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Chapter 7: Problem 87

7.87 What is the number of different orbitals in each of the following subshells? a) \(3 d\) b) \(4 f\) c) \(4 p\) d) \(5s\)

Short Answer

Expert verified
3d has 5 orbitals, 4f has 7, 4p has 3, and 5s has 1 orbital.

Step by step solution

01

Understand the Subshell Notation

The given subshells, represented with notations like \(3d\), \(4f\), \(4p\), and \(5s\), have two parts: the number indicates the principal quantum number \(n\), and the letter corresponds to the azimuthal quantum number \(l\). The letters \(s\), \(p\), \(d\), and \(f\) correspond to \(l = 0, 1, 2, \text{and} \,3\) respectively.
02

Recall Orbital Calculation Formula

The number of orbitals within a subshell can be given by \(2l + 1\), where \(l\) is the azimuthal quantum number. Each value of \(l\) corresponds to a particular subshell type: \(s\), \(p\), \(d\), and \(f\).
03

Applying to 3d Subshell

For the \(3d\) subshell, \(l = 2\) (because \(d\) corresponds to \(l = 2\)). The number of orbitals is calculated as \(2(2) + 1 = 5\).
04

Applying to 4f Subshell

For the \(4f\) subshell, \(l = 3\) (since \(f\) corresponds to \(l = 3\)). Thus, the number of orbitals is \(2(3) + 1 = 7\).
05

Applying to 4p Subshell

For the \(4p\) subshell, \(l = 1\) (because \(p\) corresponds to \(l = 1\)). The number of orbitals is \(2(1) + 1 = 3\).
06

Applying to 5s Subshell

For the \(5s\) subshell, \(l = 0\) (since \(s\) corresponds to \(l = 0\)). This gives us \(2(0) + 1 = 1\).

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Key Concepts

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

Principal Quantum Number
The principal quantum number, denoted by the symbol \( n \), plays a crucial role in the quantum model of the atom. It essentially represents the main energy level occupied by an electron.
  • This quantum number takes on positive integer values (\( n = 1, 2, 3, \) and so on).
  • Higher values of \( n \) indicate that the electron is farther from the nucleus and, therefore, possesses higher energy.
Each principal quantum number corresponds to a major shell. For example, for the subshell notation "3d," the number "3" signifies that all orbitals in this subshell are part of the third principal shell.
Determining \( n \) is fundamental because it helps define the size and energy of an atom's electron cloud.
Subshell Orbitals
Subshell orbitals are the divisions within a principal energy level that provide more detailed information about electron distribution.
An electron in any given principal shell can reside in one of several subshells, determined by the azimuthal quantum number \( l \).- Subshells are labeled \( s, p, d, \) and \( f \), with numerical values for \( l \) corresponding to 0, 1, 2, and 3 respectively.- Since each subshell (determined by \( l \)) has a unique shape or form, they impact the electron's angular distribution within an atom.
To find the number of orbitals contained in a subshell, you can use the formula \( 2l + 1 \). For example, given a "3d" subshell:- The "d" indicates \( l = 2 \), leading to \( 2(2) + 1 = 5 \) orbitals within that subshell.
This calculation helps pinpoint not just where electrons might be, but also the orientation possibilities they may have within an atom.
Azimuthal Quantum Number
The azimuthal quantum number, represented by \( l \), details the shape and angular momentum associated with an electron's orbit within a subshell.
Primarily, it defines the shape of the orbital and helps categorize subshells.
  • \( l = 0 \) corresponds to an \( s \) subshell with a spherical shape.
  • \( l = 1 \) indicates a \( p \) subshell, which has a dumbbell shape.
  • \( l = 2 \) represents a \( d \) subshell, featuring more complex shapes.
  • \( l = 3 \) denotes an \( f \) subshell with multi-lobed shapes.
The azimuthal quantum number also governs the number of orbitals in each subshell. As seen with the number of orbitals in a given subshell (like in a "4f" subshell where \( l = 3 \)), the formula \( 2l + 1 \) confirms there are seven orbitals.
Understanding \( l \) is key for predicting electron arrangements, helping chemists and physicists explain and anticipate chemical and physical properties of elements.

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Most popular questions from this chapter

Light of wavelength \(1.03 \times 10^{-7} \mathrm{~m}\) is emitted when an electron in an excited level of a hydrogen atom undergoes a transition to the \(n=1\) level. What is the region of the spectrum of this light? What is the principal quantum number of the excited level?

The term degeneracy means the number of different quantum states of an atom or molecule having the same energy. For example, the degeneracy of the \(n=2\) level of the hydrogen atom is 4 (a \(2 s\) quantum state, and three different \(2 p\) states). What is the degeneracy of the \(n=5\) level?

A hydrogen-like ion has a nucleus of charge \(+Z e\) and a single electron outside this nucleus. The energy levels of these ions are \(-Z^{2} R_{\mathrm{H}} / n^{2}\) (where \(Z=\) atomic number). Calculate the wavelength of the transition from \(n=3\) to \(n=2\) for \(\mathrm{He}^{+}\), a hydrogen-like ion. In what region of the spectrum does this emission occur?

Give a brief wave description of light. What are two characteristics of light waves?

Ozone in the stratosphere absorbs ultraviolet light of wavelengths shorter than \(320 \mathrm{nm}\), thus filtering out the most energetic radiation from sunlight. During this absorption, an ozone molecule absorbs a photon, which breaks an oxygen-oxygen bond, yielding an oxygen molecule and an oxygen atom: $$ \mathrm{O}_{3}(g)+h \nu \longrightarrow \mathrm{O}_{2}(g)+\mathrm{O}(g) $$ (Here, \(h \nu\) denotes a photon.) Suppose a flask of ozone is irradiated with a pulse of UV light of wavelength \(275 \mathrm{nm}\). Assuming that each photon of this pulse that is absorbed breaks up one ozone molecule, calculate the energy absorbed per mole of \(\mathrm{O}_{2}\) produced, giving the answer in \(\mathrm{kJ} / \mathrm{mol}\)
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