repulsion and results in greater stability.
To apply Hund's Rule:

• Place one electron in each orbital of the same energy before starting to pair.
• Ensure that unpaired electrons in degenerate orbitals have parallel spins to reduce repulsive interactions.

An example from the exercise is Cr^{3+}. After removing three electrons, we end up with three unpaired electrons in the 3d orbitals. According to Hund's Rule, these electrons will occupy different d orbitals with their spins aligned, ensuring maximum stability.

Pauli Exclusion Principle

The Pauli Exclusion Principle is a principle stating that no two electrons within an atom can have the same set of four quantum numbers, ensuring that each electron in an atom has its unique "address." The principle plays a critical role in determining electron configurations and understanding atomic structure.
Key aspects of Pauli Exclusion Principle include:

• Each atomic orbital can hold a maximum of two electrons.
• The two electrons occupying the same orbital must have opposite spins, usually denoted as \( +\frac{1}{2} \) and \( -\frac{1}{2} \).

In the electron configurations mentioned in the exercise, this principle ensures that within each orbital, especially in multi-electron atoms, electrons avoid having the same set of quantum numbers, leading to the unique distribution seen in the configurations of Ge, Cl, Cr^{3+}, and Br^-.