Question

What is the maximum number of orbitals that can be identified by each of the following sets of quantum numbers? When "none" is the correct answer, explain your reasoning. (a) \(n=3, \ell=0, m_{\ell}=+1\) (b) \(n=5, \ell=1\) (c) \(n=7, \ell=5\) (d) \(n=4, \ell=2, m_{\ell}=-2\)

Short answer

(a) None. (b) Three. (c) Eleven. (d) One.

Step-by-step solution

Understanding Quantum Numbers

Quantum numbers describe the properties of atomic orbitals and the electrons in those orbitals. The principal quantum number \(n\) determines the size and energy level of the orbital. The azimuthal quantum number \(\ell\) determines the shape of the orbital and ranges from 0 to \(n-1\). The magnetic quantum number \(m_{\ell}\) indicates the orientation of an orbital and ranges from \(-\ell\) to \(+\ell\).

Analyze Set (a)

For \(n=3, \ell=0, m_{\ell}=+1\): The possible values for \(m_{\ell}\) range from \(-\ell\) to \(+\ell\). Since \(\ell=0\) can only have \(m_{\ell}=0\), a value of \(+1\) is not possible. Thus, there are no orbitals.

Analyze Set (b)

For \(n=5, \ell=1\): The magnetic quantum number \(m_{\ell}\) ranges from \(-1\) to \(+1\) (i.e., three possible values: \(-1, 0, +1\)). Therefore, there are three orbitals corresponding to these values.

Analyze Set (c)

For \(n=7, \ell=5\): The magnetic quantum number \(m_{\ell}\) ranges from \(-5\) to \(+5\), giving 11 possible values (\(-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5\)). Thus, there are eleven orbitals with these quantum numbers.

Analyze Set (d)

For \(n=4, \ell=2, m_{\ell}=-2\): The magnetic quantum number would be \(-2\). As \(\ell=2\) ranges for \(m_{\ell}\) from \(-2\) to \(+2\), \(m_{\ell}=-2\) is valid. Therefore, there is one orbital corresponding to this set of quantum numbers.

Key concepts

Principal Quantum Number

The principal quantum number, denoted by \( n \), is the first quantum number and a fundamental aspect of the quantum mechanical model of atoms. It represents the main energy level or shell in which an electron resides within an atom. The values of \( n \) are positive integers (1, 2, 3, etc.), where a higher number signifies electrons that are further from the nucleus and in higher energy levels.

This quantum number is crucial because it determines the size and energy of the atomic orbital. Larger \( n \) values typically correspond to larger orbitals with higher energy, as electrons are farther from the nucleus's attractive force. It's essential to understand that each energy level can contain more than one type of orbital, defined by other quantum numbers, including the azimuthal and magnetic quantum numbers. By defining \( n \), we can predict the electron configuration and energy distribution within an atom, directly influencing chemical properties and reactivity.

Azimuthal Quantum Number

The azimuthal quantum number, symbolized by \( \ell \), describes the shape of the atomic orbital and is sometimes referred to as the angular momentum quantum number. It takes values that range from 0 to \( n-1 \), where \( n \) is the principal quantum number. Each value of \( \ell \) corresponds to a specific type of orbital shape:

• \( \ell = 0 \): s orbital (spherical shape)
• \( \ell = 1 \): p orbital (dumbbell shape)
• \( \ell = 2 \): d orbital (cloverleaf shape)
• \( \ell = 3 \): f orbital (complex shape)

These shapes are critical because they define how electrons occupy space around the nucleus and form electron clouds of various geometry. Understanding the azimuthal quantum number helps predict an atom's bonding behavior due to the spatial distribution of its electrons. Notably, different \( \ell \) values not only determine the shape but also align with the energy levels due to electron repulsion forces acting within multi-electron atoms.

Magnetic Quantum Number

The magnetic quantum number, given by \( m_{\ell} \), further specifies the orientation of an orbital in space and allows distinct orbitals of the same shape to orient differently in a 3D space. It can take on integer values ranging from \(-\ell\) to \(+\ell\), providing multiple orbitals for each \( \ell \) value.

For example, if \( \ell = 1 \), then \( m_{\ell} \) can be \(-1, 0, \) or \(+1 \). These correspond to the different orientations of a p orbital (\( p_x, p_y, p_z \)). The magnetic quantum number is essential in specifying how these orbitals are arranged within an atom, particularly when placed in an external magnetic field, where these different orientations influence the electron dynamics. The number of possible \( m_{\ell} \) values also directly corresponds to the number of orbitals available within a subshell, influencing how electrons are distributed across these spaces.

Atomic Orbitals

Atomic orbitals are regions in an atom's electron cloud where electrons are likely to be found. Each orbital corresponds to a specific set of quantum numbers, providing unique characteristics such as size, shape, and orientation. The combination of principal, azimuthal, and magnetic quantum numbers precisely dictates the nature and energy of these orbitals.

Orbitals with the same \( n \) and \( \ell \) but different \( m_{\ell} \) values are considered degenerate, meaning they have the same energy level in absence of an external field. Such degeneracy affects electron configuration, orbital filling order, and ultimately an atom's chemical properties.

• The s orbitals are spherical and have the simplest shape.
• p orbitals have a more complex, dumbbell-like appearance.
• d and f orbitals exhibit even more geometrical complexity with multiple lobes.

The conceptualization of atomic orbitals helps chemists and physicists predict how atoms interact in bonding and reactions, providing insights into the structure of complex molecules and materials.