Problem 33 - Chemistry and Chemical Reactivity

Question

Explain bricfly 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

All given sets are invalid because they violate the rules of quantum numbers.

Step 1: Understanding quantum numbers

Quantum numbers describe the properties of an electron in an atom. They include the principal quantum number \( n \), the azimuthal quantum number \( \ell \), and the magnetic quantum number \( m_{\ell} \). Each has specific rules: \( n \) must be positive integers, \( \ell \) can be any integer from 0 up to \( n-1 \), and \( m_{\ell} \) ranges from \(-\ell\) to \(\ell\).

Step 2: Analyzing part (a) set

For \( n=2 \), the possible values of \( \ell \) are 0 and 1 (since \( \ell \) must be less than \( n \)). If \( \ell = 2 \), which is incorrect because it must be less than \( n=2 \). Thus, the set (a) is not possible because \( \ell \) cannot be equal to \( n \).

Step 3: Analyzing part (b) set

For \( n=3 \) and \( \ell=0 \), the only possible value for \( m_{\ell} \) is 0, as \( m_{\ell} \) must be between \( -\ell \) and \( \ell \). Given \( m_{\ell}=-2 \), it falls outside this range, making this set invalid.

Step 4: Analyzing part (c) set

For \( n=6 \) and \( \ell=0 \), \( m_{\ell} \) must be 0, since \( m_{\ell} \) can only be zero when \( \ell \) itself is zero. Here, \( m_{\ell} = 1 \) is not possible, making this set also invalid.

Key Concepts

Principal Quantum Number

The principal quantum number, denoted by \( n \), is a fundamental parameter in quantum mechanics that signifies the energy level or shell in which an electron resides within an atom. Imagine it as a measure of how far an electron is from the nucleus.

• The value of \( n \) must be a positive integer like 1, 2, 3, and so on.
• Higher values of \( n \) indicate electrons are further out from the nucleus and have higher energy.

In the context of our exercise, recall that \( n = 2 \) offers possibilities for \( \ell \), from 0 up to \( n-1 \), hence 0 and 1 are valid. If \( n = 3 \), \( \ell \) can be 0, 1, or 2, defining different shapes of orbitals for this energy level. Recognizing these relations helps in visualizing the electron's placement and energy within an atom.

Azimuthal Quantum Number

The azimuthal quantum number, symbolized as \( \ell \), provides insight into the shape of an electron's orbital. This quantum number is crucial for predicting the angular momentum of electrons within an atom.

• \( \ell \) ranges from 0 to \( n-1 \).
• Each value of \( \ell \) corresponds to different orbital shapes: \( \ell = 0 \) for s orbitals, \( \ell = 1 \) for p orbitals, and so on.

In parts (a) and (b) of the exercise, \( \ell \) exceeded its allowed range, thus violating the rules. Always ensure the \( \ell \) value fits the condition \( \ell < n \). This adherence reinforces the structure specific to electron configurations, such that \( n=2 \), \( \ell \) validly includes only s and p orbitals.

Magnetic Quantum Number

The magnetic quantum number, represented as \( m_{\ell} \), specifies the orientation of the electron's orbital in space, providing a three-dimensional perspective on electron position. Its value can be any integer between the negative and positive values of \( \ell \).

• If \( \ell = 0 \), \( m_{\ell} \) can only be 0.
• If \( \ell = 1 \), \( m_{\ell} \) can be -1, 0, or +1.

The role of \( m_{\ell} \) is crucial in defining how an atom behaves in a magnetic field. In the exercise, incorrect values like \( m_{\ell} = -2 \) for \( \ell = 0 \) and \( m_{\ell} = 1 \) for also \( \ell = 0 \) were highlighted as impossible. By understanding the relationship between \( \ell \) and \( m_{\ell} \), students can correctly identify valid sets of quantum numbers, linking theoretical insight to practical electron placement.