Problem 59 - Chemistry

Question

List the hydrogen orbitals in increasing order of energy.

Step-by-Step Solution

Answer

The hydrogen orbitals in increasing order of energy are 1s, 2s = 2p, 3s = 3p = 3d, and so on.

Step 1: Understanding Atomic Orbitals

An atomic orbital is a region in an atom where there is a high probability of finding an electron. Each orbital is associated with a specific energy level (principal quantum number, n) and shape (orbital type, l). Here, we have to list hydrogen atomic orbitals (which host electrons).

Step 2: Identifying Principal Quantum Numbers

The principal quantum number (n) tells us the energy level of the electron and the size of the orbital. The minimum value of 'n' is 1. Increasing values of n mean higher energy and the orbital is located further from the nucleus.

Step 3: Listing the Hydrogen Orbitals

Hydrogen atomic orbitals are typically labeled as 1s, 2s, 2p, 3s, 3p, 3d and so on, where the number refers to the principal quantum number n, and the letter refers to the type of orbital (s, p, or d). For hydrogen (or any one-electron ion), the energy of the orbital increases with n, regardless of the type of orbital. Therefore, the order of energy increasing order of hydrogen atomic orbitals is 1s, 2s = 2p, 3s = 3p = 3d, and so on.

Key Concepts

Principal Quantum Number

The principal quantum number, often denoted by the symbol \( n \), identifies the primary electron shell occupied by an electron. It serves as a crucial indicator of the energy level of an electron within an atom and directly correlates with the size of the atomic orbital.
While the principal quantum number can begin at 1 and extends upwards to higher integers, its value determines :

• The energy level: A higher \( n \) implies more energy.
• The distance from the nucleus: Larger \( n \) values indicate orbitals further from the nucleus.

In essence, as \( n \) increases, the electron is more energetically excited and resides further from the atomic nucleus. This plays a pivotal role in defining the electron distribution within atoms and impacts how they interact chemically with each other.

Hydrogen Atomic Orbitals

Hydrogen, with its single electron, provides a fundamental study model for atomic orbitals. For any hydrogen-like atoms, the orbitals are designated from lower to higher energy levels by combinations of the principal quantum number \( n \) and the letter designating the shape of the orbitals: \'s\' (spherical), \'p\' (dumbbell-shaped), \'d\', etc.
Each type of orbital has a distinct configuration:

• 1s: The smallest and most tightly bound, spherical orbital closest to the nucleus.
• 2s & 2p: Larger than 1s, providing more space for the electron, both occupying the second energy level.
• 3s, 3p & 3d: Further expanding orbitals found in the third energy level, offering more complex shapes.

In hydrogen, the energy of these orbitals solely depends on \( n \), illustrating a simple and calculable sequence of electron accommodation as \( n \) increases.

Orbital Energy Levels

In hydrogen atoms, the energy of orbitals is determined entirely by the principal quantum number \( n \). This uniqueness arises because hydrogen only contains one electron, eliminating electron-electron repulsion that complicates energy considerations in multi-electron atoms.
The predictable energy sequence of hydrogen orbitals is:

• 1s: The lowest energy level.
• 2s = 2p: The second energy level, indicative of equal energy for these orbitals.
• 3s = 3p = 3d: The third energy level, representing equivalent energy among them.

Each step up in \( n \) represents a quantized jump in energy. The ordered energy levels are foundational for predicting electron transitions and understanding hydrogen's atomic spectra, showcasing a staircase-like progression as \( n \) increases.