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Chapter 7: Problem 55

(a) How many subshells are present in the \(n=4\) shell? (b) How many orbitals are in the \(3 d\) subshell? (c) What is the maximum value of \(\ell\) that is allowed in the shell with \(n=3\) ? (d) What are the values of \(n\) and \(\ell\) for a \(3 p\) subshell? Give all allowed values of the \(m_{\ell}\) quantum number for this subshell.

Short Answer

Expert verified
(a) 4 subshells; (b) 5 orbitals; (c) \(\ell=2\); (d) \(n=3\), \(\ell=1\), \(m_{\ell}=-1,0,+1\).

Step by step solution

01

Understanding Subshells in Shells

The principal quantum number, \(n\), defines the shell level of an electron. For a shell with \(n=4\), the possible values of the azimuthal quantum number \(\ell\) range from 0 to \(n-1\). Each value of \(\ell\) corresponds to a different subshell: \(\ell=0\) for \(s\), \(\ell=1\) for \(p\), \(\ell=2\) for \(d\), and \(\ell=3\) for \(f\). Therefore, the \(n=4\) shell contains four subshells: \(4s, 4p, 4d,\) and \(4f\).
02

Counting Orbitals in a Subshell

The number of orbitals in a subshell is determined by the possible values of the magnetic quantum number \(m_{\ell}\), which ranges from \(-\ell\) to \(+\ell\). For a \(3d\) subshell, \(\ell=2\), and hence \(m_{\ell}\) can be \(-2, -1, 0, +1, +2\). There are five magnetic quantum numbers, so the \(3d\) subshell contains five orbitals.
03

Determining Maximum \(\ell\) for a Given Shell

For a shell with principal quantum number \(n=3\), the maximum value for the azimuthal quantum number \(\ell\) is \(n-1=3-1=2\). This means the allowed values for \(\ell\) are 0, 1, and 2.
04

Identifying Quantum Numbers and Values for a Subshell

For a \(3p\) subshell, \(n=3\) and \(\ell=1\) (since \(p\)-orbitals correspond to \(\ell=1\)). The magnetic quantum number \(m_{\ell}\) takes integer values from \(-\ell\) to \(+\ell\), so the allowed values are \(-1, 0, +1\).

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Key Concepts

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

Understanding Subshells
In the quantum world, subshells play a crucial role in the arrangement of electrons within an atom. Each shell, defined by the principal quantum number \( n \), contains subshells, which are determined by the azimuthal quantum number \( \ell \). This number dictates the shape and energy of the electron orbital.

Let's break it down:
  • The principal quantum number \( n \) is a positive integer that indicates the main energy level of an electron in an atom. For example, when \( n = 4 \), the shell level is 4.
  • The azimuthal quantum number \( \ell \) can vary from 0 to \( n-1 \). Example values include:\( \ell = 0 \) for "s," \( \ell = 1 \) for "p," \( \ell = 2 \) for "d," and \( \ell = 3 \) for "f."
  • Each value of \( \ell \) corresponds to a different subshell, defining the type of orbital electrons reside in, such as \( 4s, 4p, 4d, \text{and} \, 4f \) when \( n = 4 \).
This hierarchy reflects how electrons fill the most stable, low-energy configuration first, explaining the presence of multiple subshells within a shell.
All About Orbitals
Orbitals are specific regions around an atom's nucleus where electrons are most likely to be found. These regions are defined by the sets of quantum numbers. Orbitals are key concepts in predicting and understanding the behavior of electrons in atoms.

Here are the essentials:
  • Each subshell is made up of one or more orbitals. For instance, a "p" subshell (\( \ell = 1 \)) has three orbitals because the magnetic quantum number \( m_\ell \) can take the values \(-1, 0, \) and \(+1\).
  • The number of orbitals in a subshell is determined by the range \( -\ell \) to \( +\ell \) of the \( m_\ell \) quantum number. So, a "d" subshell (\( \ell = 2 \)) includes five orbitals: \(-2, -1, 0, +1, +2\).
  • These orbitals represent different orientations and probabilities of where to find an electron, with each one having a unique shape and spatial orientation.
Through understanding orbitals, we gain insights into how electrons fill an atom and the energies associated with different atomic configurations.
Principles of Quantum Mechanics
Quantum mechanics is the fundamental theory governing the behavior of particles at microscopic scales, like electrons in atoms. It offers a framework to understand phenomena that classical physics cannot explain.

A few key principles include:
  • The quantization of energy levels: Electrons can only exist in specific energy states known as levels or shells, as determined by quantum numbers \( n, \ell, \text{and} \, m_\ell \).
  • Uncertainty principle: There's an intrinsic limitation in knowing both the position and momentum of an electron simultaneously, affecting how we describe electron positions.
  • Wave-particle duality: Electrons exhibit both particle-like and wave-like behavior, which is why they occupy orbitals as probability clouds rather than fixed paths.
These principles combine to form the quantum model of the atom, explaining much of the chemical behavior and bonding of elements.
Building Electron Configurations
Electron configuration describes the arrangement of electrons in an atom's energy levels, subshells, and orbitals. It is crucial for understanding an element's chemical properties and predicting how it will react.

Here's how it works:
  • The Pauli Exclusion Principle: No two electrons in an atom can have the same set of quantum numbers, meaning each electron has a unique "address" in terms of its shell, subshell, and orbital.
  • Aufbau Principle: Electrons fill orbitals starting from the lowest energy level available and move to higher ones as lower levels fill up, following a specific order (such as \( 1s, 2s, 2p, 3s, \) etc.).
  • Hund's Rule: In subshells with multiple orbitals (like "p" or "d"), electrons fill each orbital singly before pairing up, minimizing repulsions and energy.
By using these rules, we can write electron configurations that help visualize and predict an atom's interactions, bonding tendencies, and physical properties.

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Most popular questions from this chapter

In each part, sketch the contour surface for the orbital described. (a) \(n=2, \ell=0\) (b) \(3 p_{x}\) (c) \(4 d_{x y}\) (d) \(n=2, \ell=1\) (e) \(n=1, \ell=0\)

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