Problem 81 - A Complete Resource Book in Chemistry for JEE Main

Question

For a d-electron, the orbital angular momentum is (a) \(\sqrt{6} \mathrm{~h}\) (b) \(\sqrt{2 h}\) (c) \(\mathrm{h}\) (d) \(2 \mathrm{~h}\)

Step-by-Step Solution

Answer

The orbital angular momentum for a d-electron is \( \sqrt{6} \mathrm{~h} \), which corresponds to option (a).

Step 1: Recall the Orbital Angular Momentum Formula

The formula to calculate the orbital angular momentum for an electron in the shell is \( L = \sqrt{l(l+1)}\hbar \), where \( l \) is the azimuthal quantum number (or angular momentum quantum number) and \( \hbar \) is the reduced Planck's constant (\( h/(2\pi) \)).

Step 2: Identify the Quantum Number for a d-Electron

For a d-orbital, the azimuthal quantum number \( l \) is 2.

Step 3: Substitute the Value of l into the Formula

Substitute \( l = 2 \) into the formula for orbital angular momentum: \[ L = \sqrt{2(2+1)}\hbar = \sqrt{6}\hbar \].

Step 4: Compare with Given Options

The calculated orbital angular momentum is \( \sqrt{6} \mathrm{~h} \), which matches option (a). Therefore, the correct answer is (a) \( \sqrt{6} \mathrm{~h} \).

Key Concepts

Quantum Numbers

In the world of quantum mechanics, particles like electrons in an atom are described by quantum numbers. Quantum numbers are used to specify the properties of atomic orbitals and the properties of electrons in those orbitals. They provide a numerical value for the electron's energy, spin, angular momentum, and magnetic orientation. There are four primary quantum numbers that give a complete description of an electron within an atom:

• Principal Quantum Number (n) : This number represents the main energy level of the electron and its average distance from the nucleus. Larger values of n indicate larger orbitals and higher energy levels.
• Azimuthal Quantum Number (l ): Also known as the angular momentum quantum number, it defines the shape of the orbital and has values ranging from 0 up to n-1 for each principal quantum number n .
• Magnetic Quantum Number (m ) : This describes the orientation of the orbital in space relative to the other orbitals, with values ranging from -l to l.
• Spin Quantum Number (s ) : It specifies the direction of the electron's spin, with possible values of +1/2 or -1/2.

By using all these quantum numbers together, you can fully describe the state of an electron in an atom.

Azimuthal Quantum Number

The azimuthal quantum number, represented by the symbol l , plays a crucial role in determining the geometry of an electron's orbital. It directly influences the shape of the orbital, an essential aspect of modern chemistry and atomic theory.

• The value of l ranges from 0 to n-1 , where n is the principal quantum number.
• The specific value of l assigns a letter symbol to the orbital type: 0 is s , 1 is p , 2 is d , and 3 is f.
• Higher values of l indicate orbitals with more complex shapes.

In essence, the azimuthal quantum number is like a blueprint for the electron cloud’s architecture around the nucleus. It indicates how the wave function or probability cloud is distributed in different directions in space, leading to the characteristic shapes of the s , p , d , and f orbitals.

d-Orbitals

The d-orbitals are a set of orbitals with an azimuthal quantum number l of 2. They are vital in the formation of transition metals and complex ions in chemistry.
These orbitals are fascinating because of their complex shapes and are generally found in atoms that have electrons in their third or higher energy levels.

• There are five d-orbitals, and due to their higher angular momentum, they exhibit unique shapes like cloverleaf or double-lobed geometries.
• These orbitals can accommodate up to ten electrons since each orbital holds a maximum of two electrons with opposite spins.
• The complex shapes of d-orbitals allow them to have directional preferences, important in the bonding and arrangement of atoms in metallic and magnetic materials.

Understanding d-orbitals helps explain many phenomena in chemistry, such as the color, magnetism, and reactivity of transition metal compounds, a key topic in advanced chemistry studies.