Problem 55 - Chemistry The Central Science In Si Units

Question

(a) For \(n=4,\) what are the possible values of \(l ?(\mathbf{b})\) For \(l=2\), what are the possible values of \(m_{l} ?\) (c) If \(m_{l}\) is \(2,\) what are the possible values for \(l ?\)

Step-by-Step Solution

Answer

(a) 0, 1, 2, 3; (b) -2, -1, 0, 1, 2; (c) l = 2

Step 1: Understanding Quantum Numbers

Quantum numbers describe the properties of atomic orbitals and the properties of electrons in orbitals. The principal quantum number, denoted as \( n \), determines the energy level. The azimuthal quantum number, \( l \), depends on \( n \) and defines the shape of the orbital.

Step 1: Compute Possible Values of l for n = 4

The azimuthal quantum number \( l \) ranges from 0 to \( n-1 \). Therefore, for \( n = 4 \), the possible values of \( l \) are 0, 1, 2, and 3.

Step 2: Compute Possible Values of m_l for l = 2

The magnetic quantum number \( m_l \) can take integer values from \(-l\) to \( +l \). For \( l = 2 \), \( m_l \) can be -2, -1, 0, 1, or 2.

Step 3: Determine Possible Values of l for m_l = 2

For \( m_l = 2 \), the values of \( l \) must be 2 or greater because \( |m_l| \) must be less than or equal to \( l \). Therefore, the possible value for \( l \) is 2.

Key Concepts

Principal Quantum Number

The principal quantum number is a fundamental concept in quantum mechanics that represents the energy level an electron occupies in an atom. Denoted by the symbol \( n \), it directly correlates with the size and energy of the orbital. Here's what you need to know about it:

• The principal quantum number can be any positive integer starting from 1 (e.g., 1, 2, 3, ...).
• It defines the shell of the electron, with higher values of \( n \) corresponding to higher energy levels and larger orbitals.
• For each value of \( n \), the azimuthal quantum number \( l \) can take values from 0 to \( n-1 \).

By understanding the principal quantum number, students can predict the potential energy and size of electron clouds in an atom. This is crucial for comprehending more complex quantum scenarios.

Azimuthal Quantum Number

The azimuthal quantum number, also known as the angular momentum quantum number, is symbolized by \( l \) and describes the shape of the electron's orbital. It's essential for defining the types of subshells within a principal energy level:

• The azimuthal quantum number can take integer values ranging from 0 to \( n-1 \), where \( n \) is the principal quantum number.
• Each value of \( l \) corresponds to a specific type of orbital shape: \( l = 0 \) represents an s orbital, \( l = 1 \) represents a p orbital, \( l = 2 \) represents a d orbital, and \( l = 3 \) represents an f orbital.
• The number of possible values for \( l \) increases with higher principal quantum numbers. For example, when \( n = 4 \), \( l \) could be 0, 1, 2, and 3, allowing for s, p, d, and f orbitals.

The azimuthal quantum number provides insight into the angular distribution of the electron cloud and helps differentiate between various electron subshells within the same energy level.

Magnetic Quantum Number

The magnetic quantum number, represented by \( m_l \), complements the principal and azimuthal quantum numbers by providing information about the orientation of an electron's orbital in space. This is how it works:

• The value of \( m_l \) depends on the azimuthal quantum number \( l \) and ranges from \( -l \) to \( +l \). This means if \( l = 2 \), the possible values for \( m_l \) are -2, -1, 0, 1, and 2.
• The magnetic quantum number is crucial for determining how orbitals are oriented within a magnetic field, affecting electron configurations and chemical bonding.
• It essentially describes how many orbitals there are for a given subshell. For example, a p subshell (\( l = 1 \)) has three orbitals, corresponding to \( m_l = -1, 0,\) and \( +1 \).

The concept of the magnetic quantum number is integral in quantum mechanics for understanding the spatial orientation of orbitals, which plays a pivotal role in the behavior of electrons in atoms.