Question

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

Short answer

The Quantum numbers for \(2 p\) are \(n = 2, l = 1, m_l = -1, 0, 1, m_s = +½, -½\). For \(3 s\) are \(n = 3, l=0, m_l = 0, m_s = +½, -½\). And for \(5 d\), they are \(n=5, l=2, m_l = -2, -1, 0, 1, 2, m_s = +½, -½\).

Step-by-step solution

Identification of the quantum numbers for \(2 p\)

For a \(2 p\) orbital, quantum numbers are as following: Principal quantum number (\(n\)) is 2 as it denotes the main energy level of an electron in an atom. Azimuthal quantum number (\(l\)) i.e., the one indicates the shape of the electron's orbital is 1. This corresponds to \(p\) orbitals. Magnetic quantum number (\(m_l\)) varies between -\(l\) and \(l\). In this case, it can be -1, 0, or 1. The fourth quantum number called the Spin quantum number (\(m_s\)) can only be +½ or -½ that state the spin of an electron.

Identification of the quantum numbers for \(3 s\)

In a \(3 s\) orbital: Principal quantum number (\(n\)) is 3. Azimuthal quantum number (\(l\)) is 0, for \(s\) type orbital. The corresponding Magnetic quantum number (\(m_l\)) is 0 because the range of the \(m_l\) values is from -\(l\) to \(l\). The Spin quantum number (\(m_s\)) again, can be either +½ or -½.

Identification of the quantum numbers for \(5 d\)

For a \(5 d\) orbital, the quantum numbers are as follows: Principal quantum number (\(n\)) is 5. The Azimuthal quantum number (\(l\)) is 2 because 'd' type orbital corresponds to \(l=2\). For 'd' type orbitals the Magnetic quantum number (\(m_l\)) ranges from -2 to 2 hence it can have five possible values: -2, -1, 0, 1, or 2. The Spin quantum number (\(m_s\)) again can be either +½ or -½

Key concepts

Principal Quantum Number

The principal quantum number, denoted as \( n \), is one of the four quantum numbers assigned to electrons in atoms. It primarily indicates the main energy level or shell of an electron.

• Higher values of \( n \) mean greater energy and a larger radius of the electron's orbit.
• In practical terms, \( n \) determines the size and energy level similar to how floors work in a building (e.g., 1st floor, 2nd floor).
• In the exercise, the principal quantum numbers for the orbitals are: \( n = 2 \) for the \(2p\) orbital, \( n = 3 \) for the \(3s\) orbital, and \( n = 5 \) for the \(5d\) orbital.

In essence, \( n \) provides the broadest classification, telling us where an electron resides in terms of energy levels.

Azimuthal Quantum Number

The azimuthal quantum number, denoted as \( l \), is essential in understanding the shape of electron orbitals within a given energy level.

• It is determined by the equation: \( l = 0, 1, 2, \.\.\., (n-1) \).
• The azimuthal quantum number defines subshells or sub-levels like \( s \), \( p \), \( d \), and \( f \).
• Each value of \( l \) corresponds to specific types of subshells: \( l = 0 \) for \( s \), \( l = 1 \) for \( p \), \( l = 2 \) for \( d \), and \( l = 3 \) for \( f \).

For the orbitals given in the exercise, the azimuthal quantum numbers are: \( l = 1 \) for the \(2p\) orbital, indicating a \(p\) orbital shape; \( l = 0 \) for the \(3s\) orbital, indicating an \(s\) shape; and \( l = 2 \) for the \(5d\) orbital, indicating a \(d\) shape.

Magnetic Quantum Number

The magnetic quantum number, \( m_l \), indicates the orientation of an orbital around the nucleus and arises due to the azimuthal quantum number.

• This number can take on values between \( -l \) and \( +l \), including zero.
• The range of possible values for \( m_l \) thereby determines the number of orbitals in a subshell.
• For example, a \(p\) subshell (\( l = 1 \)) can have values of \( m_l \): -1, 0, and 1, resulting in three orbitals.

In our exercise, the magnetic quantum numbers are defined as follows: For the \(2p\) orbital, \( m_l \) can be -1, 0, or 1. For the \(3s\) orbital, \( m_l = 0 \). In the case of the \(5d\) orbital, \( m_l \) ranges from -2 to 2.

Spin Quantum Number

Spin quantum number, \( m_s \), refers to the intrinsic form of angular momentum carried by electrons. It's a bit like how a top spins on its axis.

• This number can only have two possible values: \( +\frac{1}{2} \) and \( -\frac{1}{2} \).
• These numbers symbolize the two possible spin states of an electron, commonly referred to as "spin up" and "spin down."
• The presence of these spin states is the reason electrons pair up in orbitals.

Regardless of the orbital, each electron's spin must be distinct within the same orbital. In the context of this exercise, for all orbitals \((2p, 3s, 5d)\), the spin quantum numbers for electrons can be either \( +\frac{1}{2} \) or \( -\frac{1}{2} \). This manifests as two electrons per orbital with opposite spins.