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 is 0 because both have angular momentum \(\sqrt{2} \hbar\).

Step 1: Understand Angular Momentum of Electron in an Orbital

The angular momentum of an electron in an 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: Determine Azimuthal Quantum Number for \(2\mathrm{p}\) Orbital

The principal quantum number, \( n \), for a \(2\mathrm{p}\) orbital is 2, and the azimuthal quantum number, \( l \), for a \mathrm{p} orbital is 1.

Step 3: Calculate Angular Momentum for \(2\mathrm{p}\) Orbital

Using the formula \( L = \sqrt{l(l+1)} \hbar \), substitute \( l = 1 \):\[ L_{2\mathrm{p}} = \sqrt{1(1+1)} \hbar = \sqrt{2} \hbar \]

Step 4: Determine Azimuthal Quantum Number for \(4\mathrm{p}\) Orbital

Similarly, for the \(4\mathrm{p}\) orbital, the principal quantum number \( n \) is 4, and the azimuthal quantum number \( l \) remains 1 since it is a \mathrm{p} orbital.

Step 5: Calculate Angular Momentum for \(4\mathrm{p}\) Orbital

Using the same formula \( L = \sqrt{l(l+1)} \hbar \) and substituting \( l = 1 \):\[ L_{4\mathrm{p}} = \sqrt{1(1+1)} \hbar = \sqrt{2} \hbar \]

Step 6: Calculate the Difference in Angular Momentum

The angular momenta for \(2\mathrm{p}\) and \(4\mathrm{p}\) are both \(\sqrt{2} \hbar\). Therefore, the difference is:\[ L_{4\mathrm{p}} - L_{2\mathrm{p}} = \sqrt{2} \hbar - \sqrt{2} \hbar = 0 \]

Key Concepts

Azimuthal Quantum Number

In the world of quantum mechanics, the azimuthal quantum number, symbolized by \( l \), plays a crucial role in determining the shape of an electron's orbital. This number is one of the four quantum numbers used to describe an electron's state in an atom. It is sometimes referred to as the angular momentum quantum number because it relates directly to the angular momentum of electrons.

• For \( p \) orbitals, like those in the 2\( p \) and 4\( p \) configurations discussed in the exercise, \( l \) is equal to 1.
• The azimuthal quantum number ranges from 0 to \( n-1 \), where \( n \) is the principal quantum number.
• This number influences the shape and type of orbital, determining whether it is a spherical \( s \), dumbbell-shaped \( p \), or other more complex forms like \( d \) and \( f \).

These shapes define how the electron cloud surrounds the nucleus, impacting the electron cloud's energy and other properties. For both 2\( p \) and 4\( p \) orbitals, since they are \( p \) orbitals, \( l \) is consistently 1, which is why their angular momentum calculations yield the same result despite different principal quantum numbers.

Orbitals in Quantum Mechanics

Orbitals in quantum mechanics describe regions around an atom's nucleus where electrons are likely to be found. Unlike the defined paths of electrons in classical physics, orbitals represent probabilities. This means that rather than moving in neat orbits, electrons are more spread out, present in a cloud-like region.

• Each type of orbital (\( s, p, d, \) and \( f \)) has a distinct shape that influences the atom's chemistry and bonding patterns.
• \( p \) orbitals, for example, have a dumbbell shape and can hold up to six electrons across three spatial orientations (\( p_x, p_y, \, or \, p_z \)).
• These orbitals are defined by quantum numbers, which provide a comprehensive picture of an electron's unique state.

Understanding these shapes and the electron probabilities is crucial for predicting how atoms will interact, bond, and form the myriad of substances we encounter in the natural world. In quantum calculations, as seen in the exercise, the orbital type directly affects properties like angular momentum.

Quantum Numbers in Chemistry

Quantum numbers in chemistry are pivotal for understanding electronic arrangements within atoms. They provide a complex labeling system that captures the intricate behavior of electrons in atoms. Four quantum numbers define an electron's probable location and energy state:

• Principal Quantum Number (\( n \)): Determines the size and energy level of the electron's orbital. Larger \( n \) values correspond to higher energy levels and larger orbitals.
• Azimuthal Quantum Number (\( l \)): Relates to the shape of the orbital and is explained in more detail in its own section.
• Magnetic Quantum Number (\( m_l \)): Describes the orientation of an orbital in space relative to the other orbitals.
• Spin Quantum Number (\( m_s \)): Relates to the electron's intrinsic angular momentum or "spin," which can be either +1/2 or -1/2.

These numbers work together, providing a comprehensive framework for predicting how atoms will form bonds and reaction pathways. By decoding these numbers, chemists can understand the electronic configurations that underlie chemical properties and trends across the periodic table, ensuring they can predict reactions and new compound formations effectively.