Problem 53 - Chemistry

Question

Give the values of the quantum numbers associated with the following orbitals: (a) \(2 p,\) (b) \(3 s,\) (c) \(5 d\).

Step-by-Step Solution

Answer

The quantum numbers for the orbitals are: (a) 2p: [n=2, l=1, m=-1, 0, 1], (b) 3s: [n=3, l=0, m=0], (c) 5d: [n=5, l=2, m=-2, -1, 0, 1, 2]

Step 1 Orbital 2p

Identify the quantum numbers from the orbital \(2 p\). Here, 2 is the principal quantum number \(n=2\) and p corresponds to the angular momentum quantum number \(l=1\). Since p orbital can have -1, 0, 1 as the magnetic quantum number (m), the quantum numbers for 2p are [n=2, l=1, m=-1, 0, 1]

Step 2 Orbital 3s

Identify the quantum numbers from the orbital \(3 s\). Here, 3 is the principal quantum number \(n=3\) and s corresponds to the angular momentum quantum number \(l=0\). Since s orbital can have 0 as the magnetic quantum number (m), the quantum numbers for 3s are [n=3, l=0, m=0]

Step 3 Orbital 5d

Identify the quantum numbers from the orbital \(5 d\). Here, 5 is the principal quantum number \(n=5\) and d corresponds to the angular momentum quantum number \(l=2\). Since d orbital can have -2, -1, 0, 1, 2 as the magnetic quantum number (m), the quantum numbers for 5d are [n=5, l=2, m=-2, -1, 0, 1, 2]

Key Concepts

Principal Quantum Number

In the realm of quantum mechanics, the principal quantum number, symbolized by \( n \), is crucial in determining the energy level of an electron within an atom. Think of it as a shell that houses electrons, where each shell corresponds to a particular energy level. Higher values of \( n \) signify that electrons reside in shells further away from the nucleus, leading to greater energy.

• For example, the 2p orbital signifies \( n = 2 \), meaning electrons are in the second energy level or second shell.
• Similarly, in the 3s orbital, \( n = 3 \) indicates electrons are in the third shell.
• For the 5d orbital, \( n = 5 \) places electrons in the fifth energy level.

The principal quantum number not only indicates the energy level but also determines the number of possible sublevels within that shell. Understanding \( n \) provides a fundamental insight into how electrons are organized within an atom, influencing its chemical behavior.

Angular Momentum Quantum Number

The angular momentum quantum number, represented by \( l \), describes the shape of the electron cloud or orbital. It is always linked to a specific principal quantum number and can have integer values ranging from 0 to \( n - 1 \).

• An \( s \)-orbital corresponds to \( l = 0 \), like in the 3s orbital.
• A \( p \)-orbital relates to \( l = 1 \), seen in the 2p orbital.
• For a \( d \)-orbital, \( l = 2 \), as found in the 5d orbital.

These different shapes come with various levels of complexity and hold a certain number of electrons.
The angular momentum quantum number helps further describe how electrons are spatially distributed around the nucleus. By specifying \( l \), we can understand not only the shape but also the energy sublevel within a given principal quantum shell.

Magnetic Quantum Number

The magnetic quantum number, denoted by \( m \) or \( m_l \), details the orientation of an orbital within a magnetic field. For each value of \( l \), \( m \) can take values ranging from \(-l\) to \(+l\), encompassing zero.

• For a \( p \)-orbital (\( l = 1 \)), \( m \) can be \(-1, 0, \text{or} 1\), which applies to the 2p orbital.
• In the case of an \( s \)-orbital (\( l = 0 \)), \( m \) is always 0, as seen in the 3s orbital.
• For \( d \)-orbitals (\( l = 2 \)), \( m \) varies from \(-2\) to \(+2\), corresponding to the 5d orbital.

The magnetic quantum number provides insight into how these orbitals can be oriented in three-dimensional space. Each orbital orientation can house a pair of electrons with opposite spins. Understanding \( m \) is essential for visualizing electron distributions in atoms, particularly in complex transition elements where the magnetic properties of electrons have broader implications.