Question

Discuss the four quantum numbers. Explain the interrelationship among the four quantum numbers. Also explain the significance of a quantum number.

Short answer

Answer: The four quantum numbers are: 1. Principle quantum number (n) - Represents the main energy level or shell of an electron. 2. Azimuthal quantum number (l) - Determines the shape of the electron's orbital or subshell. 3. Magnetic quantum number (m_l) - Defines the orientation of the electron's orbital within a subshell. 4. Spin quantum number (m_s) - Indicates the intrinsic angular momentum or "spin" of an electron (either +1/2 or -1/2).

Step-by-step solution

1. Introduction to Quantum Numbers

Quantum numbers are a set of four numerical values that give a complete description of an electron in an atom. These numbers help determine the size, shape, and orientation of electron orbitals, as well as the electron's spin direction. Each quantum number has specific restrictions and allowed values to ensure a stable and valid electron configuration within the atom.

2. Principle Quantum Number (n)

The principle quantum number (n) defines the main energy level or shell of an electron. It is always a positive integer starting from 1, where n = 1 represents the first energy level closest to the nucleus. As n increases, the electron's energy also increases, and the electron resides in a shell farther from the nucleus. The value of n also determines the maximum number of electrons allowed in a shell, which can be calculated using the formula: 2n^2.

3. Azimuthal Quantum Number (l)

The azimuthal quantum number (l) defines the shape of an electron's orbital or subshell. It has integer values ranging from 0 to (n-1). For a given value of n, there will be n possible values for l, such as s, p, d, and f orbitals with l values of 0, 1, 2, and 3, respectively. Each specific value of l corresponds to a particular shape of the electron cloud surrounding the nucleus. The total number of electrons in a subshell can be calculated using the formula: 2(2l + 1).

4. Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) defines the orientation of an electron's orbital within a subshell. For a given value of l, m_l can take on integer values ranging from -l to +l, including zero. For example, if l = 2 (a d-orbital), m_l can have values of -2, -1, 0, 1, 2. Each value corresponds to a specific orientation of the orbital in three-dimensional space.

5. Spin Quantum Number (m_s)

The spin quantum number (m_s) represents the intrinsic angular momentum or "spin" of an electron and has only two possible values: +1/2 or -1/2. This indicates that an electron can spin clockwise (+1/2) or counterclockwise (-1/2) around its own axis. Since each orbital can hold a maximum of two electrons, the two electrons within an orbital must have opposite spins to comply with the Pauli Exclusion Principle, which states that no two electrons can have the same set of quantum numbers.

6. Interrelationship among Quantum Numbers

The four quantum numbers are interrelated in such a way that they describe an electron's unique position, energy, and movement within an atom. The value of n determines the allowed values of l, and together, n and l determine the range for m_l values. Additionally, m_s values are affected by the occupancy of electron orbitals, with all orbitals being able to accommodate two electrons with opposite spins.

7. Significance of Quantum Numbers

Quantum numbers have significant importance in understanding the electron configuration and properties of atoms. They help classify the elements in the periodic table based on their electron configurations, explain trends and periodic properties, and determine an atom's chemical reactivity. Quantum numbers are also useful in predicting the behavior of atoms in chemical reactions and understanding the spectra of light emitted by excited atoms.