Problem 38 - Chemistry and Chemical Reactivity

Question

Explain briefly why each of the following is not a possible set of quantum numbers for an electron in an atom. In each case, change the incorrect value (or values) to make the set valid. (a) \(n=2, \ell=2, m_{\ell}=0, m_{s}=+1 / 2\) (b) \(n=2, \ell=1, m_{\ell}=-1, m_{\mathrm{s}}=0\) (c) \(n=3, \ell=1, m_{\ell}=-2, m_{\mathrm{s}}=+1 / 2\)

Step-by-Step Solution

Answer

(a) Change \(\ell=2\) to \(\ell=1\); (b) Change \(m_{s}=0\) to \(\pm\frac{1}{2}\); (c) Change \(m_{\ell}=-2\) to \(-1\), \(0\), or \(+1\).

Step 1: Understanding Quantum Numbers

Quantum numbers define the energy level and position of an electron in an atom. They include the principal quantum number \(n\), the azimuthal quantum number \(\ell\), the magnetic quantum number \(m_{\ell}\), and the spin quantum number \(m_{s}\). Each must fall within specific allowable ranges for a valid set of quantum numbers.

Step 2: Checking Set (a)

For set (a) \(n=2, \ell=2, m_{\ell}=0, m_{s}=+\frac{1}{2}\), the azimuthal quantum number \(\ell\) must be less than \(n\). Here, \(\ell=2\) is not valid because it cannot be equal to or greater than \(n\). To correct it, \(\ell\) should be \(0\) or \(1\).

Step 3: Correcting Set (a)

Changing \(\ell = 2\) to \(\ell = 1\) makes set (a) valid: \(n=2, \ell=1, m_{\ell}=0, m_{s}=+\frac{1}{2}\).

Step 4: Checking Set (b)

For set (b) \(n=2, \ell=1, m_{\ell}=-1, m_{s}=0\), the spin quantum number \(m_{s}\) must be \(+\frac{1}{2}\) or \(-\frac{1}{2}\), not \(0\).

Step 5: Correcting Set (b)

To fix set (b), change \(m_{s}=0\) to \(m_{s}=+\frac{1}{2}\) or \(m_{s}=-\frac{1}{2}\). A valid set is \(n=2, \ell=1, m_{\ell}=-1, m_{s}=+\frac{1}{2}\).

Step 6: Checking Set (c)

For set (c) \(n=3, \ell=1, m_{\ell}=-2, m_{s}=+\frac{1}{2}\), the magnetic quantum number \(m_{\ell}\) should range from \(-\ell\) to \(+\ell\), here being \(-1\), \(0\), or \(+1\) for \(\ell=1\). Since \(m_{\ell}=-2\) is not in this range, it is incorrect.

Step 7: Correcting Set (c)

Change \(m_{\ell}=-2\) to any value within the range \(-1\) to \(+1\). A valid set is \(n=3, \ell=1, m_{\ell}=-1, m_{s}=+\frac{1}{2}\).

Key Concepts

Principal Quantum Number

The principal quantum number, denoted as \( n \), is fundamentally important in quantum mechanics for describing an electron's position in an atom. It is also known as the "shell" number and indicates the size and energy level of the orbital in which the electron resides.

It takes on any positive integer value (1, 2, 3,...), and as \( n \) increases, the electron's orbital becomes larger and farther from the nucleus. This results in greater energy levels because the electron is at a higher state of potential energy. However, the energy difference between levels decreases as \( n \) becomes larger.

• The minimum allowable value is \( n = 1 \), which corresponds to the closest shell to the nucleus.
• Higher \( n \) values mean the electron has higher energy and is more loosely bound to the atom.

Understanding \( n \) is crucial because it limits the other quantum numbers. \( \ell \), the azimuthal quantum number, must be less than \( n \).

Azimuthal Quantum Number

The azimuthal quantum number, represented as \( \ell \), provides information about the shape of the electron's orbital. This shape determines the electron's "sub-level," and associated letters are used to identify different shapes, specifically: s, p, d, f corresponding to \( \ell = 0, 1, 2, 3 \) respectively.

In each principal quantum level \( n \), \( \ell \) can take on integer values from 0 up to \( n-1 \). This rule sets the number and shape of possible subshells for any given shell. For example, if \( n = 3 \), the possible \( \ell \) values are 0, 1, and 2, which are s, p, and d sub-levels.

• \( \ell = 0 \) corresponds to s orbitals which are spherical.
• \( \ell = 1 \) corresponds to p orbitals, which are dumbbell-shaped.
• \( \ell = 2 \) corresponds to d orbitals, which are more complex.

The azimuthal quantum number plays a crucial role in predicting an atom's electron configuration and its contribution to chemical bonding and properties.

Magnetic Quantum Number

The magnetic quantum number, noted as \( m_\ell \), describes the orientation of the electron's orbital in space. Essentially, \( m_\ell \) denotes the specific cloud of electron density within a given sub-level, or azimuthal quantum number \( \ell \).

It can take on integral values ranging from \(-\ell\) to \(+\ell\), including zero. For example:

• If \( \ell = 1 \), \( m_\ell \) can be -1, 0, or +1, which corresponds to the three orientations of p orbitals.
• If \( \ell = 2 \), \( m_\ell \) can be -2, -1, 0, 1, or 2, covering d orbital orientations.

This orientation aspect of \( m_\ell \) helps in determining how many orbitals are available in a given subshell, influencing the atom's magnetic properties and possible electron arrangements. It's closely tied to electron configurations and their interpretation in magnetic and spectral lines.

Spin Quantum Number

The spin quantum number, represented as \( m_s \), signifies the intrinsic spin of an electron in its orbital. Unlike the other quantum numbers which are derived from the electronic wave functions, the spin quantum number deals with the spin, granting it a unique role in defining electron behavior.

An electron spin is a fundamental property, much like charge or mass. It can take one of two values, +1/2 or -1/2, which is often illustrated with arrows\( \uparrow \) or \( \downarrow \) for convenience.

• \( m_s = +1/2 \) signifies "spin up," which is a clockwise rotation.
• \( m_s = -1/2 \) signifies "spin down," representing a counterclockwise rotation.

The spin quantum number simplifies many models and is crucial in the Pauli exclusion principle, which states that no two electrons can have the same set of all four quantum numbers. This requirement is critical in defining electron configurations and shell filling orders in atoms.