Problem 59 - Chemistry: The Central Science in SI Units, Global Edition

Question

A certain orbital of the hydrogen atom has \(n=4\) and \(l=3\). (a) What are the possible values of \(m_{l}\) for this orbital? (b) What are the possible values of \(m_{s}\) for the orbital?

Step-by-Step Solution

Answer

For a hydrogen atom's orbital with n = 4 and l = 3: (a) The possible values of the magnetic quantum number \(m_{l}\) are -3, -2, -1, 0, 1, 2, and 3. (b) The possible values of the electron spin quantum number \(m_{s}\) are \(+\frac{1}{2}\) and \(-\frac{1}{2}\).

Step 1: Determine the possible values of m_l

Since the given value for the azimuthal quantum number (l) is 3, the magnetic quantum number (m_l) can have integer values ranging from -3 to 3. We will write all integers from -3 to 3, inclusive: \[m_{l} = -3, -2, -1, 0, 1, 2, 3\]

Step 2: Determine the possible values of m_s

For the electron spin quantum number (m_s), there are two possible values which describe the two possible orientations of the electron spin. These values are +1/2 and -1/2. We can write these values as: \[m_{s} = +\frac{1}{2}, -\frac{1}{2}\] In conclusion, for a hydrogen atom's orbital with n = 4 and l = 3: (a) The possible values of the magnetic quantum number (m_l) are -3, -2, -1, 0, 1, 2, and 3. (b) The possible values of the electron spin quantum number (m_s) are +1/2 and -1/2.

Key Concepts

Hydrogen Atom

The hydrogen atom is one of the most fundamental systems in quantum mechanics. It's composed of a single electron and a single proton. This simplicity makes it a cornerstone for understanding more complex atoms. The electron in a hydrogen atom is found in various energy levels, which correspond to different orbital structures defined by quantum numbers. These energy levels are determined by the principal quantum number, denoted by \(n\). For each value of \(n\), there are electrons occupying specific orbitals characterized by the azimuthal quantum number \(l\) and other quantum numbers.
This setup is not just theoretical. It's reflected in the spectral lines we see when hydrogen emits light. Each of those lines corresponds to an electron transitioning between energy levels, closely tied to the quantum numbers defining its state.

Magnetic Quantum Number

The magnetic quantum number, represented by \(m_{l}\), is a vital component in quantum mechanics, primarily when detailing atomic orbitals. It indicates the orientation of an orbital within a given subshell. The range of \(m_{l}\) is determined by the azimuthal quantum number \(l\). Specifically, \(m_{l}\) can take on any integer value from \(-l\) to \(+l\), inclusive.
For example, if \(l=3\), as in the exercise, the possible values for \(m_{l}\) are -3, -2, -1, 0, 1, 2, and 3. Each of these represents a unique spatial orientation of the electron's orbital. The different possible orientations are crucial when atoms are exposed to a magnetic field, as the splitting of spectral lines into several components often depends on these orientations.

Electron Spin

Electron spin is one of the four quantum numbers describing the state of an electron in an atom. While other quantum numbers relate to the electron's position and path, spin specifically refers to its intrinsic angular momentum. Symbolized by \(m_{s}\), the electron spin quantum number can have a value of either \(+\frac{1}{2}\) or \(-\frac{1}{2}\). These represent the two possible spin orientations of an electron: 'spin-up' or 'spin-down'.
Spin is a fundamental property, and it's crucial to the concept of "spin" itself, which is based on quantum mechanics rather than classical physics. A significant outcome of electron spin is the Pauli Exclusion Principle, which states that no two electrons in an atom can have the same set of four quantum numbers. This principle helps explain the structure of the periodic table and the electronic configurations of atoms.

Azimuthal Quantum Number

The azimuthal quantum number, denoted by \(l\), is integral to understanding the shape and energy of orbitals within an atom. It determines the orbital angular momentum and is sometimes referred to as the orbital quantum number. The value of \(l\) ranges from 0 to \(n-1\), where \(n\) is the principal quantum number. This means for a given energy level, you can have multiple ionic shapes or subshells.
Each value of \(l\) corresponds to a different subshell type, often labeled with letters as follows: \(l=0\) is the "s" subshell, \(l=1\) is "p", \(l=2\) is "d", and \(l=3\) corresponds to the "f" subshell, as seen in this problem with \(n=4, l=3\). Understanding \(l\) is crucial because it influences not only the shape of the electron cloud but also the chemical bonding and properties of an element.