Question

The orbital angular momentum for an electron re volving in an orbit is given by \(\mathrm{V} /(l+1) \mathrm{h} / 2 \pi\). This momentum for an s electron will be given by: (a) \(+\frac{1}{2} \cdot \mathrm{h} / 2 \pi\) (b) Zero (c) \(\mathrm{h} / 2 \pi\) (d) \(\sqrt{2} \cdot \mathrm{h} / 2 \pi\)

Short answer

The angular momentum for an s electron is zero, thus option (b) Zero is correct.

Step-by-step solution

Understand the Problem

We have to determine the orbital angular momentum of an s electron. Angular momentum is described by the formula \( L = \frac{V}{(l+1)\frac{h}{2\pi}} \), where \( l \) is the azimuthal quantum number.

Identify the Azimuthal Quantum Number

In quantum mechanics, the azimuthal quantum number \( l \) determines the shape of the orbital. For an 's' orbital, \( l = 0 \). This means we will use \( l = 0 \) in our angular momentum formula.

Substitute into the Formula

Substitute \( l = 0 \) into the formula for angular momentum: \( L = \frac{V}{(l+1)\frac{h}{2\pi}} = \frac{V}{(0+1)\frac{h}{2\pi}} = \frac{V}{\frac{h}{2\pi}} = V \cdot \frac{2\pi}{h} \).

Interpret the Options

Since the standard formula for orbital angular momentum is \( L = \sqrt{l(l+1)} \frac{h}{2\pi} \) and for \( l = 0 \), \( L = 0 \). Therefore, the correct answer is "Zero" compared to the adapted expression since \( l = 0 \) makes the angular momentum zero.

Key concepts

Azimuthal Quantum Number

In quantum mechanics, the azimuthal quantum number, often represented by the letter \( l \), is a crucial concept. This number helps us understand the shape and angular momentum of electron orbitals. Each type of orbital—whether it be s, p, d, or f—has a unique azimuthal quantum number associated with it.

• The 's' orbital has an azimuthal quantum number \( l = 0 \).
• The 'p' orbital has \( l = 1 \).
• The 'd' orbital has \( l = 2 \).
• The 'f' orbital has \( l = 3 \).

The azimuthal quantum number is essential as it not only indicates the shape of the orbital (for instance, spherical for s-orbitals) but it also influences the orbital's angular momentum. Whenever an electron transitions to different orbitals, its azimuthal quantum number helps determine the total angular momentum, calculated as \( L = \sqrt{l(l+1)} \frac{h}{2\pi} \).
Thus, since the 's' orbital has \( l = 0 \), its orbital angular momentum is zero.

s Orbital

The 's' orbital is the simplest type of orbital in quantum mechanics, characterized by its spherical shape. It is very different from p, d, or f orbitals, which have more complex shapes. The spherical symmetry of the s orbital means that the probability of finding an electron is the same in all directions at a given distance from the nucleus.
Every energy level in an atom has one s orbital:

• The 1s orbital is the closest to the nucleus and holds up to 2 electrons.
• The 2s orbital is similar but larger and located at a higher energy level.
• Higher energy levels (3s, 4s, etc.) follow the same pattern, each larger than the last.

For an s orbital, the azimuthal quantum number \( l = 0 \), contributing to its spherical shape. This is significant because it simplifies the calculation of certain quantum properties, like angular momentum, making it zero for s electrons. Putting these pieces together, the s orbital is fundamentally important as it defines a starting point for understanding electron configurations across all elements.

Quantum Mechanics

Quantum mechanics is the science that describes the behavior of particles at atomic and subatomic levels, like electrons, protons, and neutrons. It’s a branch of physics that defies classical physics' rules by introducing concepts such as probability, uncertainty, and quantization.

• Probability replaces certainty: Instead of knowing exactly where an electron is, quantum mechanics describes probabilities of where it might be.
• Quantization: Electrons exist in discrete energy levels or "quantized" states, not in a smooth continuum.
• Wave-particle duality: Particles like electrons exhibit properties of both waves and particles.
• Uncertainty Principle: It's impossible to know both the position and momentum of a particle precisely at the same time.

These concepts lay the groundwork for understanding how atoms and molecules behave. Especially relevant to orbital angular momentum, quantum mechanics allows us to predict the various states and shapes of electron clouds around the nucleus. This is where quantum numbers, including the azimuthal quantum number, become essential tools for describing these states and making predictions about an atom’s behavior.