Problem 54 - Chemistry The Central Science

Question

Give the values for \(n, l\), and \(m_{l}\) for (a) each orbital in the \(2 p\) subshell, (b) each orbital in the \(5 d\) subshell.

Step-by-Step Solution

Answer

(a) For each orbital in the \(2p\) subshell, the quantum numbers are: - Orbital 1: \(n=2, l=1, m_l=-1\) - Orbital 2: \(n=2, l=1, m_l=0\) - Orbital 3: \(n=2, l=1, m_l=1\) (b) For each orbital in the \(5d\) subshell, the quantum numbers are: - Orbital 1: \(n=5, l=2, m_l=-2\) - Orbital 2: \(n=5, l=2, m_l=-1\) - Orbital 3: \(n=5, l=2, m_l=0\) - Orbital 4: \(n=5, l=2, m_l=1\) - Orbital 5: \(n=5, l=2, m_l=2\)

Step 1: Principal Quantum Number n

For the \(2p\) subshell, the principal quantum number, \(n = 2\).

Step 2: Azimuthal Quantum Number l

In the \(2p\) subshell ("p" subshell), the azimuthal quantum number, \(l = 1\).

Step 3: Magnetic Quantum Number m_l

For the p orbital (\(l=1\)), the magnetic quantum number, \(m_l\), can have values of -1, 0, and 1. So, the orbital quantum numbers for the \(2p\) subshell are: - Orbital 1: \(n=2, l=1, m_l=-1\) - Orbital 2: \(n=2, l=1, m_l=0\) - Orbital 3: \(n=2, l=1, m_l=1\) (b) Each orbital in the \(5d\) subshell:

Step 4: Principal Quantum Number n

For the \(5d\) subshell, the principal quantum number, \(n = 5\).

Step 5: Azimuthal Quantum Number l

In the \(5d\) subshell ("d" subshell), the azimuthal quantum number, \(l = 2\).

Step 6: Magnetic Quantum Number m_l

For the d orbital (\(l=2\)), the magnetic quantum number, \(m_l\), can have values of -2, -1, 0, 1, and 2. So, the orbital quantum numbers for the \(5d\) subshell are: - Orbital 1: \(n=5, l=2, m_l=-2\) - Orbital 2: \(n=5, l=2, m_l=-1\) - Orbital 3: \(n=5, l=2, m_l=0\) - Orbital 4: \(n=5, l=2, m_l=1\) - Orbital 5: \(n=5, l=2, m_l=2\)

Key Concepts

Understanding the Principal Quantum Number

The principal quantum number, denoted as \( n \), is like the building level in a multi-story building, where each level provides space for electrons. It tells us about the size and energy level of an electron's orbital. If you think of an atom as a hotel, the principal quantum number would be the floor where you would find your room or electrons. The higher the value of \( n \), the larger and more energetic the electron's orbital becomes.

For instance:

• The \( 2p \) subshell has a principal quantum number of \( n = 2 \).
• The \( 5d \) subshell has a principal quantum number of \( n = 5 \).

Each increase in \( n \) implies that the electrons are further from the nucleus. This helps in understanding how electrons are distributed at various energy levels in an atom.

Exploring the Azimuthal Quantum Number

The azimuthal quantum number, symbolized by \( l \), introduces the idea of shape to an atom's structure. It is determined by the type of subshell, and it provides more detail about the shape of an electron cloud within an orbital. Imagine drawing a circle and then deciding whether it'll be a simple loop, a loop with squiggles, or layered loops; the azimuthal quantum number is the deciding factor for these intricacies.

Here's how it works:

• In a "p" type subshell (like \( 2p \)), \( l = 1 \). The orbitals here are dumbbell shaped.
• In a "d" type subshell (like \( 5d \)), \( l = 2 \). These orbitals tend to have more complex shapes, like four clover leaves or a donut with a belt around it.

The different values of \( l \) not only explain the shape but also influence the subshell's energy. The azimuthal quantum number thus tells us about the electron's rotational momentum and the orbital's shape.

Unraveling the Magnetic Quantum Number

The magnetic quantum number, denoted as \( m_{l} \), is like the precise address of an electron within an orbital, telling us about its orientation in space. Think of it as the preference of the electron to sit at a particular plot on its level. This quantum number is affected by the azimuthal quantum number, \( l \), and determines the number of orbitals and their specific alignment along a magnetic field.

For example:
In a "p" orbital where \( l = 1 \), the possible values of \( m_{l} \) are -1, 0, and 1, representing three orientations of the dumbbell shape orbitals.

• \( 2p \) allows for \( m_{l} = -1, 0, 1 \).

In a "d" orbital where \( l = 2 \), \( m_{l} \) can take on values -2, -1, 0, 1, and 2, displaying five different orientations.

• \( 5d \) results in \( m_{l} = -2, -1, 0, 1, 2 \).

Understanding \( m_{l} \) is crucial for comprehending how electrons behave in different magnetic fields, and how orbitals are spatially arranged in an atom.