QUESTION NO. 8

(a) State and explaio Pauli’s Exclusion principle.

Pauli’s Exclusion Principle is a fundamental concept in quantum mechanics that describes the behavior of electrons (and other fermions) within an atom. It was formulated by the physicist Wolfgang Pauli in 1925.

Statement of the Principle:

No two electrons (or any other fermions) in an atom can have the same set of quantum numbers.

Explanation:

To understand Pauli’s Exclusion Principle, it is helpful to know the quantum numbers that describe the state of an electron in an atom:

1. Principal Quantum Number ((n)): Defines the energy level or shell in which the electron resides.
2. Angular Momentum Quantum Number ((l)): Defines the shape of the orbital (subshell) within the energy level.
3. Magnetic Quantum Number ((m_l)): Defines the orientation of the orbital within the subshell.
4. Spin Quantum Number ((m_s)): Defines the intrinsic angular momentum (spin) of the electron, which can be either +½ or -½.

According to Pauli’s Exclusion Principle:

• No two electrons in an atom can have the same set of all four quantum numbers. This means that within a given atom, each electron must have a unique combination of (n), (l), (m_l), and (m_s).

Implications of the Principle:

1. Electron Configuration: The principle explains the arrangement of electrons in different orbitals. It accounts for the fact that electrons fill orbitals in a way that each orbital can hold a maximum of two electrons, one with spin +½ and the other with spin -½.
2. Atomic Structure: It helps explain the structure of the periodic table and the distribution of electrons in various energy levels and orbitals. This arrangement impacts the chemical properties of elements.
3. Stability of Matter: The principle contributes to the stability of matter. Without it, all electrons could theoretically collapse into the lowest energy state, which would not align with observed matter stability and atomic structure.
4. Spectral Lines: It explains the observed spectral lines and the fine structure of atomic spectra. The exclusion principle affects the energy levels and transitions between them, leading to discrete spectral lines.

In summary, Pauli’s Exclusion Principle is crucial for understanding the arrangement of electrons in atoms, the structure of the periodic table, and the behavior of matter at the quantum level. It is a foundational principle in quantum mechanics and atomic physics.

(b) Write down electronic configuration of the following:
Mn25, M042, Fe2+ 26 16 Cu2+ S2 29

Here’s the electronic configuration for each of the given species:

1. Manganese (Mn)

• Atomic Number: 25
• Ground-State Configuration: [ \text{Mn} = [\text{Ar}] \, 4s^2 \, 3d^5 ]

2. Molybdenum (Mo) in the 4+ oxidation state (Mo(^{4+}))

• Atomic Number: 42
• Ground-State Configuration: [ \text{Mo} = [\text{Kr}] \, 5s^1 \, 4d^5 ] For Mo(^{4+}), we remove 4 electrons: [ \text{Mo}^{4+} = [\text{Kr}] \, 4d^4 ]

3. Iron (Fe) in the 2+ oxidation state (Fe(^{2+}))

• Atomic Number: 26
• Ground-State Configuration: [ \text{Fe} = [\text{Ar}] \, 4s^2 \, 3d^6 ] For Fe(^{2+}), we remove 2 electrons: [ \text{Fe}^{2+} = [\text{Ar}] \, 3d^6 ]

4. Sulfur (S) in the 2- oxidation state (S(^{2-}))

• Atomic Number: 16
• Ground-State Configuration: [ \text{S} = [\text{Ne}] \, 3s^2 \, 3p^4 ] For S(^{2-}), we add 2 electrons: [ \text{S}^{2-} = [\text{Ne}] \, 3s^2 \, 3p^6 ]

5. Copper (Cu) in the 2+ oxidation state (Cu(^{2+}))

• Atomic Number: 29
• Ground-State Configuration: [ \text{Cu} = [\text{Ar}] \, 4s^1 \, 3d^{10} ] For Cu(^{2+}), we remove 2 electrons (first from the 4s orbital, then from the 3d orbital if needed): [ \text{Cu}^{2+} = [\text{Ar}] \, 3d^9 ]

Summary:

• Mn: ([ \text{Ar}] \, 4s^2 \, 3d^5 )
• Mo(^{4+}): ([ \text{Kr}] \, 4d^4 )
• Fe(^{2+}): ([ \text{Ar}] \, 3d^6 )
• S(^{2-}): ([ \text{Ne}] \, 3s^2 \, 3p^6 )
• Cu(^{2+}): ([ \text{Ar}] \, 3d^9 )

(c) Explain Azimuthal and Magnetic quantum numbers with suitable examples.

Azimuthal Quantum Number ((l)) and Magnetic Quantum Number ((m_l)) are two of the four quantum numbers used to describe the properties and locations of electrons in an atom. They provide information about the shape and orientation of atomic orbitals.

Azimuthal Quantum Number ((l))

Definition:

• The azimuthal quantum number, also known as the angular momentum quantum number, determines the shape of the electron’s orbital and relates to the angular momentum of the electron.
• It can have integer values from (0) to (n-1), where (n) is the principal quantum number.

Symbol and Values:

• The symbol for the azimuthal quantum number is (l).
• Values: (l = 0, 1, 2, 3, \ldots, n-1).

Orbital Types:

• For each value of (l), there is a specific type of orbital:
• (l = 0): s-orbital (spherical shape)
• (l = 1): p-orbital (dumbbell shape)
• (l = 2): d-orbital (cloverleaf shape)
• (l = 3): f-orbital (complex shape)

Example:

• For an electron in the 3p orbital:
• Principal Quantum Number ((n)) = 3
• Azimuthal Quantum Number ((l)) = 1 (p-orbital)

Magnetic Quantum Number ((m_l))

Definition:

• The magnetic quantum number determines the orientation of the orbital within a given subshell (or angular momentum state).
• It can have integer values ranging from (-l) to (+l), including zero.

Symbol and Values:

• The symbol for the magnetic quantum number is (m_l).
• Values: (m_l = -l, -(l-1), \ldots, 0, \ldots, (l-1), +l).

Examples:

1. s-orbital ((l = 0)):

• For (l = 0), (m_l) can only be (0).
• This means there is only one orientation for an s-orbital.

1. p-orbital ((l = 1)):

• For (l = 1), (m_l) can be (-1, 0, +1).
• This indicates that there are three possible orientations for p-orbitals:
  • (p_x), (p_y), and (p_z).

1. d-orbital ((l = 2)):

• For (l = 2), (m_l) can be (-2, -1, 0, +1, +2).
• This means there are five possible orientations for d-orbitals.

1. f-orbital ((l = 3)):

• For (l = 3), (m_l) can be (-3, -2, -1, 0, +1, +2, +3).
• This indicates seven possible orientations for f-orbitals.

Summary:

• Azimuthal Quantum Number ((l)): Determines the shape of the orbital. Values range from (0) to (n-1). Examples: (s)-orbitals ((l = 0)), (p)-orbitals ((l = 1)), (d)-orbitals ((l = 2)), (f)-orbitals ((l = 3)).
• Magnetic Quantum Number ((m_l)): Determines the orientation of the orbital. Values range from (-l) to (+l), including zero. Examples: For (p)-orbitals ((l = 1)), (m_l) can be (-1, 0, +1); for (d)-orbitals ((l = 2)), (m_l) can be (-2, -1, 0, +1, +2).