1. Define:

Sure! Here’s a breakdown of each of the requested concepts:

a) Quantum Mechanical Numbers

Quantum mechanical numbers describe the quantum state of an electron in an atom. There are four quantum numbers:

1. Principal Quantum Number (n):

   • Defines the main energy level or shell of an electron.
   • It can take positive integer values: \( n = 1, 2, 3, \ldots \).
   • The larger the value of \( n \), the higher the energy level and the further from the nucleus the electron is likely to be.

2. Azimuthal Quantum Number (l):

   • Defines the shape of the electron's orbital.
   • It can take integer values from \( 0 \) to \( n-1 \).
   • The values of \( l \) correspond to different orbital types: \( 0 \) (s), \( 1 \) (p), \( 2 \) (d), \( 3 \) (f), etc.

3. Magnetic Quantum Number (m_l):

   • Defines the orientation of the orbital in space.
   • It can take integer values ranging from \( -l \) to \( +l \), including zero.
   • For example, if \( l = 1 \) (p orbital), \( m_l \) can be \( -1, 0, +1 \), representing three different orientations.

4. Spin Quantum Number (m_s):

   • Describes the intrinsic spin of the electron, which can be thought of as the electron's intrinsic angular momentum.
   • It can take on one of two values: \( +\frac{1}{2} \) (spin up) or \( -\frac{1}{2} \) (spin down).

b) Hund’s Rule

Hund’s Rule states that for a given set of orbitals, the lowest energy configuration is achieved when the maximum number of unpaired electrons occupy degenerate (equal energy) orbitals before any pairing occurs. This means that each orbital in a subshell is singly occupied before any orbital is doubly occupied.

Example:

Consider the electron configuration of carbon, which has six electrons: \( 1s^2 , 2s^2 , 2p^2 \).

• The 2p subshell can hold a total of six electrons (three orbitals).
• According to Hund's Rule, the two electrons in the 2p level will occupy different orbitals before pairing up. Thus, the configuration in the 2p subshell will be depicted as:
  • \( \uparrow , , \uparrow , , , , , (2p_x , \text{and} , 2p_y) \)

c) Pauli Exclusion Principle

The Pauli Exclusion Principle states that no two electrons in an atom can have the same set of four quantum numbers. In other words, each electron must have a unique configuration that distinguishes it from every other electron in the atom.

Example:

In a helium atom (He), which has two electrons, the electron configuration is \( 1s^2 \). The two electrons occupy the same 1s orbital, but they have opposite spins to comply with the Pauli Exclusion Principle:

• The first electron can be described by the quantum numbers \( (n=1, l=0, m_l=0, m_s=+\frac{1}{2}) \).
• The second electron will then have \( (n=1, l=0, m_l=0, m_s=-\frac{1}{2}) \).

This ensures that both electrons have different spin quantum numbers, hence adhering to the exclusion principle.