Question

For each of the following, give the sublevel designation, the allowable \(m_{i}\) values, and the number of orbitals: (a) \(n=2, l=0\) (b) \(n=3, l=2\) (c) \(n=5, l=1\)

Short answer

(a) 2s, m_l=0, 1 orbital (b) 3d, m_l=-2,-1,0,1,2, 5 orbitals (c) 5p, m_l=-1,0,1, 3 orbitals

Step-by-step solution

Determine the sublevel designation

The sublevel designation is identified by the principal quantum number () and the azimuthal quantum number (). Each value of corresponds to a specific sublevel: = 0 (s), = 1 (p), = 2 (d), = 3 (f), and so on.

Find the allowable m_{l} values

The magnetic quantum number () ranges from - to +, including 0. The total number of values for a given is 2 + 1.

Calculate the number of orbitals

The number of orbitals in a sublevel is equal to the number of allowable values, which is 2 + 1.

Solution for (a) n=2, l=0

- Sublevel: 2s- Allowable values: 0 (since = 0)- Number of orbitals: 1 (since 2*0 + 1 = 1)

Solution for (b) n=3, l=2

- Sublevel: 3d- Allowable values: -2, -1, 0, 1, 2 (since ranges from -2 to 2)- Number of orbitals: 5 (since 2*2 + 1 = 5)

Solution for (c) n=5, l=1

- Sublevel: 5p- Allowable values: -1, 0, 1 (since ranges from -1 to 1)- Number of orbitals: 3 (since 2*1 + 1 = 3)

Key concepts

Sublevel Designation

To understand sublevel designation, we start with the principal quantum number (n) and the azimuthal quantum number (l). The principal quantum number (n) indicates the main energy level or shell, while the azimuthal quantum number (l) describes the shape of the orbital within that shell. Each value of l corresponds to a specific sublevel:

• l = 0 corresponds to the s sublevel
• l = 1 corresponds to the p sublevel
• l = 2 corresponds to the d sublevel
• l = 3 corresponds to the f sublevel

This means for each n and l pair, we have a specific designation, such as 2s, 3d, or 5p. For example, when n=2 and l=0, the sublevel designation is 2s. When n=5 and l=1, the sublevel designation is 5p. Understanding these designations helps in identifying the types of orbitals and their arrangement within an atom.

Magnetic Quantum Number

The magnetic quantum number (m_l) is crucial for knowing the orientation of an orbital within a sublevel. The values of m_l depend directly on the azimuthal quantum number (l). For a given l, m_l can range from -l to +l, including zero. This makes sense when you think about the different possible orientations of an orbital in a three-dimensional space. For example:

• If l = 0 (an s sublevel), then m_l can only be 0, since there is only one orientation for an s orbital.
• If l = 1 (a p sublevel), then m_l can be -1, 0, or +1, giving us three possible orientations for p orbitals.
• If l = 2 (a d sublevel), m_l can range from -2 to +2, providing five possible orientations for d orbitals.

Using this information, we can determine the allowable m_l values for any given sublevel. For example, when n=3 and l=2, the sublevel is 3d, and the m_l values range from -2 to +2. This helps us understand how orbitals within a sublevel differ from one another in their spatial orientation.

Number of Orbitals

Calculating the number of orbitals in a sublevel is straightforward once you understand the magnetic quantum number (m_l). The number of distinct m_l values for a given l is given by the formula 2l + 1. This formula accounts for all possible orientations of an orbital within a sublevel. For example:

• If l = 0 (an s sublevel), there is 2*0 + 1 = 1 orbital.
• If l = 1 (a p sublevel), there are 2*1 + 1 = 3 orbitals.
• If l = 2 (a d sublevel), there are 2*2 + 1 = 5 orbitals.

With this method, you can determine the number of orbitals in any sublevel. For instance, for n=2 and l=0, the sublevel is 2s, which has 1 orbital. For n=3 and l=2, the sublevel is 3d, which has 5 orbitals. Consequently, knowing the number of orbitals in each sublevel helps in visualizing how electrons are distributed in an atom's electron cloud and predicting an atom's chemical behavior.