Problem 105 - Chemistry The Molecular Nature of Matter

Question

If the value of \(m_{\ell}\) for an electron in an atom is \(-4,\) what is the smallest value of \(\ell\) that the electron could have? What is the smallest value of \(n\) that the electron could have?

Step-by-Step Solution

Answer

The smallest value of \(\ell\) that the electron could have is 4, and the smallest value of \(n\) that the electron could have is 5.

Step 1: Understanding Quantum Numbers

Identify the quantum numbers involved. The quantum number represented by \(m_{\ell}\) is the magnetic quantum number, which depends on the azimuthal or angular momentum quantum number \(\ell\). The magnetic quantum number \(m_{\ell}\) can take on values including zero and ranging from \-\ell\ to \(\ell\), in whole steps.

Step 2: Determining the Minimum Value of \(\ell\)

Find the smallest value of \(\ell\) for which the given \(m_{\ell}\) is valid. For \(m_{\ell} = -4\), the smallest \(\ell\) would be 4, because \(\ell\) must be an integer and at least as large as the absolute value of \(m_{\ell}\).

Step 3: Determining the Minimum Value of \(n\)

Identify the smallest possible value for the principle quantum number \(n\). Since \(\ell\) ranges from 0 up to \(n-1\), knowing that the smallest \(\ell\) is 4 allows us to determine that the smallest \(n\) must be 5, as this is the smallest integer greater than \(\ell\).

Key Concepts

Principle Quantum Number

The principal quantum number, denoted by the symbol \( n \), is the first and foremost quantum number that one encounters in the study of quantum mechanics. It is intrinsically linked to the size and energy of the electron's orbit in an atom. The principal quantum number can take on any positive integer value, starting from 1. The greater the value of \( n \), the higher the energy level and the larger the atom’s orbit or shell. This quantum number essentially determines the electron's shell and, consequently, its distance from the nucleus.

For example, when calculating the minimum value of \( n \) for an electron with a given \( m_{\ell} \) value, one should remember that \( n \) must always be greater than the azimuthal quantum number \( \ell \). Thus, with \( \ell = 4 \), the smallest possible value for \( n \) becomes 5. By understanding the role of \( n \), students can better grasp how electrons populate an atom's various energy levels and why certain quantum numbers are interrelated.

Azimuthal Quantum Number

Diving deeper into the electron's arrangement within an atom, we encounter the azimuthal quantum number, which is represented by \( \ell \). It informs us about the subshell, or the shape of the electron's orbital within a principal energy level. The azimuthal quantum number can range from 0 to \( n-1 \) for each electron, where \( n \) is the principal quantum number. Different values of \( \ell \) correspond to different orbital shapes and are often designated by letter codes: s (0), p (1), d (2), and f (3).

When a student is given a magnetic quantum number, like \( m_{\ell} = -4 \), they must find the smallest positive value of \( \ell \) that allows this \( m_{\ell} \) value to exist. Since \( m_{\ell} \) spans from \(-\ell\) to \(\ell\), a \( m_{\ell} \) of \(-4\) necessitates that \( \ell \) be at least 4 in order to include both positive and negative values up to the magnitude of 4.

Magnetic Quantum Number

Finally, let's focus on the magnetic quantum number, labeled as \( m_{\ell} \). This quantum number is a sub-level within the azimuthal quantum number that describes the orientation of an electron's orbital around the nucleus. It reflects the response of an orbital to an external magnetic field, hence the term 'magnetic'. The allowed values of \( m_{\ell} \) are integers ranging from \(-\ell\) to \(\ell\), including zero. This determines how many orbitals there are within a subshell and their orientation.

In our exercise, the student is dealing with a magnetic quantum number of \(-4\). This value sets limits on the azimuthal quantum number, as \( \ell \) must be equal to or greater than the absolute value of \( m_{\ell} \), leading to the answer of \( \ell \geq 4 \). To facilitate understanding, one can visualize the magnetic quantum number as specifying the specific 'slots' available within an orbital for an electron to reside.