Question

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

Short answer

(a) 2p: n=2, l=1, ml=-1,0,1; (b) 3s: n=3, l=0, ml=0; (c) 5d: n=5, l=2, ml=-2,-1,0,1,2.

Step-by-step solution

Understanding Quantum Numbers

Quantum numbers are sets of values that help define the state of an electron in an atom. There are four quantum numbers: the principal quantum number (n), the azimuthal (or angular momentum) quantum number (l), the magnetic quantum number (ml), and the spin quantum number (ms). For this exercise, we need to determine n, l, and possible ml values for the given orbitals.

Determine Quantum Numbers for 2p Orbital

The 2p orbital has a principal quantum number, n, of 2. The azimuthal quantum number, l, for a p orbital is 1. The magnetic quantum numbers, ml, can be -1, 0, or 1. So for the 2p orbital, the quantum numbers are n=2, l=1, and ml=-1, 0, 1.

Determine Quantum Numbers for 3s Orbital

The 3s orbital has a principal quantum number, n, of 3. For an s orbital, the azimuthal quantum number, l, is 0. The only possible magnetic quantum number, ml, for l=0 is 0. Thus, the quantum numbers for the 3s orbital are n=3, l=0, and ml=0.

Determine Quantum Numbers for 5d Orbital

For the 5d orbital, the principal quantum number, n, is 5. The azimuthal quantum number, l, for a d orbital is 2. The magnetic quantum numbers, ml, can be -2, -1, 0, 1, or 2. Therefore, the quantum numbers for the 5d orbital are n=5, l=2, and ml=-2, -1, 0, 1, 2.

Key concepts

Principal Quantum Number

The principal quantum number, often represented by the symbol \( n \), is the first quantum number and plays a crucial role in defining the electron's location within an atom. It primarily indicates the main energy level, or shell, of an electron. This number can be any positive integer such as 1, 2, 3, and so on. The larger the value of \( n \), the farther the electron is, on average, from the nucleus and hence, the higher the energy level.

• The primary function of the principal quantum number is to determine the distance of the electron cloud from the nucleus, indicating its potential energy level.
• An electron in the 2nd shell, for example, has \( n = 2 \).
• The principal quantum number also influences the number of subshells that can be present in a given principal energy level.

Hence, understanding \( n \) is the foundation for determining other quantum numbers and electron configurations.

Azimuthal Quantum Number

The azimuthal quantum number, also known as angular momentum quantum number and denoted as \( l \), is instrumental in determining the shape of an electron's orbital. While the principal quantum number governs the size of the electron cloud, the azimuthal quantum number focuses on the orbital's angular distribution. This number can take on any integer value from 0 up to \( n-1 \), where \( n \) is the principal quantum number.

• If \( n = 2 \), then \( l \) can be 0 or 1.
• Each value of \( l \) corresponds to a specific type of orbital, where \( l = 0 \) represents an s orbital, \( l = 1 \) corresponds to a p orbital, \( l = 2 \) to a d orbital, and higher values for f, g, and other types.
• This quantum number helps in understanding the subshells within a principal energy level.

By defining the shape of the electron cloud, \( l \) offers insight into the electron's angular relationship within the atom.

Magnetic Quantum Number

The magnetic quantum number, represented by \( m_l \), gives further detail about the orientation of the orbital in space. This quantum number is crucial when discussing the spatial arrangement of orbitals.

• The values of \( m_l \) range from \(-l\) to \(+l\), including zero. Hence, it hinges directly on the value of the azimuthal quantum number \( l \).
• For an orbital with \( l = 1 \), such as a p orbital, \( m_l \) could be -1, 0, or 1, indicating the three different orientations of the p orbital in space.
• This quantum number helps in understanding how different orbitals can occupy the same energy level but differ in their spatial orientation.

\( m_l \) thus allows us to visualize how electrons are "nested" within their respective subshells, providing a more complete picture of atomic structure.