Question

Which quantum number is related to a. the orientation of the orbital in space? b. the size of the orbital? c. the magnetic moment produced by an electron d. the shape of the orbital?

Short answer

a. The orientation of the orbital in space is related to the magnetic quantum number (\(m_l\)). b. The size of the orbital is related to the principal quantum number (\(n\)). c. The magnetic moment produced by an electron is related to the spin quantum number (\(m_s\)). d. The shape of the orbital is related to the azimuthal quantum number (angular momentum quantum number, \(l\)).

Step-by-step solution

Identify the quantum number for orientation

The quantum number related to the orientation of the orbital in space is the magnetic quantum number, denoted as \(m_l\). b. the size of the orbital?

Identify the quantum number for size

The quantum number related to the size of the orbital is the principal quantum number, denoted as \(n\). c. the magnetic moment produced by an electron

Identify the quantum number for magnetic moment

The quantum number related to the magnetic moment produced by an electron is the spin quantum number, denoted as \(m_s\). d. the shape of the orbital?

Identify the quantum number for shape

The quantum number related to the shape of the orbital is the azimuthal quantum number, also known as the angular momentum quantum number, denoted as \(l\).

Key concepts

Magnetic Quantum Number

The magnetic quantum number, symbolized by \(m_l\), is a crucial quantum number that helps define the orientation of an electron's orbital in space. It originates from the azimuthal quantum number \(l\) and takes on integer values ranging from \(-l\) to \(+l\).
This range indicates how an orbital can orient itself differently in a three-dimensional space.When electrons occupy sublevels, the value of \(m_l\) determines the direction the orbital can face with respect to a three-dimensional plane. For instance, in a \(p\)-sublevel where \(l = 1\), \(m_l\) can have values of \(-1, 0,\) or \(+1\), representing three distinct orientations.
This means the \(p\) orbital can point along the \(x\), \(y\), or \(z\) axis. Such orientation is instrumental in bonding characteristics and the interaction of electrons with external magnetic fields.

Principal Quantum Number

The principal quantum number, designated by \(n\), is fundamental in shaping the atom's electron configuration. It primarily indicates the size and energy level of an electron's orbital.
The value of \(n\) is a positive integer \(1, 2, 3, \ldots\), and corresponds to the shells or energy levels where electrons reside around an atom's nucleus.The larger the value of \(n\), the larger the orbital size and the higher the energy level. For example, \(n = 1\) reflects an electron in the closest orbit to the nucleus, and \(n = 3\) indicates a higher, more distant shell with greater energy.
Each principal energy level can accommodate a specific number of electrons, determined by the formula \(2n^2\), providing insight into an atom's electron capacity for each shell.
This quantum number sets the framework for electron configuration and the periodic trends seen across elements.

Spin Quantum Number

The spin quantum number, identified by \(m_s\), reflects an electron's intrinsic angular momentum, or spin, and affects its magnetic moment. This quantum number is not derived from other quantum numbers but is an independent characteristic of electrons.
It can assume one of two values: \(\pm \frac{1}{2}\). These values signify the two possible spin states of an electron within an orbital, often referred to as "spin-up" and "spin-down."The concept of electron spin is essential in explaining fine structure in spectral lines and magnetic properties of materials. Electrons in the same orbital must have opposite spins, as per the Pauli Exclusion Principle.
This results in minimizing repulsion between electrons and stabilizing the atom.
As electrons have charges, when they exhibit spin, they produce a magnetic moment. This aspect is vital in phenomena like electron pairing, magnetism, and many types of spectroscopy used to study molecular and atomic properties.

Azimuthal Quantum Number

The azimuthal quantum number, denoted as \(l\), also known as the angular momentum quantum number, is pivotal in determining the shape of an electron's orbital. It captures how the orbital is defined spatially and thus influences the electron cloud's geometry.
This quantum number takes integer values from \(0\) to \(n-1\), where \(n\) is the principal quantum number.The value of \(l\) is associated with the subshells within a principal energy level and corresponds to the orbital shapes:

• \(l = 0\) signifies \(s\) orbitals, which are spherical.
• \(l = 1\) corresponds to \(p\) orbitals, which are dumbbell-shaped.
• \(l = 2\) identifies \(d\) orbitals, known for more complex shapes.
• \(l = 3\) refers to \(f\) orbitals, with even more intricate shapes.

Together with other quantum numbers, \(l\) helps to describe how electrons are distributed in atoms and informs the chemical properties and reactivity of elements. Understanding \(l\) provides foundational insight into chemical bonding and atomic structure.