Correct Magnetic Quantum Number Set For N=2 Subshells

In the realm of quantum mechanics, understanding the electronic structure of atoms is pivotal for comprehending chemical behavior. Specifically, the quantum numbers – namely, the principal quantum number (n), the azimuthal quantum number (l), and the magnetic quantum number (m_l) – define the state of an electron within an atom. This article aims to elucidate the correct set of values for the magnetic quantum number (m_l) for subshells when the principal quantum number (n) is 2. We will dissect the quantum mechanical principles that govern these numbers and evaluate the given options to identify the accurate set of m_l values.

At the heart of atomic structure lies the concept of quantized energy levels. Electrons in an atom can only exist in specific energy states, defined by a set of quantum numbers. These numbers provide a detailed description of an electron's characteristics, such as its energy, angular momentum, and spatial orientation. To address the question, it's crucial to first understand what each quantum number signifies.

• Principal Quantum Number (n): This number determines the energy level or shell of an electron. It is a positive integer (n = 1, 2, 3, ...), where higher numbers indicate higher energy levels. For instance, n = 1 corresponds to the ground state, and n = 2 represents the first excited state.
• Azimuthal Quantum Number (l): Also known as the angular momentum or orbital quantum number, l defines the shape of the electron's orbital and its orbital angular momentum. Its values range from 0 to n - 1. Each l value corresponds to a specific subshell: l = 0 (s orbital, spherical), l = 1 (p orbital, dumbbell-shaped), l = 2 (d orbital, more complex shapes), and l = 3 (f orbital, even more complex shapes).
• Magnetic Quantum Number (m_l): The magnetic quantum number m_l describes the orientation of the orbital in space. It can take integer values from -l to +l, including 0. Therefore, for a given l, there are 2l + 1 possible m_l values, each representing a different orbital within the subshell.

Given that the principal quantum number (n) is 2, we can determine the possible values for the azimuthal quantum number (l). The values of l range from 0 to n - 1, so when n = 2, l can be either 0 or 1.

• When l = 0 (s subshell): The magnetic quantum number m_l can only have one value, which is 0. This corresponds to a single s orbital, which is spherically symmetrical.
• When l = 1 (p subshell): The magnetic quantum number m_l can take values from -l to +l, including 0. Thus, m_l can be -1, 0, or +1. These three values correspond to the three p orbitals (p_x, p_y, and p_z), which are dumbbell-shaped and oriented along the x, y, and z axes, respectively.

Therefore, for n = 2, the possible values of m_l are -1, 0, and 1. This set of values corresponds to the p subshell, which has three orbitals.

Now, let's evaluate the options provided in the question:

A. $-1, 0, 1$ B. $-1, -2, 0, 1, 2$ C. $-1, -2, -3, 0, 1, 2, 3$ D. $-1, -2, -3, -4, 0, 1, 2, 3, 4$

Based on our understanding of quantum numbers:

• Option A (-1, 0, 1): This set of m_l values corresponds to the p subshell (l = 1). As we discussed, when l = 1, m_l can indeed be -1, 0, or +1. Therefore, this is a correct set of values for one of the subshells of n = 2.
• Option B (-1, -2, 0, 1, 2): This set is incorrect because it includes values of m_l that are not possible when n = 2. The maximum value of l when n = 2 is 1, and therefore the values of m_l cannot exceed -1 or +1.
• Option C (-1, -2, -3, 0, 1, 2, 3): This set is also incorrect. These values would be appropriate for a subshell with l = 3, which is an f subshell. However, when n = 2, the maximum value of l is 1, so an f subshell is not possible.
• Option D (-1, -2, -3, -4, 0, 1, 2, 3, 4): This set is incorrect for similar reasons as Options B and C. These values would correspond to a subshell with l = 4, which is a g subshell. This is not possible when n = 2.

In summary, the correct set of values for the magnetic quantum number (m_l) for one of the subshells of n = 2 is A. -1, 0, 1. This corresponds to the p subshell (l = 1), which has three orbitals with different spatial orientations. Understanding these quantum numbers and their relationships is essential for predicting and explaining the electronic structure and behavior of atoms and molecules. This foundational knowledge is crucial for further studies in chemistry, physics, and material science. Grasping these principles allows scientists and students to visualize and interpret the microscopic world that governs macroscopic properties and reactions.

By meticulously examining each quantum number and its implications, we can accurately describe the state of electrons within atoms and build a comprehensive understanding of chemical phenomena. The interplay between n, l, and m_l dictates the unique properties of each element and compound, making the study of quantum numbers indispensable in the field of chemistry.