Question

What are the possible values for the quantum numbers \(n, \ell\), and \(m_{\ell} ?\)

Short answer

Possible values for the quantum numbers are: - Principal quantum number (n): \(n = 1, 2, 3, ...\) - Azimuthal quantum number (ℓ): \(ℓ = 0, 1, ..., n-1\) - Magnetic quantum number (m_ℓ): \(m_{ℓ} = -ℓ, -ℓ+1, ... , 0, ... , ℓ-1, ℓ\)

Step-by-step solution

Understand quantum numbers

Quantum numbers are numbers that describe the unique state of an electron in an atom. There are three main quantum numbers: 1. Principal quantum number (n): Determines the energy level of an electron in an atom and the size of its orbital. It can have any positive integer value starting from 1 (n=1, 2, 3, ...). 2. Azimuthal quantum number (ℓ): Determines the shape of the orbital and is related to the angular momentum of an electron. It depends on the value of n and can have integer values ranging from 0 to n-1 (ℓ=0, 1, ..., n-1). 3. Magnetic quantum number (m_ℓ): Determines the orientation of the orbital in space and is related to the projection of the angular momentum along with a specified axis. It depends on the value of ℓ and can have integer values ranging from -ℓ to +ℓ (m_ℓ=-ℓ, -ℓ+1, ... , 0, ... , ℓ-1, ℓ).

Finding the possible values of n

The principal quantum number can have any positive integer value. So the possible values of n can be expressed as: \(n = 1, 2, 3, ...\)

Finding the possible values of ℓ

The azimuthal quantum number (ℓ) is related to the value of n and ranges from 0 to n-1. For each value of n, possible values of ℓ can be expressed as: \(ℓ = 0, 1, ..., n-1\)

Finding the possible values of m_ℓ

The magnetic quantum number (m_ℓ) is related to the value of ℓ and ranges from -ℓ to +ℓ. For each value of ℓ, possible values of m_ℓ can be expressed as: \(m_{ℓ} = -ℓ, -ℓ+1, ... , 0, ... , ℓ-1, ℓ\)

Listing the possible quantum number values

The possible values for the quantum numbers n, ℓ, and m_ℓ can be summarized as: - Principal quantum number (n): \(n = 1, 2, 3, ...\) - Azimuthal quantum number (ℓ): \(ℓ = 0, 1, ..., n-1\) - Magnetic quantum number (m_ℓ): \(m_{ℓ} = -ℓ, -ℓ+1, ... , 0, ... , ℓ-1, ℓ\)

Key concepts

Principal Quantum Number Explained

Imagine being in a vast library, the shelves labeled with numbers that help you find books at different heights. In the world of atoms, electrons are a bit like books and the shelves are energy levels determined by the principal quantum number, often denoted by the symbol \(n\). The higher the number, the higher the energy level and the larger the orbital where an electron can be found.

With each step up, starting from 1, the electron resides further from the nucleus — meandering in shells that can hold more energy and more electrons. Think of \(n=1\) as the ground floor, \(n=2\) a little higher, and so on, with no upper limit in theory. However, due to the limitations in energy that an electron can possess, we do not see electrons with an arbitrarily high \(n\) value.

So, the principal quantum number sets the stage, sketching out the size of the electron's 'room' — it chalks the boundaries for an electron's personal space within an atom.

Azimuthal Quantum Number Demystified

Now, within each energy level, or 'floor', defined by the principal quantum number, there are various types of 'rooms' with different shapes — enter the azimuthal quantum number (\(\ell\)). It's akin to the various room shapes in a hotel, ranging from a cozy circular room to an elongated suite.

These 'rooms' are in fact subshells, represented by s, p, d, and f, corresponding to the values of \(\ell = 0, 1, 2, 3\), respectively. The value of \(\ell\) is contingent on \(n\); for instance, if \(n=2\), \(\ell\) could be 0 or 1 (s or p subshell). But it could never exceed \(n-1\), ensuring that every room fits neatly within its floor.

It's crucial to understand that the azimuthal quantum number is not just about shapes; it relates to the intricate dance of electrons — their angular momentum. Each increase in \(\ell\) adds more loops and swirls to the electron's pathway, effectively changing its motion style.

Magnetic Quantum Number: The Orientation Guide

When it comes to the magnetic quantum number, represented by \(m_{\ell}\), one moves from the realm of size and shape to orientation — it's as if each room on our hotel floor could be rotated to face different directions. Each \(m_{\ell}\) value describes an electron's orbital orientation within a subshell relative to an external magnetic field.

For a given \(\ell\), \(m_{\ell}\) spans from \(\ell\) to \( -\ell\). If our subshell (room shape) is p-type (\(\ell=1\)), \(m_{\ell}\) could be -1, 0, or 1. This range of values accounts for the fact that you could rotate your hotel room left, right, or keep it as is. In quantum terms, each \(m_{\ell}\) value corresponds to a specific orbital that can house up to two electrons, who each get their unique spin direction, further distinguishing them.

Thus, the magnetic quantum number does not influence energy or shape but directs how different orbitals are positioned around the nucleus, providing a unique address for each electron's residence in atomic real estate.