Problem 73 - Chemistry: Principles and Reactions

Question

No currently known elements contain electrons in \(g(\ell=4)\) orbitals in the ground state. If an element is discovered that has electrons in the g or- bital, what is the lowest value for \(\mathbf{n}\) in which these \(\mathrm{g}\) orbitals could exist? What are the possible values of \(\mathbf{m}_{\ell} ?\) How many electrons could a set of \(\mathrm{g}\) orbitals hold?

Step-by-Step Solution

Answer

Answer: The lowest principal quantum number for g orbitals is 5, and they can hold a maximum of 18 electrons.

Step 1: Find the lowest value of n for g orbitals

Given that \(\ell=4\) for g orbitals, we need to determine the smallest possible value of n. Since n can only be a positive integer, and \(\ell\) ranges from 0 to n-1, the smallest possible value of n is when \(\ell = n-1\). In this case, we have: \(n-1 = 4\) Now, add 1 to both sides to find the value of n: \(n = 4 + 1\) \(n = 5\) So, the lowest possible value of n where g orbitals can exist is 5.

Step 2: Find the possible values of \(m_{\ell}\) for g orbitals

To find the possible values of \(m_\ell\), we need to use the range from \(-\ell\) to \(+\ell\). Since \(\ell = 4\), the values of \(m_\ell\) range from \(-4\) to \(+4\): \(m_\ell = -4,-3,-2,-1,0,1,2,3,4\) So, there are a total of 9 possible values of \(m_\ell\) for g orbitals.

Step 3: Determine the number of electrons that can be held in g orbitals

The number of electrons that can be held in a set of orbitals can be found by the formula: Number of electrons = \((2\ell + 1) * 2\) Here, we multiply by 2 because each orbital can hold a maximum of two electrons with opposite spins. For g orbitals, we have \(\ell = 4\), so we have: Number of electrons = \((2\cdot4 + 1) * 2\) Number of electrons = \((9)*2\) Number of electrons = 18 Therefore, a set of g orbitals can hold a maximum of 18 electrons.

Key Concepts

Electronic Configuration

Electronic configuration refers to how electrons are distributed in an atom's orbitals, which are determined by quantum numbers. It provides the atom's 'address' showing where each electron 'lives'. Knowing the electronic configuration is crucial as it influences the atom's chemical properties and behavior.

Every atom follows a specific order of filling its orbitals, known as the Aufbau principle. This principle suggests that orbitals are filled in order of increasing energy, starting from the lowest. So, electrons occupy the nearest orbital to the nucleus first before moving to higher energy levels.

Standard Notation:

• The notation for electronic configuration lists the energy level and sublevel (orbital type: s, p, d, f, g) and indicates the number of electrons in these orbitals using superscripts.
• For instance, the configuration of a chlorine atom is 1s² 2s² 2p⁶ 3s² 3p⁵.

When a new type of orbital, such as the g orbital, becomes involved, it indicates the discovery of new elements or requires considering excited states. This ties into determining where such configurations might begin, as the question of g orbitals suggests.

Atomic Orbitals

Atomic orbitals are regions of space around an atom's nucleus where electrons have a high probability of being found. They are defined by quantum numbers and come in different shapes (s, p, d, f, and g, etc.). Each shape and orientation is determined by the values of the principal quantum number (n) and the azimuthal quantum number (ℓ).

For orbitals:

• Principal Quantum Number (n): This number determines the energy level and size of the orbital, where a higher n indicates larger orbitals further from the nucleus. For instance, g orbitals first appear when n=5.
• Azimuthal Quantum Number (ℓ): It defines the shape. For g orbitals, ℓ=4.
• Magnetic Quantum Number (mℓ): It describes the orientation in space and has values between -ℓ and +ℓ. With ℓ=4, g orbitals have 9 possible orientations: -4 through +4, showing how they align differently.

Understanding these attributes helps in visualizing how electrons inhabit these orbitals and how they contribute to the chemical bonding and properties of atoms.

Quantum Mechanics

Quantum mechanics is the fundamental theory in physics that provides the foundation for understanding atomic and subatomic processes. It explains the behavior of electrons in atoms, which do not follow the rules of classical physics, such as predictable pathways.

Essential Quantum Concepts:

• Electrons exhibit both particle and wave-like properties. This dual nature is crucial to explaining their behavior in orbitals.
• Uncertainty Principle: Proposed by Heisenberg, it states that one cannot simultaneously know both the position and momentum of an electron with absolute precision. Thus, electron locations are described probabilistically.
• Quantum Numbers: Indicate properties of electrons: energy level (n), shape (ℓ), orientation (mℓ), and spin (ms). Each combination of these numbers points to a unique state of an electron in an atom.

Quantum mechanics challenges everyday intuition, but it is essential for describing why and how electrons arrange themselves, giving rise to the concept of electronic configuration.