Problem 144 - A Complete Resource Book in Chemistry for JEE Main

Question

What is the difference in the angular momentum of an electron present in \(2 \mathrm{p}\) and that present in \(4 \mathrm{p}\) orbital?

Step-by-Step Solution

Answer

The difference in angular momentum is 0.

Step 1: Understand Angular Momentum

The angular momentum of an electron in an atomic orbital is given by the formula \( L = \sqrt{l(l+1)} \hbar \), where \( l \) is the azimuthal quantum number and \( \hbar \) is the reduced Planck's constant.

Step 2: Identify Quantum Numbers

For the \( 2p \) and \( 4p \) orbitals, the azimuthal quantum number \( l \) is 1, because \( p \) orbitals always have \( l = 1 \). This applies to both orbitals.

Step 3: Calculate Angular Momentum for \( 2p \)

Substitute \( l = 1 \) into the angular momentum formula: \[ L_{2p} = \sqrt{1(1+1)} \hbar = \sqrt{2} \hbar \].

Step 4: Calculate Angular Momentum for \( 4p \)

Similarly, substitute \( l = 1 \) for the \( 4p \) orbital: \[ L_{4p} = \sqrt{1(1+1)} \hbar = \sqrt{2} \hbar \].

Step 5: Find the Difference

The angular momentum for both \( 2p \) and \( 4p \) orbitals is \( \sqrt{2} \hbar \). Therefore, the difference in angular momentum is \[ L_{4p} - L_{2p} = \sqrt{2} \hbar - \sqrt{2} \hbar = 0 \].

Key Concepts

Quantum Numbers

Quantum numbers are essential in the field of quantum mechanics as they describe the unique quantum state of an electron. They help in characterizing electronic orbitals where electrons reside. There are four types of quantum numbers:

• Principal Quantum Number (\( n \)): Indicates the energy level and size of the orbital; higher values mean larger orbitals.
• Azimuthal Quantum Number (\( l \)): Describes the shape of the orbital.
• Magnetic Quantum Number (\( m_l \)): Represents the orientation of the orbital in space.
• Spin Quantum Number (\( m_s \)): Denotes the direction of the electron's spin.

Each electron in an atom has its own set of these four quantum numbers, making it unique. These numbers are not solely abstract figures; they deeply influence the region where an electron is likely to be found, dictating the atom's chemical behavior.

Atomic Orbitals

Atomic orbitals are regions in an atom where electrons have a high probability of being found. These orbitals are described by quantum numbers and vary in shape and size depending on the energy level and type of orbital:

• \( s \)-orbitals: Spherical in shape with l=0.
• \( p \)-orbitals: Dumbbell-shaped with l=1.
• \( d \)-orbitals: More complex shapes with l=2.
• \( f \)-orbitals: Even more complex, with l=3.

The shape and energy of these orbitals affect how atoms bond and interact with one another. For example, the overlap of \( p \)-orbitals between two atoms can lead to the formation of a covalent bond. Understanding the shapes and orientations of atomic orbitals is fundamental to mastering chemistry and understanding molecular structures.

Azimuthal Quantum Number

The azimuthal quantum number, often symbolized as \( l \), is one of the four quantum numbers that are vital in identifying an electron's state in an atom. It plays a crucial role in determining the shape of the electron orbital:

• \( l = 0 \) corresponds to \( s \)-orbitals, which are spherical.
• \( l = 1 \) refers to \( p \)-orbitals, which have a dumbbell shape.
• \( l = 2 \) indicates \( d \)-orbitals, with more complex shapes.
• \( l = 3 \) is associated with \( f \)-orbitals, showcasing even more complex structures.

The value of \( l \) ranges from 0 to \( n-1 \), where \( n \) is the principal quantum number of the electron's shell. For instance, in the \( 2p \) and \( 4p \) orbitals discussed earlier, \( l \) is 1, denoting their \( p \)-orbital structure. This azimuthal quantum number not only shapes the orbitals but also impacts their energy levels and how they interact with other orbitals within an atom or during chemical reactions.