Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 7: Problem 55

For each of the following, give the sublevel designation, the allowable \(m_{l}\) values, and the number of orbitals: (a) \(n=4, l=2\) (b) \(n=5, l=1\) (c) \(n=6, l=3\)

Short Answer

Expert verified
(a) 4d, \(m_{l}\) = -2, -1, 0, +1, +2, 5 orbitals. (b) 5p, \(m_{l}\) = -1, 0, +1, 3 orbitals. (c) 6f, \(m_{l}\) = -3, -2, -1, 0, +1, +2, +3, 7 orbitals.

Step by step solution

01

Identify Sublevel Designation for Part (a)

For part (a), given parameters are: principal quantum number, \(n=4\) and the azimuthal quantum number, \(l=2\). Utilize the relationship between \(l\) and respective sublevel designation where \(l=0\) denotes \(s\), \(l=1\) denotes \(p\), \(l=2\) denotes \(d\), and \(l=3\) denotes \(f\). Henceforth, for \(l=2\), the sublevel designation is \(d\). Therefore, the sublevel designation is 4d.
02

Identify Allowable \(m_{l}\) Values for Part (a)

For \(l=2\), the values of \(m_{l}\) can range from \(-l\) to \(+l\), i.e., \(m_{l} = -2, -1, 0, +1, +2\).
03

Identify Number of Orbitals for Part (a)

The number of orbitals in a sublevel is determined by the number of possible \(m_{l}\) values. Here, there are 5 possible values of \(m_{l}\): \(-2, -1, 0, +1, +2\). Thus, there are 5 orbitals.
04

Identify Sublevel Designation for Part (b)

For part (b), given parameters are: \(n=5\) and \(l=1\). The sublevel designation for \(l=1\) is \(p\). Therefore, the sublevel designation is 5p.
05

Identify Allowable \(m_{l}\) Values for Part (b)

For \(l=1\), the values of \(m_{l}\) can range from \(-l\) to \(+l\), i.e., \(m_{l} = -1, 0, +1\).
06

Identify Number of Orbitals for Part (b)

The number of orbitals in a sublevel is given by the number of possible \(m_{l}\) values. Here, there are 3 possible values of \(m_{l}\): \(-1, 0, +1\). Consequently, there are 3 orbitals.
07

Identify Sublevel Designation for Part (c)

For part (c), given parameters are: \(n=6\) and \(l=3\). The sublevel designation for \(l=3\) is \(f\). Therefore, the sublevel designation is 6f.
08

Identify Allowable \(m_{l}\) Values for Part (c)

For \(l=3\), the values of \(m_{l}\) can range from \(-l\) to \(+l\), i.e., \(m_{l} = -3, -2, -1, 0, +1, +2, +3\).
09

Identify Number of Orbitals for Part (c)

The number of orbitals in a sublevel is determined by the number of possible \(m_{l}\) values. Here, there are 7 possible values of \(m_{l}\): \(-3, -2, -1, 0, +1, +2, +3\). Thus, there are 7 orbitals.

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

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

principal quantum number
The principal quantum number, denoted as \(n\), plays a crucial role in defining the energy level of an electron in an atom. It essentially tells us the electron's shell or orbit around the nucleus.
The values of \(n\) are positive integers (1, 2, 3, ...) and as \(n\) increases, the electron's energy and its distance from the nucleus also increase. For example, when \(n=1\), the electron is in the first shell or energy level, closest to the nucleus.
  • \(n = 1\): closest shell to the nucleus, lowest energy
  • \(n = 2\): second shell, more energy than the first
  • \(n = 3\): third shell, even more energy, farther from nucleus
In the provided exercise, we encounter principal quantum numbers of 4, 5, and 6, corresponding to higher energy levels further from the nucleus.
azimuthal quantum number
The azimuthal quantum number, denoted as \(l\), defines the shape of the electron's orbital within a given principal quantum number (\(n\)). It is also known as the angular momentum quantum number.
The values of \(l\) range from 0 to \(n-1\). Each value of \(l\) corresponds to a specific type of orbital, which are:
  • \(l = 0\): \(s\)-orbital (spherical shape)
  • \(l = 1\): \(p\)-orbital (dumbbell shape)
  • \(l = 2\): \(d\)-orbital (cloverleaf shape)
  • \(l = 3\): \(f\)-orbital (complex shape)
For instance, in the exercise steps, if \(n = 4\) and \(l = 2\), then the sublevel is 4d.
magnetic quantum number
The magnetic quantum number, represented as \(m_l\), determines the orientation of an orbital in space within a given sublevel. The values of \(m_l\) depend on the azimuthal quantum number, \(l\).
The possible values for \(m_l\) range from \(-l\) to \(+l\), including zero. This means, for any value of \(l\), there will be \(2l + 1\) possible values of \(m_l\).
In simpler terms:
  • If \(l = 0\): \(m_l = 0\)
  • If \(l = 1\): \(m_l = -1, 0, +1\)
  • If \(l = 2\): \(m_l = -2, -1, 0, +1, +2\)
For example, with \(l = 2\) in part (a) of the exercise, \(m_l\) can take the values -2, -1, 0, +1, +2.
orbitals
Orbitals are regions around the nucleus where the probability of finding an electron is highest. Each orbital can hold a maximum of two electrons. The type and number of orbitals depend on the quantum numbers.
The number of orbitals in a particular sublevel is directly given by the magnetic quantum number values \(m_l\). For each valid \(m_l\) value, there is one orbital.
To illustrate:
  • For \(l = 0\): one s orbital
  • For \(l = 1\): three p orbitals (since \(m_l = -1, 0, +1\))
  • For \(l = 2\): five d orbitals (\(m_l = -2, -1, 0, +1, +2\))
  • For \(l = 3\): seven f orbitals (\(m_l = -3, -2, -1, 0, +1, +2, +3\))
In the steps given in the exercise, we see that \(n=4, l=2\) leads to 5 d orbitals.

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

Only certain transitions are allowed from one energy level to another. In one- electron species, the change in \(I\) for an allowed transition is \(\pm 1 .\) For example, a \(3 p\) electron can move to a \(2 s\) orbital but not to a \(2 p\). Thus, in the UV series, where \(n_{\text {final }}=1\), allowed transitions can start in a \(p\) orbital \((l=1)\) of \(n=2\) or higher, not in an \(s(l=0)\) or \(d(l=2)\) orbital of \(n=2\) or higher. From what orbital do each of the allowed transitions start for the first four emission lines in the visible series \(\left(n_{\text {final }}=2\right) ?\)

A sodium flame has a characteristic yellow color due to emission of light of wavelength \(589 \mathrm{nm}\). What is the mass equivalence of one photon with this wavelength \(\left(1 \mathrm{~J}=1 \mathrm{~kg} \cdot \mathrm{m}^{2} / \mathrm{s}^{2}\right) ?\)

For each of the following, give the sublevel designation, the allowable \(m_{i}\) values, and the number of orbitals: (a) \(n=2, l=0\) (b) \(n=3, l=2\) (c) \(n=5, l=1\)

In his explanation of the threshold frequency in the photoelectric effect, Einstein reasoned that the absorbed photon must have a minimum energy to dislodge an electron from the metal surface. This energy is called the work function (\phi) of the metal. What is the longest wavelength of radiation (in \(\mathrm{nm}\) ) that could cause the photoelectric effect in each of these metals: (a) calcium, \(\phi=4.60 \times 10^{-19}\) \(\mathrm{J} ;\) (b) titanium, \(\phi=6.94 \times 10^{-19} \mathrm{~J} ;\) (c) sodium, \(\phi=4.41 \times 10^{-19} \mathrm{~J} ?\)

The photoelectric effect is illustrated in a plot of the kinetic energies of electrons ejected from the surface of potassium metal or silver metal at different frequencies of incident light. (a) Why don't the lines begin at the origin? (b) Why don't the lines begin at the same point? (c) From which metal will light of shorter wavelength eject an electron? (d) Why are the slopes equal?
See all solutions

Recommended explanations on Chemistry Textbooks

Kinetics

Read Explanation

Organic Chemistry

Read Explanation

Physical Chemistry

Read Explanation

Nuclear Chemistry

Read Explanation

Ionic and Molecular Compounds

Read Explanation

The Earths Atmosphere

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.

Sign-up for free