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Chapter 7: Problem 64

State which of the following sets of quantum numbers would be possible and which impossible for an electron in an atom. a. \(n=0, l=0, m_{l}=0, m_{s}=+\frac{1}{2}\) b. \(n=1, l=1, m_{l}=0, m_{s}=+\frac{1}{2}\) c. \(n=1, l=0, m_{l}=0, m_{s}=-\frac{1}{2}\) d. \(n=2, l=1, m_{l}=-2, m_{s}=+\frac{1}{2}\) e. \(n=2, l=1, m_{l}=-1, m_{s}=+\frac{1}{2}\)

Short Answer

Expert verified
Possible sets: (c) and (e). Impossible sets: (a), (b), and (d).

Step by step solution

01

Understand Quantum Numbers

An electron in an atom is described by four quantum numbers: principal quantum number \(n\), azimuthal (or angular momentum) quantum number \(l\), magnetic quantum number \(m_l\), and spin quantum number \(m_s\). These quantum numbers have specific rules:- \(n\) must be a positive integer (\(n = 1, 2, 3, \ldots\))- \(l\) can range from 0 to \(n-1\)- \(m_l\) can range from \(-l\) to \(+l\)- \(m_s\) can only be \(+\frac{1}{2}\) or \(-\frac{1}{2}\).
02

Evaluate Option (a)

For option (a) \(n=0, l=0, m_{l}=0, m_{s}=+\frac{1}{2}\): The principal quantum number \(n\) cannot be zero, it must be a positive integer. Therefore, this set is impossible.
03

Evaluate Option (b)

For option (b) \(n=1, l=1, m_{l}=0, m_{s}=+\frac{1}{2}\): Given \(n=1\), the azimuthal quantum number \(l\) must satisfy \(0 \leq l < n\). Therefore \(l\) can only be 0, but here \(l=1\). This set is impossible.
04

Evaluate Option (c)

For option (c) \(n=1, l=0, m_{l}=0, m_{s}=-\frac{1}{2}\): Here, \(l\) is correctly 0, which is the only possible value when \(n=1\). The \(m_{l}\) value is valid, as it satisfies \(-l \leq m_{l} \leq l\), and \(m_{s}\) is valid as it is either \(+\frac{1}{2}\) or \(-\frac{1}{2}\). Therefore, this set is possible.
05

Evaluate Option (d)

For option (d) \(n=2, l=1, m_{l}=-2, m_{s}=+\frac{1}{2}\): Given \(l=1\), the range for \(m_{l}\) is from \(-1\) to \(+1\). But \(m_{l} = -2\) falls outside this range. Therefore, this set is impossible.
06

Evaluate Option (e)

For option (e) \(n=2, l=1, m_{l}=-1, m_{s}=+\frac{1}{2}\): Here, \(l=1\) satisfies the rule \(0 \leq l < n\), and \(m_{l}=-1\) lies in the valid range of \(-1\) to \(+1\). The \(m_{s}\) value \(+\frac{1}{2}\) is also valid. Therefore, this set is possible.

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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, often denoted by the symbol \( n \), provides crucial information about the energy level of an electron within an atom. It must always be a positive integer, like 1, 2, 3, and so on. This quantum number indicates the main shell in which an electron resides and significantly affects the size and energy of that shell.
  • Higher values of \( n \) correspond to higher energy levels and larger atomic orbitals.
  • When \( n \) is increased, the electron is further from the nucleus.
  • As a guideline, the possible values for \( n \) are 1, 2, 3, etc., but never 0 or negative integers.
One important rule to remember is that the value of \( n \) dictates the possible values of the next quantum number, \( l \). Therefore, understanding the principal quantum number helps lay the foundation for identifying the possible electron configurations in an atom.
Azimuthal Quantum Number
The azimuthal quantum number, represented as \( l \), describes the subshell or sublevel within a principal energy level and tells us about the shape of the orbital. This number can take any integer value from 0 to \( n-1 \), where \( n \) is the principal quantum number.
  • For example, if \( n = 2 \), then \( l \) could be 0 or 1.
  • An \( l \) value of 0 corresponds to an "s" orbital, \( l = 1 \) to a "p" orbital, \( l = 2 \) to a "d" orbital, and so forth.
The azimuthal quantum number thus determines not just the shape of the orbital but also influences the orbital's angular momentum. This makes \( l \) crucial when predicting the chemical bonding and properties of elements.
Magnetic Quantum Number
The magnetic quantum number, denoted by \( m_l \), specifies the orientation of an orbital within a particular subshell. This quantum number can assume integer values ranging from \(-l\) to \(+l\), where \( l \) is the azimuthal quantum number.
  • For instance, if \( l = 1 \), then \( m_l \) can be \(-1\), 0, or 1.
  • This indicates that for a "p" orbital, which has \( l = 1 \), there are three possible orientations.
This quantum number is pivotal for determining how individual orbitals are oriented in space compared to external magnetic fields. A proper understanding of \( m_l \) helps determine the magnetic properties of atoms and their interactions under various physical and chemical conditions.
Spin Quantum Number
The spin quantum number, symbolized as \( m_s \), describes the intrinsic spin of an electron, a property not related to its motion around the nucleus. This quantum number can only have two possible values: \(+\frac{1}{2}\) or \(-\frac{1}{2}\). These values indicate the two possible "spin" states of an electron within the same orbital.
  • The differing spin states prevent electrons from having identical sets of quantum numbers, complying with the Pauli Exclusion Principle.
  • The electron spin contributes to the magnetic moment of an atom because of its tiny magnetic field.
Understanding \(m_s\) is crucial for explaining phenomena like electron pairing in orbitals and the overall magnetism of molecules and materials. It plays a key role in defining the electronic structure beyond just the three-dimensional placement of orbitals.

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