Problem 57 - Chemistry: The Central Science in SI Units, Global Edition

Question

Give the numerical values of \(n\) and \(l\) corresponding to each of the following orbital designations: (a) \(3 p\), (b) \(2 s,(\mathbf{c}) 4 f,\) (d) \(5 d\).

Step-by-Step Solution

Answer

The numerical values of $n$ and $l$ for each orbital designation are as follows: (a) \(3p\) : \(n = 3\) and \(l = 1\) (b) \(2s\) : \(n = 2\) and \(l = 0\) (c) \(4f\) : \(n = 4\) and \(l = 3\) (d) \(5d\) : \(n = 5\) and \(l = 2\)

Step 1: Identify n and l for 3p orbital

In the orbital designation, the number preceding the orbital type gives the value of n. Here, n = 3. Next, look at the orbital type, which is p. p orbitals correspond to l = 1. Therefore, for a 3p orbital, we have n = 3 and l = 1. (b) 2s

Step 2: Identify n and l for 2s orbital

In the orbital designation, the number preceding the orbital type gives the value of n. Here, n = 2. Next, look at the orbital type, which is s. s orbitals correspond to l = 0. Therefore, for a 2s orbital, we have n = 2 and l = 0. (c) 4f

Step 3: Identify n and l for 4f orbital

In the orbital designation, the number preceding the orbital type gives the value of n. Here, n = 4. Next, look at the orbital type, which is f. f orbitals correspond to l = 3. Therefore, for a 4f orbital, we have n = 4 and l = 3. (d) 5d

Step 4: Identify n and l for 5d orbital

In the orbital designation, the number preceding the orbital type gives the value of n. Here, n = 5. Next, look at the orbital type, which is d. d orbitals correspond to l = 2. Therefore, for a 5d orbital, we have n = 5 and l = 2.

Key Concepts

Principal Quantum Number

Quantum numbers are used to describe the properties of atomic orbitals and the properties of electrons in those orbitals. The principal quantum number, denoted as \( n \), is one of the critical aspects of this. It provides information about the energy level that an electron is in, and it essentially describes the size of the orbital. Higher values of \( n \) mean the electron is further away from the nucleus and the energy of an electron increases as \( n \) increases.

• The integer values of \( n \) can be 1, 2, 3, 4, etc., and these are called the electron shells.
• For example, when you see a designation like "3p," the "3" indicates that the electron is in the third shell.
• As \( n \) increases, the orbital becomes larger, and electrons are generally found in higher energy states.

In summary, \( n \) tells us the general distance from the nucleus and the energy of the electron within that orbital.

Angular Momentum Quantum Number

Another important quantum number is the angular momentum quantum number, indicated by \( l \). This quantum number primarily describes the shape of the orbital. Each principal quantum number \( n \) has \( n \) possible values for \( l \), ranging from \( 0 \) to \( n-1 \).

• If \( l = 0 \), the orbital is spherical, known as an "s" orbital.
• If \( l = 1 \), the orbital is dumbbell-shaped, referred to as a "p" orbital.
• If \( l = 2 \), the orbital is more complex and known as a "d" orbital.
• If \( l = 3 \), the orbital is even more complex, known as an "f" orbital.

For example, in a "4f" orbital, \( n \) is 4, and \( l \) is 3, indicating the orbital's complex shape and the relatively high energy level of the electrons within. This quantum number helps us understand both the shape and the type of orbital present in each principal energy level.

Atomic Orbitals

Atomic orbitals are regions within an atom where electrons are most likely to be found. They are described by a set of quantum numbers: the principal quantum number \( n \), the angular momentum quantum number \( l \), and others not discussed here, each providing essential details about the behavior of electrons.

• "s" orbitals are spherical and can hold a maximum of 2 electrons.
• "p" orbitals have a dumbbell shape; there are three such orbitals oriented in different directions, and they can hold up to 6 electrons in total.
• "d" orbitals have more complex shapes with five orientations, allowing them to hold up to 10 electrons.
• "f" orbitals are even more intricate, with seven orientations, and can accommodate up to 14 electrons.

Each type of orbital increases the complexity and capacity for electrons, contributing to the understanding of the unique arrangement of electrons around the nucleus. This complexity reflects the atom’s potential for forming a variety of chemical bonds and interactions.