Problem 33 - 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. (a) \(n=2, \ell=2, m_{\ell}=0\) (b) \(n=3, \ell=0, m_{\ell}=-2\) (c) \(n=6, \ell=0, m_{\ell}=1\)

Step-by-Step Solution

Answer

Set (a): \(\ell\) must be < \(n\); Set (b): \(m_{\ell}\) must be 0 for \(\ell=0\); Set (c): \(m_{\ell}\) must be 0 for \(\ell=0\).

Step 1: Understanding Quantum Numbers

Quantum numbers are used to describe the properties of atomic orbitals and the electrons in those orbitals. The principal quantum number, \(n\), designates the energy level of the electron. The azimuthal quantum number, \(\ell\), represents the subshell or shape of the orbital, and must be within the range \(0 \leq \ell \leq n-1\). The magnetic quantum number, \(m_{\ell}\), indicates the orientation of the orbital and can range from \(-\ell\) to \(+\ell\).

Step 2: Analyze Set (a)

For \(n=2\), the possible values for \(\ell\) are 0 and 1, which correspond to the s and p orbitals, respectively. However, in this set (a), \(\ell=2\), which is not allowed because \(\ell\) should be less than \(n\). The value \(\ell=2\) is invalid for \(n=2\) and hence this set is not possible.

Step 3: Analyze Set (b)

For \(n=3\) and \(\ell=0\), the possible value of \(m_{\ell}\) is 0, because \(m_{\ell}\) ranges from \(-\ell\) to \(+\ell\), including zero. In set (b), \(m_{\ell}=-2\), which is not possible since \(\ell=0\) permits only \(m_{\ell}=0\). Therefore, this set is not feasible.

Step 4: Analyze Set (c)

For \(n=6\) and \(\ell=0\), the allowed \(m_{\ell}\) value should again only be 0, as \(\ell=0\) allows \(m_{\ell}\) to be only 0. However, in set (c), \(m_{\ell}=1\), which is outside the valid range for \(\ell=0\). Therefore, this set is not possible.

Key Concepts

Principal Quantum Number

The principal quantum number, denoted by \(n\), is a fundamental concept in quantum mechanics and chemistry. It describes the overall energy level of an electron within an atom. Think of it as a shell or a layer where electrons reside around the nucleus. This number can take positive integers, such as 1, 2, 3, and so on.

• \(n=1\) describes the closest shell to the nucleus, often known as the ground state.
• As \(n\) increases, electrons are located in shells that are farther away from the nucleus, meaning the energy level and the size of the electron cloud increase.

Choosing \(n\) provides us a basic "address" for where electrons can be found. It serves as a starting point for determining the other quantum numbers.

Azimuthal Quantum Number

The azimuthal quantum number, represented by \(\ell\), gives us information about the shape of the electron's orbital and is tied to the angular momentum of the electron.

• \(\ell\) can take integer values from 0 up to \(n-1\), where \(n\) is the principal quantum number.
• Each value of \(\ell\) corresponds to a specific type of orbital. For instance, \(\ell=0\) represents an "s" orbital, \(\ell=1\) signifies a "p" orbital, \(\ell=2\) stands for a "d" orbital, and so forth.

The azimuthal quantum number is critical because it tells us about the shape and the number of lobes or orientations the electron cloud takes. It's important that \(\ell\) is less than \(n\) to ensure that the energy level structure of the atom remains consistent and coherent.

Magnetic Quantum Number

The magnetic quantum number, expressed as \(m_{\ell}\), determines the orientation of the orbital in space. This number is fascinating because it shows how the shape of the orbital can be aligned in different directions within a given subshell.

• The possible values of \(m_{\ell}\) depend on \(\ell\) and range from \(-\ell\) to \(+\ell\), including zero.
• For instance, if \(\ell=1\) (a "p" orbital), \(m_{\ell}\) could be -1, 0, or +1, indicating three different orientations of the "p" orbitals.

The magnetic quantum number is essential in understanding the magnetic properties of an atom, as each orbital orientation can react differently in a magnetic field. It's crucial to ensure \(m_{\ell}\) fits within the range dictated by \(\ell\) for valid configurations.