Ch 41: Quantum Mechanics II: Atomic Structure

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 41: Quantum Mechanics II: Atomic Structure

Problem 7b

Chapter 41, Problem 7b

Consider an electron in the $N$ shell. What is the largest orbital angular momentum it could have? Express your answers in terms of $\hslash$ and in SI units.

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Identify the principal quantum number (n) for the N shell. The N shell corresponds to n = 4.

Recall that the orbital angular momentum quantum number (l) can take integer values from 0 to (n - 1). For n = 4, the possible values of l are 0, 1, 2, and 3. The largest value of l is 3.

The orbital angular momentum (L) is given by the formula:

L = √(l(l + 1))ℏ

, where ℏ is the reduced Planck's constant (

ℏ = 1.0545718 × 10

^-34 \, $\text{J·s}$).

Substitute the largest value of l (l = 3) into the formula:

L = √(3(3 + 1))ℏ = √(12)ℏ

Express the result in terms of ℏ (U) and in SI units (J·s). The largest orbital angular momentum is

L = √(12)ℏ

, where ℏ is the reduced Planck's constant.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Orbital Angular Momentum

Orbital angular momentum is a measure of the rotational motion of an electron around the nucleus in an atom. It is quantized and can be expressed using the formula L = mvr, where L is the angular momentum, m is the mass of the electron, v is its velocity, and r is the radius of its orbit. For electrons in atomic shells, the angular momentum is also related to the principal quantum number n, with the maximum value given by L = nħ, where ħ is the reduced Planck's constant.

Principal Quantum Number (n)

The principal quantum number (n) indicates the energy level of an electron in an atom and determines the size and energy of the orbital. It can take positive integer values (1, 2, 3, ...), with higher values corresponding to higher energy levels and larger orbitals. For an electron in the N shell, n equals 4, which influences the maximum possible angular momentum it can possess.

SI Units and Conversion

SI units are the standard units of measurement used in science, including physics. Angular momentum is typically expressed in joule-seconds (J·s) in SI units. To express the angular momentum of an electron in the N shell in SI units, one must convert the quantized values derived from quantum mechanics into these standard units, ensuring consistency in calculations and comparisons.

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Textbook Question

A photon is emitted when an electron in a three-dimensional cubical box of side length $8.00 \times 10^{- 11}$ m makes a transition from the $n sub X equals 2$ , $n sub Y equals 2$ , $n sub Z equals 1$ state to the $n sub X equals 1$ , $n sub Y equals 1$ , $n sub Z equals 1$ state. What is the wavelength of this photon?

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Textbook Question

Consider an electron in the $N$ shell. What is the largest orbital angular momentum this electron could have in any chosen direction? Express your answers in terms of $ℏ$ and in SI units.

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Textbook Question

Consider an electron in the $N$ shell. What is the largest spin angular momentum this electron could have in any chosen direction? Express your answers in terms of $\hslash$ and in SI units.

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Textbook Question

Consider an electron in the $N$ shell. What is the smallest orbital angular momentum it could have?

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Textbook Question

Consider an electron in the $N$ shell. For the electron in part (c), what is the ratio of its spin angular momentum in the $z$ -direction to its orbital angular momentum in the $z$ -direction? Note: Part (c) asked for the largest orbital angular momentum this electron could have in any chosen direction.

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Textbook Question

Model a hydrogen atom as an electron in a cubical box with side length $L$ . Set the value of $L$ so that the volume of the box equals the volume of a sphere of radius $a = 5.29 \times 10^{- 11}$ m, the Bohr radius. Calculate the energy separation between the ground and first excited levels, and compare the result to this energy separation calculated from the Bohr model.

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Consider an electron in the NNN shell. What is the largest orbital angular momentum it could have? Express your answers in terms of ℏ\(\hslash\)ℏ and in SI units.

Verified video answer for a similar problem:

Key Concepts

Orbital Angular Momentum

Principal Quantum Number (n)

SI Units and Conversion

Consider an electron in the $N$ shell. What is the largest orbital angular momentum it could have? Express your answers in terms of $\hslash$ and in SI units.