Problem 10 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

The total number of orbitals associated with the principal quantum number 5 is : [Main Online April 9, 2016] (a) 20 (b) 25 (c) 10 (d) 5

Step-by-Step Solution

Answer

The total number of orbitals is 25.

Step 1: Understanding Principal Quantum Numbers

The principal quantum number, denoted by \( n \), signifies the main energy level or shell of an electron within an atom. It indicates the size and energy of the orbital.

Step 2: Calculating Number of Orbitals

The number of orbitals in a given energy level \( n \) can be calculated using the formula \( n^2 \). For \( n = 5 \), the number of orbitals is \( 5^2 = 25 \).

Step 3: Confirming the Total Orbitals

Verify the calculation by listing the types of orbitals. At \( n=5 \), the possible orbitals are: 1s, 3p, 5d, 7f, and 9g, which total to \( 1 + 3 + 5 + 7 + 9 = 25 \). Therefore, there are indeed 25 orbitals associated with \( n=5 \).

Key Concepts

Principal Quantum Number

The principal quantum number, denoted as \( n \), is a fundamental concept in understanding the arrangement of electrons in atoms. It represents the main energy level, or shell, of an electron. Essentially, \( n \) tells us about the electron's distance from the nucleus, which consequently affects the size and energy of the electron's orbital. For example, the higher the principal quantum number, the further the electron is from the nucleus, and the larger and more energetic the electron's orbital is.

Each principal quantum number corresponds to an electron shell, and these vary from \( n=1 \) to higher values potentially going infinitely, though practical values are usually low. The principal quantum number plays a crucial role in determining not just the number of orbitals, but also the types of orbitals within that energy level.

Orbitals

Orbitals are regions around the nucleus where electrons are most likely to be found. The shape and number of these orbitals are defined by quantum numbers. When talking about orbitals, it's important to remember:

• Each orbital can hold a maximum of two electrons.
• The type of orbital (s, p, d, f, etc.) influences its shape and energy level.
• Orbitals are categorized based on the principal quantum number \( n \) and the azimuthal quantum number \( l \), where \( l \) defines the shape (0 = s, 1 = p, 2 = d, etc.).

The number of orbitals available for a particular principal quantum number \( n \) can be calculated using the formula \( n^2 \). For instance, when \( n = 5 \), there are \( 5^2 = 25 \) orbitals.

These various types of orbitals combine uniquely within each electron shell to accommodate electrons while minimizing energy and maintaining the atom's stability.

Electron Shells

Electron shells are groups of atomic orbitals with the same principal quantum number \( n \). These shells represent energy levels surrounding the nucleus.

In each shell, different types of orbitals (s, p, d, f, etc.) coexist. For instance, when \( n = 5 \), the electron shell will include:

• One 5s orbital
• Three 5p orbitals
• Five 5d orbitals
• Seven 5f orbitals
• Nine 5g orbitals

These contribute to a total of 25 orbitals.
Electrons within these shells possess quantized energy states, which means they can only exist in specific energy levels. This quantization is essential as it determines an atom's chemical behavior and how it bonds with other atoms.

Overall, electron shells provide the structure necessary for understanding how electrons arrange themselves in an atom, creating patterns that are observable in the periodic table. This organization helps interpret the chemical properties and reactivity of elements.