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 zero (0).

Step 1: Understanding Angular Momentum of Orbitals

The angular momentum of an electron in an atomic orbital is defined by the quantum number \( l \) (azimuthal quantum number). The angular momentum can be calculated using the formula: \[ L = \sqrt{l(l+1)} \hbar \] where \( \hbar \) is the reduced Planck's constant.

Step 2: Identify the Quantum Numbers

For both the \( 2p \) and \( 4p \) orbitals, the azimuthal quantum number \( l = 1 \) because \( p \) orbitals always have \( l = 1 \). This is true regardless of the principal quantum number \( n \).

Step 3: Calculate Angular Momentum for \( 2p \) and \( 4p \) Orbitals

Since both \( 2p \) and \( 4p \) orbitals have \( l = 1 \), calculate the angular momentum for each using \( L = \sqrt{1(1+1)} \hbar = \sqrt{2} \hbar \). Thus, the angular momentum for an electron in both the \( 2p \) and \( 4p \) orbitals is \( \sqrt{2} \hbar \).

Step 4: Determine the Difference in Angular Momentum

Since the angular momentum for both \( 2p \) and \( 4p \) orbitals is \( \sqrt{2} \hbar \), the difference in angular momentum between electrons in these orbitals is \( \sqrt{2} \hbar - \sqrt{2} \hbar = 0 \).

Key Concepts

Quantum Numbers

In quantum mechanics, quantum numbers are essential for describing the properties of electrons in an atom. They give us a detailed idea of where an electron is likely to be found and the energy state it is in. There are four fundamental quantum numbers:

• Principal Quantum Number ( igstrut n): Indicates the main energy level or shell of an electron. It determines the size and energy of the orbital.
• Azimuthal Quantum Number ( igstrut l): Relates to the shape of the orbital and is often associated with the concept of angular momentum.
• Magnetic Quantum Number ( igstrut m_l): Describes the orientation of the orbital in space.
• Spin Quantum Number ( igstrut m_s): Refers to the electron's spin direction, which can be either +1/2 or -1/2.

Understanding these numbers is crucial for visualizing how electrons are distributed in an atom and predicting the atom's chemical properties. Quantum numbers help in identifying the atomic orbitals and describe characteristics like energy, shape, and positioning of the orbitals.

Azimuthal Quantum Number

The azimuthal quantum number, represented as l, is a critical component in defining the shape and type of atomic orbitals. It is also related to the orbital angular momentum of the electron. For each principal quantum number ( n), the azimuthal quantum number can have integer values ranging from 0 to (n-1). This wide range allows for the differentiation in the complexity of electron clouds. For example:

• If l = 0, the orbital is a spherical s orbital.
• If l = 1, the orbital is a dumbbell-shaped p orbital.
• Higher values of l correspond to more complex shapes such as d and f orbitals.

The azimuthal quantum number also plays a significant role in determining the sublevel structure of an atom, influencing its chemical bonding and spectral characteristics.

Atomic Orbitals

Atomic orbitals are regions around an atom's nucleus where there is a high probability of finding an electron. They are vital for understanding atomic structure and chemical bonds. Orbitals are defined by quantum numbers, which set the energy shell, shape, and orientation:

• n: Defines the size and energy level of the orbital as well as the atom.
• l: Indicates the shape of the orbital (s, p, d, and f).

The specific arrangement and shape of orbitals determine how atoms interact and bond with each other. In simpler terms, orbitals describe the three-dimensional space where electrons are found, influencing chemical properties and reactions. Studying atomic orbitals helps us understand the electronic configuration of atoms, predicting how they will react with each other. For example, p orbitals have a characteristic dumbbell shape and play a crucial role in the forming of multiple bonds and overlapping with orbitals of other atoms to form chemical bonds, such as in the case of covalent bonds.