Problem 60 - Chemistry and Chemical Reactivity

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understanding the Quantum Numbers

Each electron in an atom is described by a set of quantum numbers. The principal quantum number \( n \) indicates the energy level. The azimuthal quantum number \( l \) indicates the shape of the orbital and is correlated with the blocks (\( s=0 \), \( p=1 \), \( d=2 \), \( f=3 \)).

Step 2: Finding the Number of Orbitals

Each set of orbitals has \( 2l + 1 \) orbitals. Calculate the number of orbitals for each case using the appropriate \( l \) value.

Step 3: (a) Calculate Orbitals for 3p

For \( 3p \), \( n=3 \) and \( l=1 \). Therefore, the number of orbitals is \( 2(1) + 1 = 3 \).

Step 4: (b) Calculate Orbitals for 4p

For \( 4p \), \( n=4 \) and \( l=1 \). Therefore, the number of orbitals is \( 2(1) + 1 = 3 \).

Step 5: (c) Calculate Orbitals for 4p_x

4p_x refers to a specific orbital. Hence, only 1 orbital corresponds to 4p_x.

Step 6: (d) Calculate Orbitals for 6d

For \( 6d \), \( n=6 \) and \( l=2 \). Therefore, the number of orbitals is \( 2(2) + 1 = 5 \).

Step 7: (e) Calculate Orbitals for 5d

For \( 5d \), \( n=5 \) and \( l=2 \). Therefore, the number of orbitals is \( 2(2) + 1 = 5 \).

Step 8: (f) Calculate Orbitals for 5f

For \( 5f \), \( n=5 \) and \( l=3 \). Therefore, the number of orbitals is \( 2(3) + 1 = 7 \).

Step 9: (g) Find Orbitals for n=5

When \( n=5 \), the possible \( l \) values are 0, 1, 2, 3, 4. The number of orbitals for each \( l \) are as follows: \( s \): 1, \( p \): 3, \( d \): 5, \( f \): 7, \( g \): 9. Total orbitals \( = 1 + 3 + 5 + 7 + 9 = 25 \).

Step 10: (h) Calculate Orbitals for 7s

For \( 7s \), \( n=7 \) and \( l=0 \). Therefore, the number of orbitals is \( 2(0) + 1 = 1 \).

Key Concepts

Orbital Designation

In quantum mechanics, the term "orbital designation" refers to the notation used to specify the type and position of an electron's orbital within an atom. This notation helps us understand where electrons are likely to be found, based on their energy levels and shapes. The designation typically consists of two parts: a principal quantum number, which denotes the energy level, and a letter, which indicates the shape of the orbital.

• The letters used are 's', 'p', 'd', and 'f', corresponding to azimuthal quantum numbers 0, 1, 2, and 3, respectively.
• For example, "3p" indicates a p-orbital (l=1) in the third energy level (n=3).

This designation helps scientists and students alike quickly identify and understand the potential energy and structure of electrons within an atom.

Principal Quantum Number

The principal quantum number, denoted by the symbol \( n \), is a fundamental component of quantum mechanics. It plays a significant role in determining the major energy level and size of an orbital. This number is always a positive integer (\( n = 1, 2, 3, \ldots \)) and provides insight into the relative energy of the electron.

• The greater the value of \( n \), the higher the energy level and the larger the orbital.
• The principal quantum number is crucial for indicating the overall electron shell in which the electron resides and directly influences the radius of an electron's orbit.

In our previous examples, such as "3p" or "4p", "3" and "4" represent the principal quantum numbers, explaining that these electrons occupy the third and fourth energy levels, respectively.

Azimuthal Quantum Number

The azimuthal quantum number, usually denoted as \( l \), is another key element in understanding electron configuration within quantum mechanics. It is directly linked to the shape of the atomic orbitals, further classifying orbitals into sublevels. The possible values of \( l \) range from 0 to \( n-1 \), where \( n \) is the principal quantum number.

• For an \( s \)-orbital (l=0), \( l \) is always 0, and it has a spherical shape.
• For a \( p \)-orbital (l=1), the shape is dumbbell-like.
• For \( d \) and \( f \) orbitals, \( l \) equals 2 and 3, showing even more complex shapes.

The azimuthal quantum number not only defines the geometry of the orbital but also helps in calculating the number of orbitals within a subshell.

Quantum Mechanics

Quantum mechanics provides the foundational framework for understanding the behavior of electrons and other subatomic particles. This branch of physics describes the wave-like properties of particles, predicting the probable locations of electrons in atoms through wave functions and probability densities.
Quantum mechanics introduces the four quantum numbers that help define the specific state of an electron:

• Principal quantum number (n)
• Azimuthal quantum number (l)
• Magnetic quantum number (m), determining the spatial orientation
• Spin quantum number (s), describing the spin direction of the electron

This theoretical framework allows for precise modeling of electronic structures, greatly enhancing our understanding and capabilities in chemistry and physics.

Electron Configuration

Electron configuration is a method used to denote how electrons are arranged within the orbitals of an atom. It provides a simple way to depict the distribution of electrons in atomic orbitals based on the Aufbau principle, Hund's rule, and Pauli-Exclusion principle.

• Aufbau Principle: Electrons fill orbitals starting with the lowest available energy levels before moving to higher ones.
• Hund's Rule: Electrons fill each orbital singly before doubling up, to maximize the number of unpaired electrons.
• Pauli-Exclusion Principle: No two electrons can have the same four quantum numbers, meaning each orbital can hold a maximum of two electrons with opposite spins.

Understanding electron configuration aids in predicting chemical properties and reactivity of elements through their electronic structure.