Problem 57 - Chemistry The Central Science

Question

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

Step-by-Step Solution

Answer

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

Step 1: (a) 3p orbital

For the 3p orbital, the principal quantum number is already provided, which is n = 3. The letter "p" corresponds to the azimuthal quantum number value l = 1. Thus, for a 3p orbital, n = 3 and l = 1.

Step 2: (b) 2s orbital

For the 2s orbital, the principal quantum number is given as n = 2. The letter "s" corresponds to the azimuthal quantum number value l = 0. Thus, for a 2s orbital, n = 2 and l = 0.

Step 3: (c) 4f orbital

For the 4f orbital, the principal quantum number is given as n = 4. The letter "f" corresponds to the azimuthal quantum number value l = 3. Thus, for a 4f orbital, n = 4 and l = 3.

Step 4: (d) 5d orbital

For the 5d orbital, the principal quantum number is given as n = 5. The letter "d" corresponds to the azimuthal quantum number value l = 2. Thus, for a 5d orbital, n = 5 and l = 2.

Key Concepts

Principal Quantum Number

The principal quantum number, symbolized by the letter \( n \), is a fundamental physical constant that helps to define the size and energy level of an electron orbital. Imagine it as a kind of address system for electrons within an atom. Electrons reside in specific energy levels, which are determined by this number. The principal quantum number can take positive integer values such as 1, 2, 3, and so on. The larger the value of \( n \), the higher the energy level and further the electron is generally from the nucleus.

In the orbital designation "3p," the number "3" in front is the principal quantum number, which shows us that the electron is in the third energy level. The principal quantum number is crucial because it not only indicates the energy level but also the general size of the electron's orbit, with larger \( n \) values indicating larger orbitals.

Azimuthal Quantum Number

The azimuthal quantum number, represented by the letter \( l \), determines the shape of the electron's orbital, and sometimes people refer to it as the angular momentum quantum number. This quantum number can have integer values from 0 to \( n-1 \), where \( n \) is the principal quantum number.

Let's look at the alphabetic designation that represents different \( l \) values:

• "s" means \( l = 0 \)
• "p" means \( l = 1 \)
• "d" means \( l = 2 \)
• "f" means \( l = 3 \)

As an example, in a "3p" orbital, the "p" indicates that the azimuthal quantum number \( l \) is 1. This tells us the shape of the orbital. "s" orbitals are spherical, "p" orbitals have a dumbbell shape, and "d" and "f" orbitals have more complex shapes.

Orbital Designations

Orbital designations play an important role in describing the specific location and energy of an electron within an atom's shell or energy level. These designations combine numbers and letters to convey this information succinctly.

In an orbital designation like "3p," "3" represents the principal quantum number \( n \), while "p" designates the azimuthal quantum number \( l \).

Here's how to break down some common orbital designations:

• **3p**: \( n=3 \), \( l=1 \)
• **2s**: \( n=2 \), \( l=0 \)
• **4f**: \( n=4 \), \( l=3 \)
• **5d**: \( n=5 \), \( l=2 \)

Knowing these designations allows you to predict the behavior and properties of electrons. Each designation refers to a unique combination of energy level and orbital shape, essential for understanding the structure of an atom.