Question

Which of the following sets of quantum numbers are allowed? What atomic orbitals do the allowed combinations correspond to? (Section 3.5) (a) \(n=2,1=2, m_{1}=2\) (b) \(n=5, l=3, m_{1}=-2\) (c) \(n=3, l=-1, m_{1}=1\) (d) \(n=2, l=1, m_{i}=1\) (e) \(n=4, l=0, m_{1}=1\)

Short answer

Valid sets are (b) 5f orbital and (d) 2p orbital.

Step-by-step solution

Understand Quantum Numbers

Quantum numbers describe the properties of atomic orbitals and the properties of electrons in orbitals. There are four quantum numbers: the principal quantum number \( n \), the azimuthal quantum number \( l \), the magnetic quantum number \( m_l \), and the spin quantum number \( m_s \). For each combination, we'll verify if the numbers are within the allowable range and describe the corresponding orbital.

Analyze Each Set of Quantum Numbers

We need to check if each set of numbers obeys the rules for quantum numbers:1. \( n \) is a positive integer.2. \( l \) can be an integer from 0 to \( n-1 \).3. \( m_l \) can be an integer from \(-l\) to \( l \).

Evaluate Set (a)

For the set \((n=2, l=2, m_l=2)\): - The allowed \( l \) values for \( n=2 \) are 0 and 1. Since \( l=2 \) is not allowed, this set is not valid.

Evaluate Set (b)

For the set \((n=5, l=3, m_l=-2)\): - The \( l \) value for \( n=5 \) can be any integer from 0 to 4. - The \( m_l \) value for \( l=3 \) should range from \(-3\) to 3. - So, this set is valid, corresponding to the 5f orbital.

Evaluate Set (c)

For the set \((n=3, l=-1, m_l=1)\): - The \( l \) value cannot be negative. - Thus, this combination is not valid.

Evaluate Set (d)

For the set \((n=2, l=1, m_l=1)\): - The \( l \) value is 1, which is allowable for \( n=2 \).- The \( m_l \) value ranges from \(-1\) to 1, so \( m_l=1 \) is valid.- Thus, this combination is valid, corresponding to the 2p orbital.

Evaluate Set (e)

For the set \((n=4, l=0, m_l=1)\): - The allowed \( l \) values for \( n=4 \) are 0, 1, 2, 3. - For \( l=0 \), the only allowed \( m_l \) value is 0. Since \( m_l=1 \), this set is not valid.

Key concepts

Atomic Orbitals

Atomic orbitals are regions around the nucleus of an atom where electrons are most likely to be found. These regions are shaped and sized according to the principal quantum number and the azimuthal quantum number (often denoted as \( n \) and \( l \), respectively).
Atomic orbitals can be visualized as clouds around the nucleus, with the density of the cloud representing the probability of finding an electron in that location.
Here are important points to remember about atomic orbitals:

• Each principal energy level (\( n \)) can contain multiple orbitals of different shapes.
• The shape of an orbital is defined by the azimuthal quantum number \( l \), ranging from 0 to \( n-1 \).
• Commonly, electron orbitals are categorized into types such as s, p, d, and f, where these correspond to \( l \) values of 0, 1, 2, and 3, respectively.

Atomic orbitals are the building blocks of understanding atomic and electronic structure in chemistry and quantum mechanics.

Quantum Mechanics

Quantum mechanics is the branch of physics that deals with the behavior of matter and energy at atomic and subatomic levels. Unlike classical mechanics, quantum mechanics provides a probabilistic rather than deterministic view of how particles like electrons behave.

In quantum mechanics:

• Electrons are described in terms of wave functions, where the square of the wave function gives the probability of locating an electron at a particular point.
• The uncertainty principle underscores that certain pairs of properties, such as position and momentum, cannot be simultaneously measured with arbitrary precision.
• The concept of quantization means electrons occupy discrete energy levels, described by quantum numbers.

This framework is fundamental for explaining phenomena that classical physics cannot, like the electron configuration of atoms and the results of the photoelectric effect.
Overall, quantum mechanics has revolutionized our understanding of how atoms and particles interact at the microscopic level.

Electron Configuration

Electron configuration refers to the arrangement of electrons in the orbitals of an atom. Understanding the configuration provides insights into the atom's chemical properties and reactivity.

Quantum numbers play a crucial role in determining the electron configuration:

• Each electron in an atom is described by a unique set of four quantum numbers.
• The principal quantum number \( n \) indicates the main energy level occupied by the electron, with larger values of \( n \) corresponding to levels that are farther from the nucleus.
• The azimuthal quantum number \( l \) specifies the shape of the orbital, with letters like s, p, d, and f denoting orbital shapes.
• The magnetic quantum number \( m_l \) defines the orientation of the orbital in space.

Electrons fill orbitals starting from the lowest energy level to higher levels, following rules such as the Aufbau principle, Hund's rule, and the Pauli exclusion principle.
This sequential filling and arrangement lead to the diverse chemical behaviors seen in elements across the periodic table.