Problem 41 - Chemistry and Chemical Reactivity

Question

Which of the following is not an allowable set of quantum numbers? Explain your answer briefly. \(n \quad \ell \quad m_{\ell} \quad m_{\mathrm{s}}\) \(\begin{array}{lllll}\text { (a) } & 2 & 0 & 0 & -\frac{1}{2}\end{array}\) \(\begin{array}{lllll}\text { (b) } & 1 & 1 & 0 & +\frac{1}{2}\end{array}\) \(\begin{array}{lllll}\text { (c) } & 2 & 1 & -1 & -\frac{1}{2}\end{array}\) \(\begin{array}{lllll}\text { (d) } 4 & 3 & +2 & -\frac{1}{2}\end{array}\)

Step-by-Step Solution

Answer

Set (b) is not allowed due to an invalid \( \ell \) value.

Step 1: Understanding Quantum Numbers

Quantum numbers describe the properties of atomic orbitals and the electrons in those orbitals. The four quantum numbers are:1. Principal quantum number \( n \): takes positive integer values \( (n = 1, 2, 3, ...) \), signifying the energy level.2. Azimuthal quantum number \( \ell \): ranges from \( 0 \) to \( n-1 \), and indicates the subshell type (\( s, p, d, f \) correspond to \( \ell = 0, 1, 2, 3 \)).3. Magnetic quantum number \( m_{\ell} \): ranges from \( -\ell \) to \( \ell \), representing orbital orientation.4. Spin quantum number \( m_s \): can be \( +\frac{1}{2} \) or \( -\frac{1}{2} \), indicating electron spin.

Step 2: Evaluating Set (a)

For set (a) \( (n = 2, \ell = 0, m_{\ell} = 0, m_s = -\frac{1}{2}) \):- \( n = 2 \) is valid.- \( \ell = 0 \), which is valid as it is within \( 0 \to (2-1) \).- \( m_{\ell} = 0 \), which is valid as it is within \( -0 \to 0 \).- \( m_s = -\frac{1}{2} \) is valid. This set of quantum numbers is allowed.

Step 3: Evaluating Set (b)

For set (b) \( (n = 1, \ell = 1, m_{\ell} = 0, m_s = +\frac{1}{2}) \):- \( n = 1 \) is valid.- \( \ell = 1 \) is invalid because \( \ell \) must be in range \( 0 \to (1-1) \), so the only allowed value for \( \ell \) is 0.Since \( \ell = 1 \) is not possible, this set of quantum numbers is not allowed.

Step 4: Evaluating Set (c)

For set (c) \( (n = 2, \ell = 1, m_{\ell} = -1, m_s = -\frac{1}{2}) \):- \( n = 2 \) is valid.- \( \ell = 1 \) is valid, within \( 0 \to 1 \).- \( m_{\ell} = -1 \) is valid, within \( -1 \to 1 \).- \( m_s = -\frac{1}{2} \) is valid.This set of quantum numbers is allowed.

Step 5: Evaluating Set (d)

For set (d) \( (n = 4, \ell = 3, m_{\ell} = +2, m_s = -\frac{1}{2}) \):- \( n = 4 \) is valid.- \( \ell = 3 \) is valid, within \( 0 \to 3 \).- \( m_{\ell} = +2 \) is valid, within \( -3 \to 3 \).- \( m_s = -\frac{1}{2} \) is valid.This set of quantum numbers is allowed.

Key Concepts

Principal Quantum Number

The principal quantum number, denoted by \( n \), is the first quantum number in the quantum mechanical model of atoms. It dictates the energy level of an electron in an atom. Think of it as the address that tells us how far the electron is from the nucleus.

• \( n \) takes positive integer values: \( 1, 2, 3, \ldots \)
• As \( n \) increases, the electron is further from the nucleus and has higher energy.
• Each value of \( n \) corresponds to a different electron shell.

Understanding \( n \) is crucial because it affects the size and energy of orbitals. For instance, an electron with \( n = 1 \) is closest to the nucleus and has lower energy than one with \( n = 3 \). This number sets the stage for the other quantum numbers.

Azimuthal Quantum Number

The azimuthal quantum number, \( \ell \), defines the subshell or shape of the orbital and is crucial for orbital configuration. It is often symbolized by the letters \( s, p, d, \) and \( f \), representing the orbital shapes associated with \( \ell \) values of 0, 1, 2, and 3, respectively.

• It ranges from 0 to \( n-1 \) for a given principal quantum number \( n \).
• For example, if \( n = 3 \), then \( \ell \) can be 0, 1, or 2.
• This number helps distinguish between orbitals within the same shell.

The value of \( \ell \) informs us about the angular momentum of an electron within an atom. This concept provides insight into the geometry of atomic orbitals and plays a vital role in chemical bonding and molecular geometry.

Magnetic Quantum Number

The magnetic quantum number, \( m_{\ell} \), clarifies the orientation of an orbital within a magnetic field. It's linked to the specific orientation of the orbital in space and gives more detail than \( \ell \).

• \( m_{\ell} \) can be any integer between \(-\ell \) and \( \ell \), including zero.
• If \( \ell = 1 \), \( m_{\ell} \) can be \( -1, 0, \text{ or } 1 \).
• This number is critical to the magnetic properties of atoms.

The different possible values of \( m_{\ell} \) determine how orbitals are arranged in space. This grouping explains how electrons occupy their respective orbitals inside a shell, impacting atom stability and interactions.

Spin Quantum Number

Spin quantum number, \( m_s \), represents the intrinsic spin of the electron, which can be considered an intrinsic form of angular momentum. Unlike other quantum numbers, it only takes two possible values.

• \( m_s = +\frac{1}{2} \) signifies 'spin up'.
• \( m_s = -\frac{1}{2} \) signifies 'spin down'.
• This number explains the electron's magnetic moment.

Electron spin is essential in quantum mechanics and determines how electrons pair up within an orbital. Electrons must have opposite spins to coexist in the same orbital, conforming to the Pauli exclusion principle. This principle is vital for understanding electronic configurations and chemical properties.