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Chapter 7: Problem 69

Explain why each of the following sets of quantum numbers would not be permissible for an electron, according to the rules for quantum numbers. a. \(n=1, l=0, m_{l}=0, m_{s}=+1\) b. \(n=1, l=3, m_{l}=+3, m_{s}=+\frac{1}{2}\) c. \(n=3, l=2, m_{l}=+3, m_{s}=-\frac{1}{2}\) d. \(n=0, l=1, m_{l}=0, m_{s}=+\frac{1}{2}\) e. \(n=2, l=1, m_{l}=-1, m_{s}=+\frac{3}{2}\)

Short Answer

Expert verified
Each set has an impermissible quantum number according to quantum rules.

Step by step solution

01

Evaluate Set a

For the set a, quantum numbers are given as \( n=1, l=0, m_{l}=0, m_{s}=+1 \). The issue lies with the spin quantum number \( m_{s} \). It must be either \(+\frac{1}{2}\) or \(-\frac{1}{2}\). The given \( m_{s}=+1 \) is not one of the permissible values.
02

Evaluate Set b

For the set b, quantum numbers are \( n=1, l=3, m_{l}=+3, m_{s}=+\frac{1}{2} \). Here, the principal quantum number \( n=1 \) indicates the first energy level where \( l \), the azimuthal quantum number, must be at least 0 and can be at most \( n-1 \). Hence, \( l=3 \) is not permissible because \( l \) can only be 0 for \( n=1 \).
03

Evaluate Set c

For the set c, quantum numbers are \( n=3, l=2, m_{l}=+3, m_{s}=-\frac{1}{2} \). Here, the magnetic quantum number \( m_{l} \) should be in the range of \(-l\) to \(+l\). Thus, \( m_{l}=+3 \) is not possible because the maximum \( m_{l} \) value for \( l=2 \) is \(+2\).
04

Evaluate Set d

For the set d, the quantum numbers are \( n=0, l=1, m_{l}=0, m_{s}=+\frac{1}{2} \). The principal quantum number \( n \) must be a positive integer, starting from 1. Therefore, \( n=0 \) is not permissible.
05

Evaluate Set e

For the set e, quantum numbers are \( n=2, l=1, m_{l}=-1, m_{s}=+\frac{3}{2} \). The spin quantum number \( m_{s} \) must be \(+\frac{1}{2}\) or \(-\frac{1}{2}\), but \(+\frac{3}{2}\) is not a valid value, making this set impermissible.

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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 \( n \), is a fundamental part of quantum mechanics. It describes the energy level or shell in which an electron resides within an atom. The value of \( n \) is always a positive integer: 1, 2, 3, and so forth.

Each increment in \( n \) signifies higher energy and a larger distance of the electron from the nucleus. For instance:
  • \( n=1 \) represents the first energy level (or shell), closest to the nucleus.
  • \( n=2 \) indicates the second energy level.
  • Higher values like \( n=3 \), \( n=4 \) denote shells further away.
In the original exercise, any set involving \( n=0 \) is impermissible because the principal quantum number must be a positive integer, starting from 1.
Azimuthal Quantum Number
The azimuthal quantum number, symbolized by \( l \), determines the subshell or the shape of the electron orbital. It is dependent on the principal quantum number \( n \).

The value of \( l \) ranges from 0 to \( n-1 \). For example, if \( n=3 \), possible values of \( l \) are 0, 1, or 2.
  • \( l=0 \) is associated with the s subshell.
  • \( l=1 \) relates to the p subshell.
  • \( l=2 \) correlates with the d subshell, and so on.
In the original question, any \( l \) value outside the permissible range for a given \( n \) is not valid. Thus, for \( n=1 \), \( l \) must be 0, making \( l=3 \) incorrect.
Magnetic Quantum Number
The magnetic quantum number, designated as \( m_l \), provides information about the orientation of the electron's orbital in space. Its values are dependent on the azimuthal quantum number \( l \).

The range of \( m_l \) spans from \(-l\) to \(+l\), including all integers in between. So for a given \( l=2 \), \( m_l \) can be -2, -1, 0, +1, or +2.
  • This means for \( l=1 \), \( m_l \) could be -1, 0, or +1.
  • For \( l=0 \), \( m_l \) must be 0.
In the original exercise, \( m_l=+3 \) was incorrect for \( l=2 \), since the highest possible value is \(+2\).
Spin Quantum Number
The spin quantum number, denoted \( m_s \), indicates the intrinsic spin and magnetic orientation of an electron. It is a fundamental part of the quantum description of an electron.

The possible values for \( m_s \) are always \( +\frac{1}{2} \) or \( -\frac{1}{2} \). This concept reflects the two possible spin states of an electron; often referred to as 'spin up' \( (+\frac{1}{2}) \) or 'spin down' \( (-\frac{1}{2}) \).
  • Each orbital can hold a maximum of two electrons, each with opposite spins.
In the given exercises, incorrect \( m_s \) values included \( +1 \) and \( +\frac{3}{2} \), both of which exceed the permissible range.
Quantum Mechanics Rules
Quantum mechanics sets specific rules for quantum numbers that describe the properties of electrons in atoms. These rules ensure that electrons exist in states defined by allowed sets of quantum numbers. Key rules include:

  • The principal quantum number \( n \) must be a positive integer, starting from 1.
  • The azimuthal quantum number \( l \) varies from 0 to \( n-1 \).
  • The magnetic quantum number \( m_l \) spans from \(-l\) to \(+l\).
  • The spin quantum number \( m_s \) can only be \( +\frac{1}{2} \) or \( -\frac{1}{2} \).
These restrictions define permissible states of electrons. If any condition fails, as shown in the exercise examples, the set of quantum numbers is invalid for electrons.

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