/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 55 (a) How many \(\ell\) values are... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 40: Problem 55

(a) How many \(\ell\) values are associated with \(n=3 ?\) (b) How many \(m_{\ell}\) values are associated with \(\ell=1 ?\)

Short Answer

Expert verified
(a) 3 \(\ell\) values for \(n=3\); (b) 3 \(m_{\ell}\) values for \(\ell=1\).

Step by step solution

01

Understand the Principal Quantum Number n

The principal quantum number, denoted as \(n\), determines the energy level and overall size of an atom's electron cloud. For a given \(n\), the possible values for the angular momentum quantum number \(\ell\) range from 0 to \(n-1\).
02

Determine the Possible \(\ell\) Values for n=3

Since \(n = 3\), the possible \(\ell\) values are integers from 0 to \(n-1\). Therefore, \(\ell\) can be \(0, 1, \) or \(2\). This means there are three possible \(\ell\) values associated with \(n = 3\).
03

Understand the Angular Momentum Quantum Number \(\ell\)

The angular momentum quantum number \(\ell\) determines the shape of the electron's orbital. For a given \(\ell\), the magnetic quantum number \(m_{\ell}\) can have values ranging from \(-\ell\) to \(+\ell\).
04

Determine the Possible \(m_{\ell}\) Values for \(\ell=1\)

When \(\ell = 1\), the \(m_{\ell}\) values can range from \(-1\) to \(+1\), including 0. Thus, the possible \(m_{\ell}\) values are \(-1, 0, \) and \(1\). Therefore, there are three possible \(m_{\ell}\) values associated with \(\ell = 1\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Principal Quantum Number
The principal quantum number, symbolized by the letter \( n \), is a fundamental concept in quantum mechanics. It primarily describes the electron's energy level within an atom. This number tells us how far an electron shell is from the nucleus of the atom.

  • \( n \) can be any positive integer starting from 1 (e.g., \( n = 1, 2, 3, \ldots \)).
  • It signifies the major electron energy levels or "shells." The higher the value of \( n \), the further the electron is from the nucleus and, typically, the higher its energy.
  • The maximum number of electrons that can be accommodated in an energy level is determined by the formula \( 2n^2 \).
In the case of \( n = 3 \), which is part of the exercise, this indicates the third energy level. Electrons in this shell can occupy different orbitals having different shapes, which are determined by another quantum number known as the angular momentum quantum number.
Angular Momentum Quantum Number
Also known as the azimuthal or orbital quantum number, the angular momentum quantum number is represented by \( \ell \). It specifies the shape of the orbital an electron occupies within an atom.

  • For each value of \( n \), \( \ell \) can take integer values from 0 to \( n-1 \).
  • The different values of \( \ell \) correspond to different kinds of orbitals. These orbitals are commonly known as \( s \), \( p \), \( d \), \( f \) (represented by \( \ell = 0, 1, 2, 3 \), respectively), and so forth.
  • Thus, for \( n = 3 \), \( \ell \) can be 0, 1, or 2, indicating the presence of \( s \), \( p \), and \( d \) orbitals.
The exercise considers \( \ell = 1 \), which is associated with \( p \) orbitals. The significance of \( \ell \) is not only in determining the shape but also in influencing the magnetic quantum number, \( m_{\ell} \).
Magnetic Quantum Number
The magnetic quantum number, denoted by \( m_{\ell} \), determines the orientation of an electron's orbital within a magnetic field. It is directly dependent on the angular momentum quantum number.

  • For a given \( \ell \), \( m_{\ell} \) can have integer values ranging from \(-\ell\) to \(+\ell\).
  • This indicates that there are \( 2\ell + 1 \) possible orientations or orbitals within a given subshell.
  • Each value of \( m_{\ell} \) represents a different orientation of an orbital in space.
In the exercise, with \( \ell = 1 \), the possible values for \( m_{\ell} \) are \(-1, 0, \) and \(+1\). Therefore, three possible orientations of the \( p \) orbitals exist for a subshell where \( \ell = 1 \). This variety of orientation impacts how electrons are distributed within the atom and plays a crucial role in the atom's overall magnetic properties.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Suppose two electrons in an atom have quantum numbers \(n=2\) and \(\ell=1\). (a) How many states are possible for those two electrons? (Keep in mind that the electrons are indistinguishable.) (b) If the Pauli exclusion principle did not apply to the electrons, how many states would be possible?

The binding energies of \(K\)-shell and \(L\)-shell electrons in copper are \(8.979\) and \(0.951 \mathrm{keV}\), respectively. If a \(K_{\alpha}\) x ray from copper is incident on a sodium chloride crystal and gives a first-order Bragg reflection at an angle of \(74.1^{\circ}\) measured relative to parallel planes of sodium atoms, what is the spacing between these parallel planes?

The active medium in a particular laser that generates laser light at a wavelength of \(694 \mathrm{~nm}\) is \(6.00 \mathrm{~cm}\) long and \(1.00 \mathrm{~cm}\) in diameter. (a) Treat the medium as an optical resonance cavity analogous to a closed organ pipe. How many standing-wave nodes are there along the laser axis? (b) By what amount \(\Delta f\) would the beam frequency have to shift to increase this number by one? (c) Show that \(\Delta f\) is just the inverse of the travel time of laser light for one round trip back and forth along the laser axis. (d) What is the corresponding fractional frequency shift \(\Delta f / f ?\) The appropriate index of refraction of the lasing medium (a ruby crystal) is \(1.75\).

An electron is in a state with \(\ell=3 .\) (a) What multiple of \(\hbar\) gives the magnitude of \(\vec{L}\) ? (b) What multiple of \(\mu_{\mathrm{B}}\) gives the magnitude of \(\vec{\mu} ?\) (c) What is the largest possible value of \(m_{\ell}\), (d) what multiple of \(\hbar\) gives the corresponding value of \(L_{z}\), and (e) what multiple of \(\mu_{\mathrm{B}}\) gives the corresponding value of \(\mu_{\mathrm{orb}, z}\) ? (f) What is the value of the semiclassical angle \(\theta\) between the directions of \(L_{z}\) and \(\vec{L}\) ? What is the value of angle \(\theta\) for \((\mathrm{g})\) the second largest possible value of \(m_{\ell}\) and \((\mathrm{h})\) the smallest (that is, most negative) possible value of \(m_{\ell}\) ?

A laser emits at \(424 \mathrm{~nm}\) in a single pulse that lasts \(0.500 \mu \mathrm{s}\). The power of the pulse is \(3.25 \mathrm{MW}\). If we assume that the atoms contributing to the pulse underwent stimulated emission only once during the \(0.500 \mu\) s, how many atoms contributed?
See all solutions

Recommended explanations on Physics Textbooks

Mechanics and Materials

Read Explanation

Modern Physics

Read Explanation

Wave Optics

Read Explanation

Kinematics Physics

Read Explanation

Oscillations

Read Explanation

Fluids

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.