Problem 24 - Chemistry: Principles and Reactions

Question

Give the number of orbitals in (a) \(\mathbf{n}=3\). (b) a 4p sublevel. (c) an f sublevel. (d) a d sublevel.

Step-by-Step Solution

Answer

Question: Determine the number of orbitals for each of the following cases: (a) n=3 energy level, (b) a 4p sublevel, (c) an f sublevel, (d) a d sublevel. Answer: (a) There are 9 orbitals in the n=3 energy level. (b) There are 3 orbitals in a 4p sublevel. (c) There are 7 orbitals in an f sublevel. (d) There are 5 orbitals in a d sublevel.

Step 1: Case (a): Number of orbitals in n=3 energy level

To find the number of orbitals in the n=3 energy level, use the formula: the total number of orbitals = \(n^2\). In this case, n=3, so the total number of orbitals is \(3^2 = 9\).

Step 2: Case (b): Number of orbitals in a 4p sublevel

To find the number of orbitals in a specific sublevel, we need to know its l value. For a p sublevel, l=1. Each value of l corresponds to (2l + 1) orbitals. In this case, l=1, so the total number of orbitals in a 4p sublevel is \((2\times1) + 1 = 3\).

Step 3: Case (c): Number of orbitals in an f sublevel

To find the number of orbitals in an f sublevel, we need to know its l value. For an f sublevel, l=3. Each value of l corresponds to (2l + 1) orbitals. In this case, l=3, so the total number of orbitals in an f sublevel is \((2\times3) + 1 = 7\).

Step 4: Case (d): Number of orbitals in a d sublevel

To find the number of orbitals in a d sublevel, we need to know its l value. For a d sublevel, l=2. Each value of l corresponds to (2l + 1) orbitals. In this case, l=2, so the total number of orbitals in a d sublevel is \((2\times2) + 1 = 5\).

Key Concepts

Atomic Orbitals

Atomic orbitals are fundamental components of the quantum model of the atom and represent regions in space where there is a high probability of finding an electron. Electrons inhabit these orbits around a nucleus, each with its distinct energy levels. The understanding of orbitals is essential for grasping how atoms form chemical bonds.

The energy level of an electron is defined by its principal quantum number, often symbolized as 'n'. As the value of 'n' increases, so does the energy level and the electron's average distance from the nucleus. Within each energy level (n), there are one or more sublevels, each containing one or more orbitals.

The shape of an orbital depends on the sublevel it is in, which could be s (spherical), p (dumbbell-shaped), d (cloverleaf-shaped), or f (complex-shaped). For instance, within the n=3 energy level, the nine orbitals are distributed across s, p, and d sublevels with one, three, and five orbitals respectively. The exercise example highlights that the n=3 level has nine orbitals based on the formula \(n^2\).

Sublevel Electron Configuration

Electron configuration within an atom's sublevels explains how electrons are distributed in the available atomic orbitals. Each orbital can hold a maximum of two electrons with opposite spins due to the Pauli exclusion principle. Sublevels are named by the azimuthal quantum number 'l', which defines the shape of the orbitals they contain: 0 (s), 1 (p), 2 (d), or 3 (f).

As you go from one level to the next, additional sublevels become available, thus increasing the number of possible orbitals. For example, a 4p sublevel (l=1) contains three orbitals, and the exercise confirms this with the calculation \(2l + 1\), which equals three when l=1. These orbitals are filled according to Hund's rule, which states that electrons will fill an unoccupied orbital before they pair up. Understanding sublevel configuration is key to predicting the chemical behavior of an element.

Quantum Model of the Atom

The quantum model of the atom is a fundamental theory that describes the atomic and subatomic levels based on the principles of quantum mechanics. Unlike the simple Bohr model, which depicts electrons in fixed orbits, the quantum model introduces the concept of atomic orbitals and probability distributions.

In quantum mechanics, electrons are described by wave functions, and the square of these functions gives the probability density of finding an electron at any point in space. This model accounts for the various energy states an electron may occupy and the restrictions on their energy levels and sublevels, which are defined by quantum numbers.

These quantum numbers include the principal (n), azimuthal (l), magnetic (m), and spin (s), each contributing to a complete description of an electron's behavior within the atom. The quantum model has excellent predictive power and explains the structure of the periodic table, spectral lines, and many other properties of atoms.

Azimuthal Quantum Number

The azimuthal quantum number, 'l', is a fundamental quantum number in chemistry that plays a significant role in determining the shape and energy of atomic orbitals. It can take on any integer value from 0 to n-1, where 'n' is the principal quantum number.

The value of 'l' defines the sublevel within a given energy level (n). For example, for 'l' values of 0, 1, 2, and 3, the corresponding sublevels are s, p, d, and f, respectively. Each sublevel shape is determined by the value of the azimuthal quantum number, affecting the distribution patterns of electrons.

The importance of the azimuthal quantum number is reflected in the exercise where the number of orbitals within a sublevel is calculated. For instance, a p sublevel with l=1 consists of three orbitals, a d sublevel with l=2 has five, and an f sublevel with l=3 contains seven. These numbers are determined by the formula \(2l + 1\), illustrating the azimuthal quantum number's impact on electron configuration.