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Chapter 3: Problem 10

The orbital angular momentum of an electron in \(2 \mathrm{~s}\) orbital is a. \(+\frac{1}{2} \frac{\mathrm{h}}{2 \pi}\) b. zero c. \(\frac{\mathrm{h}}{2 \pi}\) d. \(\sqrt{2} \frac{\mathrm{h}}{2 \pi}\)

Short Answer

Expert verified
The orbital angular momentum of an electron in a 2s orbital is zero (option b).

Step by step solution

01

Understanding Orbital Angular Momentum

Orbital angular momentum for an electron in a hydrogen-like atom is determined by the azimuthal quantum number, denoted as \( l \). The formula for orbital angular momentum is \( \sqrt{l(l+1)} \frac{h}{2\pi} \).
02

Identifying Quantum Numbers for 2s Orbital

In a 2s orbital, the principal quantum number \( n = 2 \). The azimuthal quantum number \( l \) for an s orbital is 0 because s orbitals have \( l = 0 \).
03

Calculating Orbital Angular Momentum for 2s

Since \( l = 0 \) for a 2s orbital, substitute \( l \) into the formula: \( \sqrt{0(0+1)} \frac{h}{2\pi} = \sqrt{0} \frac{h}{2\pi} = 0 \).
04

Selecting the Correct Answer

Given that the orbital angular momentum for a 2s electron is 0, the correct option from the list is (b) zero.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quantum Numbers
In quantum mechanics, quantum numbers describe specific properties of electrons and their corresponding atomic orbitals within an atom. The main quantum numbers include:
  • Principal Quantum Number (n): It determines the energy level of an electron shell and its size. Higher values of 饾憶 mean the electron is further from the nucleus and has higher energy.
  • Azimuthal Quantum Number (l): This number denotes the shape of the orbital where the electron is likely to be found. Each value is associated with a particular type of orbital (s, p, d, f).
  • Magnetic Quantum Number (m_l): It identifies the orientation of an orbital in space.
  • Spin Quantum Number (m_s): Indicates the direction of the electron's spin, either +1/2 or -1/2.
Each electron in an atom has a unique set of these quantum numbers, much like a unique address. They help in determining where electrons are located and how they behave in an atom.
Azimuthal Quantum Number
The azimuthal quantum number, often represented as 饾憴, is central to understanding the shape of an electron's orbital. It defines the subshell of an electron and hence determines the angular momentum of the orbital. At each principal energy level, the azimuthal quantum number ranges from 0 to (n - 1), where 饾憶 is the principal quantum number. Here鈥檚 a quick guide:
  • l = 0: Represents an s orbital, characterized by a spherical shape.
  • l = 1: Corresponds to a p orbital, which has a dumbbell shape.
  • l = 2: Indicates a d orbital, and they have more complex shapes.
  • l = 3: Pertains to an f orbital, which has even more intricate shapes.
For a 2s orbital, as in our problem, 饾憴 is 0. Since l is 0 for s orbitals, they do not have any angular nodes, resulting in zero angular momentum. Understanding the azimuthal quantum number is crucial for grasping orbital shapes and energy distributions in atoms.
Hydrogen-like Atom
Hydrogen-like atoms refer to any ions with only one electron. Examples include He$^+$, Li$^{2+}$, and, of course, hydrogen itself. These atoms or ions are simpler systems, making them pivotal in studying atomic structure and behaviors such as spectral lines and energy levels. Hydrogen-like atoms have many applications:
  • They allow simplification in calculations since only one electron interacts with the nucleus.
  • Using them, scientists can derive energy levels analytically, providing insights into more complex systems.
  • They serve as a model to understand spectra, as their one-electron systems emit light at characteristic frequencies.
The orbital angular momentum in these systems is particularly dependent on the azimuthal quantum number, which determines electron behavior in such atoms. Hence, a detailed understanding of hydrogen-like atoms gives crucial insights into broader quantum mechanics.

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Most popular questions from this chapter

The radius of which of the following orbits is same as that of the first Bohr's orbit of hydrogen atom? a. \(\mathrm{He}^{-}(\mathrm{n}=2)\) b. \(\mathrm{Li}^{2+}(\mathrm{n}=2)\) c. \(\mathrm{Li}^{2+}(\mathrm{n}=3)\) d. \(\mathrm{Be}^{3+}(\mathrm{n}=2)\)

Which orbital has only positive value of wave function at all distances from the nucleus? a. \(3 \mathrm{~d}\) b. \(2 p\) c. \(2 \mathrm{~s}\) d. \(1 \mathrm{~s}\)

Which statement is/are true for many electron atoms? a. The \(2 \mathrm{Px}\) and \(2 \mathrm{Py}\) orbitals have the same energy in the absence of an applied magnetic field b. The 2 s and \(2 p\) orbitals are of differing energies, whereas in a hydrogen atom they are same. c. Outer electrons penetrate the electron clouds of inner electrons d. Outer electrons experience the full nuclear charge

A photon of wavelength \(300 \mathrm{~nm}\) is absorbed by a gas and then re- emitted as two photon. One photon is red with wavelength of \(760 \mathrm{~nm}\). The wave number of the second photon will be a. \(2.20 \times 10^{7} \mathrm{~m}^{-1}\) b. \(2.02 \times 10^{6} \mathrm{~m}^{-1}\) c. \(4.04 \times 10^{\top} \mathrm{m}^{-1}\) d. \(1.01 \times 10^{6} \mathrm{~m}^{-1}\)

Select the correct statement(s): a. Gravitation has a rest mass zero but spin 2 and it is exchanged during gravitational interaction between bodies b. Rest mass of photon is zero and increases with its velocity c. Photons are carrier of energy, momentum and angular momentum between interacting particles d. Mass of a p-meson is about 276 times of an electron and it keeps nucleons together
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