Question

An electron in a certain atom is in the \(n=2\) quantum level. List the possible values of \(\ell\) and \(m_{\ell}\) that it can have.

Short answer

Possible \((\ell, m_{\ell})\) values are (0,0), (1,-1), (1,0), and (1,1).

Step-by-step solution

Understanding the primary quantum number

The primary quantum number, denoted as \(n\), defines the energy level of an electron in an atom. For this exercise, \(n\) is given as 2. This means the electron is in the second energy level.

Determining possible values of the azimuthal quantum number (\(\ell\))

The azimuthal quantum number \(\ell\) defines the shape of the orbital. It can take on integer values from 0 to \(n-1\). Therefore, if \(n = 2\), then \(\ell\) can be 0 or 1.

Determining possible values of the magnetic quantum number (\(m_{\ell}\))

The magnetic quantum number \(m_{\ell}\) defines the orientation of the orbital in space. For a given \(\ell\), the values of \(m_{\ell}\) range from \(-\ell\) to \(+\ell\).- If \(\ell = 0\), then \(m_{\ell} = 0\).- If \(\ell = 1\), then \(m_{\ell}\) can be -1, 0, or 1.

Listing all possible combinations of (\(\ell, m_{\ell}\))

From the previous steps, we compile the possible combinations of (\(\ell, m_{\ell}\)):- For \(\ell = 0\), \(m_{\ell} = 0\).- For \(\ell = 1\), \(m_{\ell}\) can be -1, 0, or 1.Thus, the combinations are (0,0), (1,-1), (1,0), and (1,1).

Key concepts

Principal Quantum Number

The principal quantum number, denoted as \( n \), acts as the foundational stepping stone for identifying the electron's position in an atom. Essentially, \( n \) dictates an electron's energy level or shell. Imagine an atom as a multi-floored building, with each floor representing a different energy level. The higher the \( n \) value, the further the electron is from the nucleus and the more energy it potentially possesses.
For example, the first floor of our imaginary building, corresponding to \( n=1 \), is closest to the nucleus and typically accommodates electrons with lower energy levels. In contrast, the second floor, with \( n=2 \), signals a higher energy level. This concept helps us understand where an electron might be, two levels away from the nucleus.

• Lower \( n \) values correspond to lower energy and closer proximity to the nucleus.
• Higher \( n \) values allow for more complex and varied electron configurations.

In the exercise, when \( n=2 \), it indicates that the electron resides in the second energy shell of the atom.

Azimuthal Quantum Number

Now, let's discuss the azimuthal quantum number, represented as \( \ell \). This quantum number is vital for describing the shape of the electron cloud or orbital within which an electron moves. It further refines our understanding by differentiating between the subshells inside the main energy level described by \( n \).
The azimuthal quantum number \( \ell \) can range from 0 to \( n-1 \). Each value of \( \ell \) denotes a different type of orbital shape:

• \( \ell = 0 \): Represents an \( s \)-shaped orbital, spherical in form.
• \( \ell = 1 \): Corresponds to a \( p \)-shaped orbital, dumbbell-shaped.

Thus, in our exercise where \( n=2 \), possible \( \ell \) values can be 0 or 1. This means our electron could occupy an \( s \) or a \( p \) orbital. Understanding \( \ell \) allows for a deeper dive into determining where, spatially, an electron can reside in its energy level.

Magnetic Quantum Number

The magnetic quantum number \( m_{\ell} \) adds another layer of detail to the electron's spatial arrangement by categorizing the orientation of an electron's orbital—imagine this as picking which direction the electron's home is facing within its subshell.
For any given azimuthal quantum number \( \ell \), \( m_{\ell} \) can take on integer values ranging from \(-\ell\) to \(+\ell\). This means

• For \( \ell = 0 \), \( m_{\ell} = 0 \) since the \( s \) orbital can only orient itself one way.
• For \( \ell = 1 \), possible values are, \( m_{\ell} = -1, 0, \) or \( 1 \), reflecting the varied orientations \( p \) orbitals can achieve.

In the exercise scenario, these enumerations translate into a set of possible pairs: (0,0) for the \( s \) orbital, and (-1, 0, 1) for the \( p \) orbitals when \( \ell = 1 \).
Collectively, the quantum numbers allow us to pinpoint much about where electrons reside and how they behave, ultimately composing the rich architectural tapestry of atomic structure.