Question

For each of the following sublevels, give the \(n\) and \(l\) values and the number of orbitals: (a) \(5 s ;\) (b) \(3 p ;\) (c) \(4 f\).

Short answer

(a) \(5s \): \( n = 5 \), \( l = 0 \), 1 orbital; (b) \(3p \): \( n = 3 \), \( l = 1 \), 3 orbitals; (c) \(4f \): \( n = 4 \), \( l = 3 \), 7 orbitals.

Step-by-step solution

Identify the Principal Quantum Number (n)

The principal quantum number, denoted as \( n \), represents the energy level of an orbital and is given directly in the notation. For each sublevel notation, identify the \( n \) value: (a) For \( 5s \), \( n = 5 \); (b) For \( 3p \), \( n = 3 \); (c) For \( 4f \), \( n = 4 \).

Determine the Azimuthal Quantum Number (l)

The azimuthal quantum number, \( l \), signifies the subshell and is assigned specific values based on the letter of the sublevel: s (\( l = 0 \)), p (\( l = 1 \)), d (\( l = 2 \)), f (\( l = 3 \)). Using this, we get: (a) For \( 5s \): \( l = 0 \); (b) For \( 3p \): \( l = 1 \); (c) For \( 4f \): \( l = 3 \).

Calculate the Number of Orbitals

The number of orbitals in a given sublevel can be determined by the formula \( 2l + 1 \). Using the \( l \) values from the previous step: (a) For \( 5s \): \( 2(0) + 1 = 1 \) orbital; (b) For \( 3p \): \( 2(1) + 1 = 3 \) orbitals; (c) For \( 4f \): \( 2(3) + 1 = 7 \) orbitals.

Key concepts

Principal Quantum Number

The principal quantum number, denoted as \( n \), plays a crucial role in the quantum mechanical model of the atom. It primarily indicates the main energy level or shell of an electron within an atom. Each principal quantum number corresponds to a different energy level. For example, an electron in the energy level \( n = 3 \) is at a higher energy state compared to an electron in \( n = 2 \).

The value of \( n \) can be any positive integer (1, 2, 3, etc.). This number not only tells us the energy level, but also gives an idea about the size of the orbital. Higher values of \( n \) correspond to orbitals that are farther from the nucleus and have higher energy.

• For \( 5s \), \( n = 5 \)
• For \( 3p \), \( n = 3 \)
• For \( 4f \), \( n = 4 \)

Azimuthal Quantum Number

Also known as the angular momentum quantum number and denoted as \( l \), the azimuthal quantum number determines the shape of the electron's orbital. Each value of \( n \) has a set of possible \( l \) values ranging from 0 to \( n-1 \). For example, if \( n = 3 \), then \( l \) can be 0, 1, or 2.
The specific letter designation (s, p, d, f) and corresponding \( l \) values are:

• s-orbital: \( l = 0 \)
• p-orbital: \( l = 1 \)
• d-orbital: \( l = 2 \)
• f-orbital: \( l = 3 \)

Using these designations, we can determine the azimuthal quantum numbers for each given sublevel:

• For \( 5s \), \( l = 0 \)
• For \( 3p \), \( l = 1 \)
• For \( 4f \), \( l = 3 \)

Orbitals

Orbitals are regions in an atom where there is a high probability of finding electrons. Each orbital is defined by the values of the quantum numbers \( n \) and \( l \).
The azimuthal quantum number \( l \) helps us determine the number of orbitals for a subshell using the formula \( 2l + 1 \). This formula captures all the possible orientations of the orbitals in space for that particular subshell. For example, in the p-subshell (where \( l = 1 \)), there are 3 orbitals because \( 2(1) + 1 = 3 \).

• For \( 5s \), \( l = 0 \), there is 1 orbital: \( 2(0) + 1 = 1 \)
• For \( 3p \), \( l = 1 \), there are 3 orbitals: \( 2(1) + 1 = 3 \)
• For \( 4f \), \( l = 3 \), there are 7 orbitals: \( 2(3) + 1 = 7 \)

By understanding the number of orbitals, we get an insight into how electrons are arranged in an atom and how they affect its chemical properties.