Question

Give the numerical values of \(n\) and \(l\) corresponding to each of the following orbital designations: (a) \(3 p,(\) b) 2 s, (c) \(4 f\), (d) \(5 d\).

Short answer

For the given orbital designations, the numerical values of n and l are as follows: (a) 3p: \(n = 3, l = 1\); (b) 2s: \(n = 2, l = 0\); (c) 4f: \(n = 4, l = 3\); (d) 5d: \(n = 5, l = 2\).

Step-by-step solution

(a) Orbital designation 3p:

The given orbital designation is 3p. The number 3 represents the principal quantum number, so n = 3. The letter p corresponds to the angular momentum quantum number l = 1. Therefore, the numerical values of n and l corresponding to the 3p orbital are \(n = 3\) and \(l = 1\).

(b) Orbital designation 2s:

The given orbital designation is 2s. The number 2 represents the principal quantum number, so n = 2. The letter s corresponds to the angular momentum quantum number l = 0. Therefore, the numerical values of n and l corresponding to the 2s orbital are \(n = 2\) and \(l = 0\).

(c) Orbital designation 4f:

The given orbital designation is 4f. The number 4 represents the principal quantum number, so n = 4. The letter f corresponds to the angular momentum quantum number l = 3. Therefore, the numerical values of n and l corresponding to the 4f orbital are \(n = 4\) and \(l = 3\).

(d) Orbital designation 5d:

The given orbital designation is 5d. The number 5 represents the principal quantum number, so n = 5. The letter d corresponds to the angular momentum quantum number l = 2. Therefore, the numerical values of n and l corresponding to the 5d orbital are \(n = 5\) and \(l = 2\).

Key concepts

Principal Quantum Number

The principal quantum number, denoted by the symbol \( n \), is fundamental in quantum mechanics as it determines the overall size and energy level of an electron's orbit within an atom. Simply put, it's like the address of an electron, giving it a "shell number." The principal quantum number is always a positive integer and can take values such as 1, 2, 3, and so on. Each increase in \( n \) indicates a higher energy level and a larger size of the electron's orbit.

Here's what you need to remember about the principal quantum number:

• As \( n \) increases, the electron is further from the nucleus, meaning it is at a higher energy level.
• It's associated directly with the rows in the periodic table, defining the shells of electrons.
• The value of \( n \) determines the number of sublevels in a shell, as each principal number has \( n \) sublevels.

For example, in the orbital designation 3p, the '3' tells us that \( n = 3 \), which means the electron is in the third energy shell.

Angular Momentum Quantum Number

The angular momentum quantum number, symbolized as \( l \), is like a "sub-address" for electrons in an atom. It provides details on the shape of the electron's orbital or which sublevel the electron occupies. This quantum number takes integer values from 0 to \( n-1 \) for a given \( n \).

Each integer value of \( l \) corresponds to a different orbital shape:

• \( l = 0 \) is called an 's' orbital
• \( l = 1 \) is a 'p' orbital
• \( l = 2 \) refers to a 'd' orbital
• \( l = 3 \) is known as an 'f' orbital

Understanding \( l \) is crucial for predicting the geometry and the orientation of the orbitals.

In the previously discussed 4f orbital designation, the 'f' implies \( l = 3 \). This means the orbital shape is more complex, often depicted in various lobes.

Orbital Designations

Orbital designations are a shorthand representation describing where an electron is likely to be found in an atom. These designations couple the principal quantum number \( n \) with the angular momentum quantum number \( l \).

Orbital designations consist of a number and a letter, like 3p or 4f:

• The number indicates the principal quantum number \( n \), establishing the energy level or shell of the electron.
• The letter corresponds to the angular momentum quantum number \( l \), signifying the shape of the orbital.

For example:- In the orbital 2s, '2' is the principal energy level, and 's' means \( l = 0 \), indicating a spherical shape.- In the 5d designation, '5' reveals that the electron is in the fifth energy level, while 'd' signifies \( l = 2 \), indicating a more complex, multi-lobed orbital.

Understanding orbital designations allows us to predict and understand the electron configuration and behavior of different elements in chemical reactions.