Problem 142 - Chemistry An Atoms First Approach

Question

Although no currently known elements contain electrons in \(g\) orbitals in the ground state, it is possible that these elements will be found or that electrons in excited states of known elements could be in \(g\) orbitals. For \(g\) orbitals, the value of \(\ell\) is 4 What is the lowest value of \(n\) for which \(g\) orbitals could exist? What are the possible values of \(m_{\ell} ?\) How many electrons could a set of \(g\) orbitals hold?

Step-by-Step Solution

Answer

The lowest value of \(n\) for which \(g\) orbitals can exist is 5. The possible values of \(m_{l}\) for \(g\) orbitals are -4, -3, -2, -1, 0, 1, 2, 3, and 4. A set of \(g\) orbitals can hold a maximum of 18 electrons.

Step 1: Understanding quantum numbers

Principal quantum number (n) determines the energy level of an electron in an atom. It can have positive integer values (n = 1, 2, 3, ...). The angular momentum quantum number (l) signifies the shape of the orbital and ranges from 0 to n-1. This means that for a specific energy level (n), there can be n different values of l (0, 1, 2, ... n-1). Finally, the magnetic quantum number (m_l) determines the orientation of the orbital in space and takes integer values from -l to +l, including 0. This means that for a given value of l, there will be 2l+1 possible values of m_l.

Step 2: Finding the lowest value of n for g orbitals

We are given that for g orbitals, the value of l is 4. As we know that l ranges from 0 to n-1, we can find the lowest value of n for which g orbitals can exist by setting l equal to n-1: \[ l = n - 1 \\ 4 = n - 1 \\ n = 5 \] So, the lowest value of n for which g orbitals can exist is 5.

Step 3: Finding the possible values of m_l

The magnetic quantum number (m_l) ranges from -l to +l, including 0. Since l is 4 for g orbitals, the possible values of m_l are: \[ m_{l} = -4, -3, -2, -1, 0, 1, 2, 3, 4 \]

Step 4: Calculating the number of electrons a set of g orbitals can hold

We know that each orbital can hold a maximum of 2 electrons (due to the Pauli Exclusion Principle). As there are 2l+1 orbitals for a given value of l, we can find the number of electrons a set of g orbitals can hold with l=4 by using this formula: \[ \text{Number of electrons} = 2(2l + 1) = 2(2(4) + 1) = 18 \] So, a set of g orbitals can hold a maximum of 18 electrons.

Key Concepts

g Orbitals

Orbitals are regions in an atom where there is a high likelihood of finding an electron. When we speak of "g orbitals," these are a specific kind of orbital with a particular angular momentum quantum number. For g orbitals, the angular momentum quantum number (\( \ell \)) is 4. The principal quantum number (\( n \)), which determines the energy level, needs to be greater than \( \ell \), so for g orbitals, \( n \) must be at least 5. That's because \( \ell \) ranges from 0 up to \( n-1 \). Although g orbitals are not yet observed in the ground state of any known element, scientists predict that they might become important for newly discovered elements, or in excited states of existing elements. The intriguing property of g orbitals is their potential to house 18 electrons due to having more sub-orbitals compared to s, p, d, and f orbitals.

Angular Momentum Quantum Number

The angular momentum quantum number, symbolized as \( \ell \), is a crucial concept in quantum mechanics as it describes the shape of an electron's orbital. It can take on integer values from 0 up to \( n-1 \), where \( n \) is the principal quantum number, denoting the energy level.Each value of \( \ell \) corresponds to a particular type of shape among an orbital family:

• \( \ell = 0 \): s orbital (spherical)
• \( \ell = 1 \): p orbital (dumbbell-shaped)
• \( \ell = 2 \): d orbital (cloverleaf or other complex shapes)
• \( \ell = 3 \): f orbital (even more complex shapes)
• \( \ell = 4 \): g orbital (futures shapes, very complex)

These different shapes allow electrons identified by these quantum numbers to occupy different regions of space, minimizing repulsion and optimizing the atom's structure.

Magnetic Quantum Number

The magnetic quantum number is denoted by \( m_\ell \) and defines the orientation of an orbital in space relative to an external magnetic field. Once \( \ell \) is determined, \( m_\ell \) takes on integer values ranging from \( -\ell \) to \( +\ell \).For g orbitals where \( \ell = 4 \), possible values for \( m_\ell \) are:

• -4, -3, -2, -1, 0, 1, 2, 3, 4

This range indicates there are 9 different orientations of g orbitals for an electron to occupy. The significance of distinct magnetic quantum numbers lies in the electron's ability to be distributed amongst these various orientations, reducing inter-electron repulsions and stabilizing the atom in complex ways.

Electron Configuration

Electron configuration is a method of noting the arrangement of electrons in an atom in terms of its orbitals. For any atom, the configuration is determined by its quantum numbers. Each electron in an atom is represented by a set of quantum numbers and fills the orbitals from lower energy to higher energy — a principle known as the Aufbau principle. Here's a brief guide to electron configuration:

• Start filling with 1s, then progress through 2s, 2p, 3s, 3p, 4s... and so forth.
• Adhere to the Pauli Exclusion Principle: no two electrons can have the same set of quantum numbers.
• Follow Hund's Rule: each orbital in a subshell gets one electron before any gets two.

It's worth noting that while there's currently no known element or electron configuration that utilizes g orbitals in their ground state, the understanding of electron configurations provides a framework for accommodating such complexities in future discoveries or excited states.