Question

How many orbitals correspond to each of the following designations? (a) \(3 p\) (b) \(4 p\) (c) \(4 p_{x}\) (d) \(6 d\) (e) \(5 d\) (f) \(5 f\) (g) \(n=5\) (h) \(7 s\)

Short answer

(a) 3; (b) 3; (c) 1; (d) 5; (e) 5; (f) 7; (g) 25; (h) 1

Step-by-step solution

Identify Orbital Designations

In quantum chemistry, orbitals are characterized by the quantum numbers: principal ( ), azimuthal ( lorall l), and magnetic ( m). For any given p value, orbitals are categorized based on these quantum numbers.

Understand the 'p' Orbitals

Both 3p and 4p indicate l = 1 because p orbitals correspond to l = 1 sublevels. The range of m for l = 1 is m = -1, 0, +1 , which means there are 3 orbitals.

Analyze Specific 'p' Orbitals

4p_{x} is a specific orbital out of the 3 'p' orbitals ( px, py, pz ). Therefore, 4p_{x} corresponds to 1 orbital.

Calculate 'd' Orbitals

6d and 5d both represent l = 2 sublevels, with m equals -2, -1, 0, +1, +2, leading to 5 different orbitals.

Count 'f' Orbitals

5f corresponds to l = 3 , where m equals -3, -2, -1, 0, +1, +2, +3. Hence, there are 7 orbitals.

Determine Orbitals for 'n=5'

For n = 5 , possible l evalues are 0 (s), 1 (p), 2 (d), 3 (f), 4 (g). For each l , sum the orbitals: 1 (s) + 3 (p) + 5 (d) + 7 (f) + 9 (g) = 25 orbitals.

Analyze the 's' Orbitals

The 7s level indicates l = 0 , yielding only 1 orbital due to m = 0 .

Key concepts

Quantum Numbers

Quantum numbers are like an address system for electrons within an atom. They help to identify the location and energy of every electron. This system consists of four quantum numbers that provide a unique description of an electron's state:

• Principal Quantum Number ( n ): Indicates the size and energy level of the orbital.
• Azimuthal Quantum Number ( l ): Determines the shape of the orbital.
• Magnetic Quantum Number ( m ): Specifies the orientation of the orbital.
• Spin Quantum Number: Describes the spin of the electron.

Together, these numbers offer a complete map to electron arrangements in atoms, contributing to our understanding of chemical reactions and molecular structures.

Principal Quantum Number

The principal quantum number, denoted as n , dictates the main energy level of an electron in an atom. It can be any positive integer, starting from 1. Think of n as representing the shell or shell number where the electron resides.
Here's what the principal quantum number tells us:

• The higher the n , the higher the energy level and the farther the electron is from the nucleus.
• Larger n values correspond to larger and more energy-consuming electron orbitals.

For instance, when n = 5 , electrons are in the fifth shell, which can accommodate different sublevels such as s, p, d, and f.

Azimuthal Quantum Number

The azimuthal quantum number, labeled as l , defines the shape of the electron's orbital. Its values range from 0 to n-1 for each energy level n . Each value of l corresponds to a specific type of subshell:

• l = 0 : s subshell (spherical shape)
• l = 1 : p subshell (dumbbell shape)
• l = 2 : d subshell (cloverleaf shape)
• l = 3 : f subshell (complex shapes)

So, for a 3p designation, n = 3 and l = 1 , indicating it's in the third energy level with a p-shaped orbital. The subshell's shape directly influences the chemical bonding and properties of atoms.

Magnetic Quantum Number

The magnetic quantum number, symbolized as m , reveals the orientation of an orbital within a subshell. For a given azimuthal quantum number l , m can take integer values between -l and +l .
This means:

• For l = 0 : m = 0 (only one orientation)
• For l = 1 : m = -1, 0, +1 (three orientations, p_x, p_y, p_z )
• For l = 2 : m = -2, -1, 0, +1, +2 (five orientations)
• For l = 3 : m = -3 to +3 (seven orientations)

For example, the 4p_x designation refers to one specific orientation within the p subshell, showing how orbitals align differently in space.