Problem 86 - Chemistry: An Atoms-Focused Approach

Question

Which of the following combinations of quantum numbers are allowed? a. \(n=3, \ell=2, m_{\ell}=0, m_{\mathrm{s}}=-\frac{1}{2}\) b. \(n=5, \ell=4, m_{\ell}=4, m_{s}=+\frac{1}{2}\) c. \(n=3, \ell=0, m_{\ell}=1, m_{s}=+\frac{1}{2}\) d. \(n=4, \ell=4, m_{\ell}=1, m_{s}=-\frac{1}{2}\)

Step-by-Step Solution

Answer

Allowed combinations are (a) and (b). Combinations (c) and (d) are not allowed due to invalid magnetic quantum numbers and angular momentum quantum numbers, respectively.

Step 1: Combination (a)

Given: \(n=3, \ell=2, m_{\ell}=0, m_{s}=-\frac{1}{2}\) The principal quantum number is 3, which is valid. \(\ell=2\), which is less than \(n=3\) and is also valid. The magnetic quantum number, \(m_{\ell}\) is 0, which is within the range -2 to 2 for \(\ell=2\) and is valid. The spin quantum number, \(m_s=-\frac{1}{2}\) is correct. Therefore, combination (a) is allowed.

Step 2: Combination (b)

Given: \(n=5, \ell=4, m_{\ell}=4, m_{s}=+\frac{1}{2}\) The principal quantum number is 5, which is valid. \(\ell=4\), which is less than \(n=5\) and is also valid. The magnetic quantum number, \(m_{\ell}\) is 4, which is within the range -4 to 4 for \(\ell=4\) and is valid. The spin quantum number, \(m_s=+\frac{1}{2}\) is correct. Therefore, combination (b) is allowed.

Step 3: Combination (c)

Given: \(n=3, \ell=0, m_{\ell}=1, m_{s}=+\frac{1}{2}\) The principal quantum number is 3, which is valid. \(\ell=0\), which is less than \(n=3\) and is also valid. However, the magnetic quantum number, \(m_{\ell}\) is 1, which is not within the range -0 to 0 for \(\ell=0\). Therefore, combination (c) is not allowed.

Step 4: Combination (d)

Given: \(n=4, \ell=4, m_{\ell}=1, m_{s}=-\frac{1}{2}\) The principal quantum number is 4, which is valid. However, \(\ell=4\) is not less than \(n=4\), and is not a valid value for l. Therefore, combination (d) is not allowed. In conclusion, combinations (a) and (b) are allowed, while combinations (c) and (d) are not allowed.

Key Concepts

Principal Quantum Number

The principal quantum number, denoted by \(n\), is a fundamental integer that determines the main energy level or shell of an electron in an atom. It primarily dictates the size of the orbital and directly relates to the energy of an electron. The larger the value of \(n\), the higher the energy level and the further the electron is from the nucleus.

• Possible values for \(n\) are positive integers like 1, 2, 3, etc.
• Each increase in \(n\) represents a new shell or energy level.

Understanding \(n\) allows us to determine how electrons fill different energy levels within an atom. For example, in our exercise, a combination with \(n=4\) means the electron is in the fourth energy level.
Identifying the correct \(n\) value helps in predicting the stability of these configurations and how they interact.

Angular Momentum Quantum Number

The angular momentum quantum number, symbolized by \(\ell\), relates to the shape of an electron's orbital and is also known as the azimuthal quantum number. It is crucial for understanding the geometry of electron paths.

• The allowed values of \(\ell\) range from 0 to \(n-1\), where \(n\) is the principal quantum number.
• Each value of \(\ell\) corresponds to a specific type of orbital: \(\ell = 0\) (s-orbital), \(\ell = 1\) (p-orbital), \(\ell = 2\) (d-orbital), etc.

In the exercise, an example where \(\ell = 4\) and \(n = 4\) was shown to be invalid because \(\ell\) should always be less than \(n\). This highlights how \(\ell\) constraints affect the possible shapes and complexity of electron distributions in different atoms.

Magnetic Quantum Number

The magnetic quantum number, represented by \(m_{\ell}\), indicates the orientation of an orbital around the nucleus and the number of orientations available. This quantum number provides an in-depth understanding of electron movement and magnetic influences.

• The range of \(m_{\ell}\) values spans from \(-\ell\) to \(+\ell\), including zero.
• Each possible \(m_{\ell}\) value corresponds to a specific orientation of the electron cloud in space.

This is crucial because it determines the orientation or directional dependence of the orbital, which has implications for chemical bonding and the atomic spectra. In our task, the incorrect \(m_{\ell} = 1\) in a case where \(\ell = 0\) was flagged as invalid, emphasizing the importance of the range criteria for valid configurations.

Spin Quantum Number

The spin quantum number, denoted by \(m_s\), reveals the intrinsic spin of an electron within an orbital. It’s a keystone in quantum mechanics for illustrating the tiny intrinsic angular momentum carried by electrons.

• Only two possible values exist for \(m_s\): \(+\frac{1}{2}\) and \(-\frac{1}{2}\).
• The two values signify the two possible spin orientations: "spin-up" and "spin-down".

This quantum number is fundamental because it explains the magnetic properties of atoms and how two electrons can coexist in the same orbital, leading to unique electron pairing properties. In our examples, both \(+\frac{1}{2}\) and \(-\frac{1}{2}\) were demonstrated as valid, underscoring this key binary nature.