Question

Write possible sets of quantum numbers for electrons in the second main energy level.

Short answer

The possible sets of quantum numbers for electrons in the second main energy level are: 1. (2, 0, 0, +1/2) 2. (2, 0, 0, -1/2) 3. (2, 1, -1, +1/2) 4. (2, 1, -1, -1/2) 5. (2, 1, 0, +1/2) 6. (2, 1, 0, -1/2) 7. (2, 1, +1, +1/2) 8. (2, 1, +1, -1/2)

Step-by-step solution

Determine the values of azimuthal quantum number (𝑙)

Since the principal quantum number (n) is 2, the possible values of the azimuthal quantum number (𝑙) are 0 and 1.

Determine the values of magnetic quantum numbers (m_𝑙)

For each value of the azimuthal quantum number (𝑙), we need to find the possible values of the magnetic quantum number (m_𝑙): 1. When 𝑙 = 0, the possible values of m_𝑙 are 0, as it ranges from -𝑙 to +𝑙, including 0. 2. When 𝑙 = 1, the possible values of m_𝑙 are -1, 0, and +1, as it ranges from -𝑙 to +𝑙, including 0.

Determine the values of spin quantum numbers (m_s)

The spin quantum number (m_s) has only two possible values, +1/2 and -1/2, for any energy level or orbital.

Combine all possible values of the four quantum numbers

Now, we can list all possible combinations of the principal quantum number (n), azimuthal quantum number (𝑙), magnetic quantum number (m_𝑙), and spin quantum number (m_s): 1. n = 2, l = 0, m_l = 0, m_s = +1/2 2. n = 2, l = 0, m_l = 0, m_s = -1/2 3. n = 2, l = 1, m_l = -1, m_s = +1/2 4. n = 2, l = 1, m_l = -1, m_s = -1/2 5. n = 2, l = 1, m_l = 0, m_s = +1/2 6. n = 2, l = 1, m_l = 0, m_s = -1/2 7. n = 2, l = 1, m_l = +1, m_s = +1/2 8. n = 2, l = 1, m_l = +1, m_s = -1/2 These are the possible sets of quantum numbers for electrons in the second main energy level.

Key concepts

Principal Quantum Number

The principal quantum number is usually denoted by the letter \( n \). It plays a crucial role in determining the overall size and energy of an electron orbital in an atom. In simple terms, it indicates the "shell" or energy level where an electron resides.
When \( n = 1 \), it represents the first energy level, the closest to the nucleus. As \( n \) increases, the electron is found in orbitals that are farther from the nucleus and experience higher energy levels.

• In our case, with \( n = 2 \), we are looking at the second main energy level.
• The value of \( n \) is always a positive integer starting from 1.

Understanding the principal quantum number is essential because it helps indicate not just the total energy of an electron, but also the number of allowable sublevels within that energy level.

Azimuthal Quantum Number

The azimuthal quantum number, often represented by \( l \), determines the shape of an electron's orbital and is also known as the angular momentum quantum number. It can take on any integer value from 0 up to \( n-1 \), where \( n \) is the principal quantum number.
This quantum number is responsible for defining the subshells within a given energy level:

• \( l = 0 \) corresponds to an "s" orbital, which is spherical.
• \( l = 1 \) represents a "p" orbital, which has a dumbbell shape.

For the second energy level (\( n = 2 \)), the azimuthal quantum number can be either 0 or 1. This means:

• There are two types of subshells available: s (\( l = 0 \)) and p (\( l = 1 \)).

This quantum number helps in understanding the orientation and complex structures of atomic orbitals.

Magnetic Quantum Number

Magnetic quantum number, symbolized as \( m_l \), describes the orientation of an orbital around the nucleus. It gives insight into how orbitals are distributed in space. This number can vary between \(-l\) and \(+l\), including zero.

• For \( l = 0 \), \( m_l \) can only be 0, indicating a single s orbital.
• For \( l = 1 \), \( m_l \) can be -1, 0, or +1, corresponding to the three possible orientations of the p orbitals (p_x, p_y, p_z).

These orientations explain how electron clouds are arranged around the nucleus and how they interact with magnetic fields. Understanding \( m_l \) values is essential for determining the electron distribution in magnetic and electrical settings.

Spin Quantum Number

The spin quantum number, denoted by \( m_s \), is rather unique compared to the other quantum numbers. It describes the intrinsic angular momentum, or "spin", of an electron within its orbital. This property can only have two possible values: \(+\frac{1}{2}\) or \(-\frac{1}{2}\). These values represent two types of spin orientations, often labeled as "spin-up" and "spin-down".
Unlike other quantum numbers, \( m_s \) does not depend on the orbital's shape or size but distinguishes electrons by their spin state. Each orbital can accommodate up to two electrons, but they must have opposite spins due to the Pauli Exclusion Principle.
In any given orbital:

• If one electron has a spin of \(+\frac{1}{2}\), the other must have \(-\frac{1}{2}\).

This quantum number is crucial for explaining more complex phenomena such as electron pairing and ferromagnetism.