Problem 40 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

The energy of the electron in the \(3 d\)-orbital is less than that in the \(4 s\)-orbital in the hydrogen atom.

Step-by-Step Solution

Answer

The energy in the \(3d\)-orbital is less than in the \(4s\)-orbital because \(n = 3 < n = 4\).

Step 1: Understanding Orbital Energies

In the hydrogen atom, the energy of an electron in an orbital depends only on the principal quantum number \(n\). The orbital with the lower \(n\) value has the lower energy.

Step 2: Identify Principal Quantum Numbers

For the \(3d\)-orbital, \(n = 3\), and for the \(4s\)-orbital, \(n = 4\). Therefore, the \(3d\)-orbital has a lower principal quantum number than the \(4s\)-orbital.

Step 3: Compare Orbital Energies

Since the energy levels in a hydrogen atom depend only on \(n\), the \(3d\)-orbital, with \(n = 3\), has lower energy than the \(4s\)-orbital with \(n = 4\).

Key Concepts

Principal Quantum Number

In the world of atomic physics, understanding the principal quantum number is pivotal. This quantum number is symbolized by the letter "n" and is essentially a positive integer. It plays a crucial role in determining the size and energy level of an atomic orbital.
The principal quantum number describes how far away an electron can be found from the nucleus. Lower values of "n" signify orbitals that are closer to the nucleus, while higher values place them further away. This number directly affects the energy of the electron; a lower principal quantum number means a lower energy level.
In hydrogen, solely the principal quantum number determines an orbital's energy. For example, the principal quantum number for the 3d orbital is 3, while it's 4 for the 4s orbital. This difference in "n" leads to significant energy differences between these orbitals.

Orbital Energy

Orbital energy is a key concept when analyzing atom electron configurations, especially for simpler elements like hydrogen. In multi-electron atoms, factors like electron shielding and repulsion complicate the energy levels. However, in hydrogen, things are uniquely straightforward.
The energy of an orbital within a hydrogen atom depends exclusively on the principal quantum number. This means orbitals with identical "n" values have equal energy levels, irrespective of their shape (s, p, d, etc.). Lower "n" values correspond to orbitals with lower energy.
This principle explains why, in a hydrogen atom, a 3d orbital has less energy than a 4s orbital. Here, the energy relation is simple: the 3d orbital's lower principal quantum number ("n = 3") makes it less energetic compared to the 4s orbital ("n = 4"). This differs from more complex atoms, where such clear-cut energy relations don't always hold.

Hydrogen Atom Electron Configuration

For hydrogen, the simplest atom, electron configuration lacks much of the complexity seen in heavier elements. Hydrogen's electron configuration is a fundamental topic in quantum mechanics.
Hydrogen has one electron, and its configuration begins and ends with this single electron entering the 1s orbital. The electron's energy in hydrogen depends solely on its principal quantum number and can occupy any orbital with "n" values in ascending order.
As a result, orbitals are filled in this orderly manner, and this is why in hydrogen, orbitals with lower "n" hold less energy. The sequential nature of this filling process—starting from 1s and extending to any other s, p, or d orbital—is a key part of understanding atomic structures. Understanding these patterns helps predict how electrons occupy available spaces in an atom.