Question

Indicate which of the following sets of quantum numbers in an atom are unacceptable and explain why: (a) \(\left(1,1,+\frac{1}{2},-\frac{1}{2}\right)\) (b) \(\left(3,0,-1,+\frac{1}{2}\right)\) (c) \(\left(2,0,+1,+\frac{1}{2}\right)\) (d) \(\left(4,3,-2,+\frac{1}{2}\right)\) (e) (3,2,+1,1)

Short answer

Unacceptable sets: (a), (b), (c), (e). Acceptable set: (d).

Step-by-step solution

Understanding Quantum Numbers

Each electron in an atom can be uniquely identified by a set of four quantum numbers: the principal quantum number \(n\), the azimuthal or angular momentum quantum number \(l\), the magnetic quantum number \(m_l\), and the spin quantum number \(m_s\). These numbers define the electron's energy level, shape, orientation, and spin of its orbital, respectively.

Review of Quantum Number Rules

Quantum number rules dictate that: \(n\geq 1\), \(l\) is an integer such that \(0 \leq l < n\), \(m_l\) is an integer ranging between \(-l\) and \(+l\), and \(m_s\) can be either \(+\frac{1}{2}\) or \(-\frac{1}{2}\). We will examine each set of quantum numbers to ensure they satisfy these constraints.

Evaluation of Set (a) (1,1,+1/2,-1/2)

In set (a), \(n = 1\), \(l = 1\). However, \(l\) should satisfy \(0 \leq l < n\). Since \(l \geq n\) here, set (a) is unacceptable.

Evaluation of Set (b) (3,0,-1,+1/2)

In set (b), \(n = 3\), \(l = 0\), \(m_l = -1\). The rule \(-l \leq m_l \leq +l\) is violated since \(m_l = -1 < -0\). Therefore, set (b) is unacceptable.

Evaluation of Set (c) (2,0,+1,+1/2)

In set (c), \(n = 2\), \(l = 0\), \(m_l = +1\). The \(m_l\) value violates \([-l, +l]\), as it should be between 0 and 0. Thus, set (c) is unacceptable.

Evaluation of Set (d) (4,3,-2,+1/2)

In set (d), \(n = 4\), \(l = 3\), \(m_l = -2\), and \(m_s = +1/2\). Every quantum number satisfies the corresponding rules. Therefore, set (d) is acceptable.

Evaluation of Set (e) (3,2,+1,1)

In set (e), the values are \(n = 3\), \(l = 2\), \(m_l = +1\), but \(m_s = 1\) violates the \(m_s\) restriction, which must be either \(+\frac{1}{2}\) or \(-\frac{1}{2}\). Set (e) is unacceptable.

Key concepts

Principal Quantum Number

The principal quantum number, denoted as \( n \), plays a crucial role in determining the energy level and size of an electron's orbital within an atom. Essentially, it indicates the shell or main energy level that an electron occupies. It is always a positive integer: \( n = 1, 2, 3, \ldots \).
The larger the value of \( n \), the greater the average distance of the electron from the nucleus, and thus, the higher its energy level. For instance, an electron in the \( n = 2 \) shell is found further from the nucleus than one in the \( n = 1 \) shell. This is why we often associate the principal quantum number with the electron's "layer" or "orbit" within the atom.
In the given sets of quantum numbers in the exercise, every principal quantum number meets the rule \( n \geq 1 \). However, it is important to note that the principal quantum number not only determines the energy level but also influences the possible values of other quantum numbers, particularly the azimuthal quantum number.

Azimuthal Quantum Number

The azimuthal quantum number, designated as \( l \), determines the shape of the electron's orbital. It is sometimes also called the angular momentum quantum number. This number can take any integer value from 0 up to \( n-1 \), where \( n \) is the principal quantum number of the electron.
Each value of \( l \) corresponds to a specific type of orbital shape:

• \( l = 0 \): s orbital, spherical in shape.
• \( l = 1 \): p orbital, dumbbell-shaped.
• \( l = 2 \): d orbital, more complex in structure.
• \( l = 3 \): f orbital, even more complex.

However, in the provided exercise, set (a) has a principal quantum number \( n = 1 \) and azimuthal quantum number \( l = 1 \), which violates the rule \( 0 \leq l < n \), since \( l \) should be less than \( n \). Thus, \( l \) values give us a clear understanding of potential orbital arrangements, tying closely with the principal quantum number.

Magnetic Quantum Number

The magnetic quantum number, symbolized by \( m_l \), reveals the orientation of an orbital in space. The value of \( m_l \) depends on the azimuthal quantum number \( l \) and is defined as taking integer values ranging from \( -l \) to \( +l \).
Therefore, if an electron has \( l = 1 \), its possible \( m_l \) values are \(-1, 0, +1\). This multi-valued possibility holds the clue to the presence of multiple orientations or "directions" for p, d, and f orbitals. This demonstrates why \( m_l \) is crucial for understanding how electrons are arranged.
In the exercise, several sets illustrate the rule violations, such as set (b) which has \( m_l = -1 \) while \( l = 0 \), making \( m_l \) outside of its possible range (since \([-0, +0]\) means \(m_l\) must be 0). Therefore, each specific \( m_l \) position aligns an electron's orbital orientation within the atom, which is vital to comprehend when addressing quantum number configurations.

Spin Quantum Number

The spin quantum number, abbreviated as \( m_s \), indicates the direction of the intrinsic angular momentum or the "spin" of an electron within an orbital. It can be either \( +\frac{1}{2} \) or \( -\frac{1}{2} \), symbolizing the two opposite spin directions an electron can adopt.
This number is unique because it doesn't depend on the principal, azimuthal, or magnetic quantum numbers, focusing purely on spin directionality.
The spin quantum number upholds the "Pauli Exclusion Principle," which states no two electrons can possess the identical set of four quantum numbers in an atom.
In the provided exercise, set (e) had an incorrect \( m_s = 1 \), showing a misunderstanding of its possible values. This error emphasizes the importance of recognizing \( m_s \) values to properly identify electron spin and validate quantum number sets in an elemental structure.