Problem 158 - A Complete Resource Book in Chemistry for JEE Main

Question

The orbital angular momentum for an electron re volving in an orbit is given by \(\sqrt{\ell} /(l+1) \mathrm{h} / 2 \pi\). This momentum for an s electron will be given by \([\mathbf{2 0 0 3}]\) (a) \(+1 / 2 \cdot \mathrm{h} / 2 \pi\) (b) zero (c) \(\mathrm{h} / 2 \pi\) (d) \(\sqrt{2} \cdot \mathrm{h} / 2 \pi\)

Step-by-Step Solution

Answer

The orbital angular momentum for an s electron is zero.

Step 1: Understanding the Problem

We are given the expression for orbital angular momentum: \( \sqrt{\ell(\ell+1)} \cdot \frac{h}{2 \pi} \). We need to find out the momentum for an s electron, which is characterized by the angular momentum quantum number \( \ell = 0 \).

Step 2: Substitute \(\ell=0\)

For an s electron, the angular momentum quantum number \( \ell = 0 \). Substitute \( \ell = 0 \) into the expression \( \sqrt{\ell(\ell+1)} \cdot \frac{h}{2 \pi} \).

Step 3: Calculate Angular Momentum

Substituting \( \ell = 0 \) gives: \( \sqrt{0(0+1)} \cdot \frac{h}{2 \pi} = \sqrt{0} \cdot \frac{h}{2 \pi} = 0 \). Therefore, for an s electron, the orbital angular momentum is zero.

Step 4: Conclusion

Based on the calculation, the orbital angular momentum for an s electron is zero, which corresponds to option (b).

Key Concepts

Angular Momentum Quantum Number

In quantum mechanics, the angular momentum quantum number is a vital concept that helps describe the environment where an electron resides within an atom. This quantum number, often symbolized as \( \ell \), dictates the shape of the orbital in which an electron moves. Angular momentum is a property intrinsic to an electron's movement, and it comes with discrete possible values.

• When \( \ell = 0 \), the electron is in an 's orbital'.
• An 's' orbital is spherical, indicating equal probability of finding the electron at any angle.
• Other possible values (when \( \ell \) is greater than zero) correspond to other orbital shapes like 'p', 'd', 'f', etc.

The expression for the magnitude of orbital angular momentum is given by: \[\sqrt{\ell(\ell+1)} \cdot \frac{h}{2 \pi}\]For the 's' electron, because \( \ell = 0 \), this simplifies to zero, since \( \sqrt{0(0+1)} = 0 \). This is why an s electron does not have orbital angular momentum.

s Electron

The 's electron' is a term used to refer to the simplest type of electron orbital in an atom. The 's' in 's-electron' signifies that it is associated with an \( \ell \) value of zero, as determined by its angular momentum quantum number. Here's what makes 's electrons' unique:

• They are located in spherical orbitals around the atomic nucleus.
• Because they are spherical, s electrons have no angular dependence, meaning their probability cloud is uniform in all directions.
• The absence of angular momentum (because the orbital angular momentum is zero) suggests these electrons do not revolve around the nucleus like tiny spinning tops.

The 's' orbital can occur in each energy level, starting from the principal quantum number \( n = 1 \). In each energy level, there is only one 's' orbital, which can hold a maximum of two electrons according to the Pauli Exclusion Principle. This foundation explains why certain elements have very predictable chemical properties based on their electron configuration.

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that describes the physical properties of nature at the scale of atoms and subatomic particles. It provides a comprehensive framework to understand the behaviors of electrons in atoms.

• Electrons are often considered as both particles and waves, leading to the development of quantum states.
• Quantum mechanics incorporates concepts such as wave functions and probability amplitudes, which predict where an electron is likely to be found.
• One of the important quantum numbers, such as angular momentum quantum number \( \ell \), play a crucial role in defining an electron's state.

The principles of quantum mechanics allow us to understand complex atomic phenomena, including electron configurations, chemical bonding, and spectral lines. It challenges classical physics by introducing the idea that certain properties cannot be precisely measured simultaneously—known as the Heisenberg Uncertainty Principle. This intricate yet fascinating field crafts a more complete picture of atomic and molecular nature, illustrating that at the smallest scales, the universe operates unlike anything in daily human experience.