Problem 59 - General Chemistry: Principles and Modern Applications

Question

What type of orbital (i.e., \(3 s, 4 p, \ldots)\) is designated by these quantum numbers? (a) \(n=5, \ell=1, m_{\ell}=0\) (b) \(n=4, \ell=2, m_{\ell}=-2\) (c) \(n=2, \ell=0, m_{\ell}=0\)

Step-by-Step Solution

Answer

a) The quantum numbers (\(n=5, \ell=1, m_{\ell}=0\)) represent a 5p orbital. b) The quantum numbers (\(n=4, \ell=2, m_{\ell}=-2\)) represent a 4d orbital. c) The quantum numbers (\(n=2, \ell=0, m_{\ell}=0\)) represent a 2s orbital.

Step 1 - Analyze the Quantum Numbers for Part (a)

For the first set of quantum numbers (\(n=5, \ell=1, m_{\ell}=0\)), the principal quantum number, \(n=5\), signifies that the electron is in the 5th energy level. The azimuthal quantum number, \(\ell=1\), corresponds to a p-orbital. The magnetic quantum number, \(m_{\ell}=0\), signifies that this orbital is aligned along one of the coordinate axes.

Step 2 - Analyze the Quantum Numbers for Part (b)

For the second set of quantum numbers (\(n=4, \ell=2, m_{\ell}=-2\)), the principal quantum number, \(n=4\), signifies that the electron is in the 4th energy level. The azimuthal quantum number, \(\ell=2\), corresponds to a d-orbital. The magnetic quantum number, \(m_{\ell}=-2\), signifies that this orbital is specifically aligned along one direction in 3D space.

Step 3 - Analyze the Quantum Numbers for Part (c)

For the third set of quantum numbers (\(n=2, \ell=0, m_{\ell}=0\)), the principal quantum number, \(n=2\), signifies that the electron is in the 2nd energy level. The azimuthal quantum number, \(\ell=0\), corresponds to an s-orbital. The magnetic quantum number, \(m_{\ell}=0\), signifies that this orbital is spherically symmetric around the nucleus.

Key Concepts

Principal Quantum Number

The Principal Quantum Number, often denoted as \( n \), is a fundamental element in understanding atomic structure. This number signifies the main energy level in which an electron is situated and can take positive integer values such as 1, 2, 3, and so on. Each increase in \( n \) represents an electron being found further from the nucleus and in higher energy environments.

• Higher \( n \) means higher energy level and larger atomic orbitals.
• Electrons in higher energy levels experience greater shielding and reduced effective nuclear charge.

For example, in our original exercise, the set of quantum numbers \( n=5, \ell=1, m_{\ell}=0 \) tells us that the electron resides within the fifth energy level of the atom. In contrast, \( n=2 \) indicates a much lower energy level closer to the nucleus, impacting the electron's behavior and chemical properties.

Azimuthal Quantum Number

The Azimuthal Quantum Number, represented by \( \ell \), helps to determine the shape and type of orbital in which an electron is found. This quantum number can take values from 0 up to \( n-1 \). Each value corresponds to a specific type of orbital:

• \( \ell=0 \) denotes an s-orbital, which is spherical in shape.
• \( \ell=1 \) corresponds to a p-orbital, which has a dumbbell shape.
• \( \ell=2 \) indicates a d-orbital, with more complex, cloverleaf shapes.
• \( \ell=3 \) represents an f-orbital, with even more intricate shapes.

By knowing the azimuthal quantum number, you can ascertain the electron's probable location in relation to the nucleus. For instance, an \( \ell \) of 1, as in a p-orbital, specifies different spatial regions compared to an \( \ell \) of 0 for an s-orbital.

Magnetic Quantum Number

Understanding the Magnetic Quantum Number \( m_{\ell} \) is crucial to recognizing the orientation of an electron's orbital. It can range from \(-\ell\) to \( \ell \), including zero. This number specifies the orbital's orientation within a magnetic field.

• Each value of \( m_{\ell} \) describes a distinct orientation of the orbital in three-dimensional space.
• For \( \ell = 1 \) (p-orbitals), \( m_{\ell} \) can be -1, 0, or +1.
• For \( \ell = 2 \) (d-orbitals), \( m_{\ell} \) could be -2, -1, 0, +1, or +2.

This variety allows electrons within the same type of orbital to exist independently according to their magnetic domains. So, in our examples, \( m_{\ell} = 0 \) for \( \ell = 0 \) s-orbitals would imply a symmetrical shape around the nucleus, unaffected by angular orientation.