Question

A possible excited state for the \(\mathrm{H}\) atom has an electron in a \(5 d\) orbital. List all possible sets of quantum numbers \(n, \ell,\) and \(m_{\ell}\) for this electron.

Short answer

Possible sets: \((5, 2, -2), (5, 2, -1), (5, 2, 0), (5, 2, +1), (5, 2, +2)\).

Step-by-step solution

Identify the principal quantum number

The principal quantum number, denoted by \( n \), determines the energy level of the electron in the atom. Since the electron is in the \( 5d \) orbital, the value of \( n \) is 5.

Determine the azimuthal quantum number

The azimuthal or angular momentum quantum number, denoted by \( \ell \), defines the subshell in which the electron resides. For a 'd' orbital, \( \ell \) is equal to 2.

List possible magnetic quantum numbers

The magnetic quantum number, denoted by \( m_{\ell} \), describes the orientation of the orbital. It can take integer values from \( -\ell \) to \( +\ell \). Since \( \ell = 2 \), \( m_{\ell} \) can be \(-2, -1, 0, +1, \) or \(+2\).

Key concepts

Principal Quantum Number

The principal quantum number, often symbolized as \( n \), is a fundamental concept in quantum mechanics that describes the energy level or shell of an electron within an atom. Think of it as the floor number in a building, where each floor can hold a certain number of rooms. Here, the rooms represent possible electron positions.

• Values for \( n \) are positive integers: \( 1, 2, 3, \ldots \)
• Higher values of \( n \) correspond to higher energy levels.
• Thus, electrons in higher energy levels are further from the nucleus.

In the context of the given exercise, the electron is situated in the \( 5d \) orbital, which means it lies in the energy level with \( n = 5 \). This denotes that the electron is in the fifth primary energy level, quite distant from the nucleus, resulting in higher potential energy.

Azimuthal Quantum Number

The azimuthal quantum number, denoted by \( \ell \), is crucial for understanding the shape of the electron cloud or orbital where an electron is found. This number provides insight into the subshell within a particular energy level. It's like selecting a specific room type on your chosen floor.

• The value of \( \ell \) ranges from \( 0 \) to \( n-1 \).
• Each \( \ell \) corresponds to a different type of orbital: \( s \), \( p \), \( d \), \( f \), etc.
• For \( \ell = 0 \), it's an \( s \) orbital; for \( \ell = 1 \), \( p \); for \( \ell = 2 \), \( d \), and so on.

In our exercise, we are dealing with a \( d \) orbital, indicating \( \ell = 2 \). This shows that the electron occupies a specific type of subshell within the fifth energy level that has a complex, multi-lobed shape.

Magnetic Quantum Number

The magnetic quantum number, represented as \( m_{\ell} \), indicates the directional orientation of an electron's orbital within a subshell. Once you've picked the floor (\( n \)) and room type (\( \ell \)), \( m_{\ell} \) allows you to select from several potential room orientations.

• The values of \( m_{\ell} \) range from \( -\ell \) to \( +\ell \).
• This includes all integer values, including zero.
• For each \( \ell \) value, multiple \( m_{\ell} \) values represent different orientations.

Given that \( \ell = 2 \) for a \( d \) orbital, \( m_{\ell} \) can take values -2, -1, 0, +1, or +2. This shows us there are five possible orientations, each defining a unique spatial orientation for the electron's probability cloud within the \( d \) subshell.