Question

What are the possible values for \(\mathbf{m}_{\ell}\) for (a) the \(\mathrm{d}\) sublevel? (b) the s sublevel? (c) all sublevels where \(\mathbf{n}=2 ?\)

Short answer

Question: List the possible values for the magnetic quantum number (ml) for (a) the d sublevel, (b) the s sublevel, and (c) all sublevels when the principal quantum number n=2. Answer: (a) For the d sublevel, ml can take the values: -2, -1, 0, 1, 2. (b) For the s sublevel, ml can only take the value: 0. (c) For n=2, possible ml values are: s sublevel - 0; p sublevel - (-1, 0, 1).

Step-by-step solution

(a) Possible values for ml for the d sublevel

For the d sublevel, the azimuthal quantum number (l) is 2. So, the possible values for the magnetic quantum number (ml) are integers ranging from -l to l. Therefore, ml can take the values: -2, -1, 0, 1, 2.

(b) Possible values for ml for the s sublevel

For the s sublevel, the azimuthal quantum number (l) is 0. So, the possible values for the magnetic quantum number (ml) are integers ranging from -l to l. Since l is 0, ml can only take the value: 0.

(c) Possible values for ml for all sublevels when n = 2

For n = 2, there are 2 possible sublevels: s and p sublevels. We have already found the possible ml values for the s sublevel, so we only need to find them for the p sublevel. For the p sublevel, the azimuthal quantum number (l) is 1. So, the possible values for the magnetic quantum number (ml) are integers ranging from -l to l. Therefore, ml can take the values: -1, 0, 1. For n = 2, we have the following possible ml values: - s sublevel: 0. - p sublevel: -1, 0, 1.

Key concepts

Azimuthal Quantum Number

The azimuthal quantum number, also known as the angular momentum quantum number or orbital quantum number, is of central importance in quantum mechanics — especially for chemistry students exploring the world of atomic structure. It is denoted by the symbol 'l' and is integral for determining the shape of an electron's orbital and consequently its energy level.

For any given principal quantum number 'n', which correlates with the electron's energy level, 'l' can take on any integer value from 0 to n-1. For instance, if an electron is in the second principal energy level (n=2), 'l' can be either 0 or 1. These numbers correspond to different sublevels or orbitals, with 'l=0' denoting the spherical s orbitals and 'l=1' corresponding to the dumbbell-shaped p orbitals.

It is this very quantum number that guides the range of the magnetic quantum number (m_l), another critical quantum number that further partitions the sublevels into individual magnetic orbitals. Each value of 'l' not only introduces a unique sublevel but also dictates the number of orientations that sublevel can have in 3D space.

Understanding the d Sublevel

Using the azimuthal quantum number, we can deduce characteristics of the d sublevel. With 'l' being 2 for the d sublevel, we can immediately tell that the shape of these orbitals is more complex compared to s and p orbitals, and that there are five possible orientations, each corresponding to the values of the magnetic quantum number m_l (-2, -1, 0, 1, 2). This is essential knowledge for predicting the distribution of electrons in atoms, which is pivotal for understanding chemical bonding and the properties of elements.

Quantum Mechanics in Chemistry

At the heart of modern chemistry lies quantum mechanics, the theoretical framework that helps us make sense of the behavior of electrons in atoms. It is the principles of quantum mechanics that chemists use to explain and predict the properties of elements and compounds.

In the shell model of the atom, electrons are thought to reside in quantized energy levels. Each of these levels is made up of one or more sublevels, which are further divided into orbitals according to the allowed values of the azimuthal quantum number. These orbitals, where electrons probabilistically exist, define the electronic configuration of an atom – an understanding that is crucial for everything from the periodic table's structure to the way atoms interact during chemical reactions.

The Schrödinger equation is a fundamental aspect of quantum mechanics that provides the wavefunctions for electrons. These wavefunctions, or orbitals, contain all the information about an electron's position and momentum. Chemists interpret these wavefunctions to determine probable electron locations, their energies, and the shapes of the orbitals they occupy, neatly linking the posit of the azimuthal and magnetic quantum numbers to observable chemical phenomena.

Understanding the principles of quantum mechanics enables chemists to predict electron arrangements and thus infer the chemical and physical properties of the elements and their compounds, which is indispensable in fields ranging from pharmacology to material science.

Electronic Sublevels

Electronic sublevels are an essential concept in chemistry that refines our understanding of electron arrangement within an atom. They are nested within electron shells, with each shell capable of containing a certain number of sublevels determined by the principal quantum number (n). The azimuthal quantum number (l) is what specifies these sublevels or orbitals.

Sublevels are categorized into four types, designated as s, p, d, and f. The s sublevel, with an azimuthal quantum number of 0 (l=0), is the simplest, containing only one spherical orbital. The p sublevel (l=1) has three degenerate orbitals oriented at right angles to each other, while the d (l=2) and f (l=3) sublevels have more complex shapes with five and seven orbitals, respectively.

The electrons fill these sublevels in a specific order dictated by their relative energies, following the principle known as 'Aufbau', or 'building up.' Electrons populate the lower energy orbitals first before moving to higher ones, and this characteristic filling pattern is what gives rise to the distinct shape of the periodic table and dictates the chemical behavior of the elements. With each sublevel holding a maximum of 2(2l+1) electrons, understanding how electrons are arranged in sublevels is pivotal in predicting an atom's magnetic properties, the shape and orientation of molecules, and the energetics of chemical reactions.

s and p Sublevels at n=2

In the context of the principal quantum number n=2, there are two sublevels: s and p. The s sublevel is spherically symmetrical and can contain a maximum of 2 electrons. The p sublevel, with its three dumbbell-shaped orbitals, can hold up to 6 electrons. These structures help us grasp the magnetic quantum numbers for these sublevels and the potential configurations of electrons within them – foundational for understanding the basics of bonding and the periodic trends in element properties.