Problem 64 - Chemistry

Question

What is the total number of electrons that can be held in all orbitals having the same principal quantum number \(n ?\)

Step-by-Step Solution

Answer

The total number of electrons that can be held in all orbitals having the same principal quantum number (\(n\)) is given by the formula \(2n^2\). For instance, for \(n = 3\), the total number of electrons possible is 18.

Step 1: Understand the Principal Quantum Number

The principal quantum number, represented by the symbol \(n\), determines the energy level (or shell) in which an electron resides in an atom. It can have positive integer values of 1, 2, 3, and so on.

Step 2: Apply the Formula

The number of electrons that can be held in all orbitals having the same principal quantum number (\(n\)) is determined by the formula \(2n^2\). This formula is derived from the quantum mechanical model of the atom, which states that each energy level can hold up to \(2n^2\) electrons.

Step 3: Calculate the Number of Electrons

Suppose we want to know the number of electrons that can be held in all orbitals when \(n = 3\). We use the formula: \(2n^2\), replacing \(n\) with 3: \(2(3^2) = 2(9) = 18\). So, for \(n = 3\), there can be 18 electrons.

Key Concepts

Quantum Mechanics

Quantum mechanics serves as the backbone of our understanding of atomic structure, explaining how particles like electrons behave at the nanoscopic level. Unlike the classical mechanics of Isaac Newton, which can predict the path of a baseball or a planet, quantum mechanics reveals a world where probabilities replace certainty. Electrons, for instance, are not found in fixed orbits but rather in 'clouds' of probabilities called orbitals, which describe where an electron is likely to be found.

Understanding quantum mechanics is fundamental for solving problems like determining the number of electrons in orbitals. Each principal quantum level () represents a discrete energy level and can host a specific number of sublevels, which further divides into orbitals. These orbitals vary in shape and orientation, providing a distinct three-dimensional space where electrons can reside.

Atomic Orbitals

Atomic orbitals are not just simple placeholders for electrons but intricate three-dimensional regions where the probability of finding an electron is highest. Each orbital is associated with a specific principal quantum number (), angular momentum quantum number (), and magnetic quantum number, which dictates their shape, subshell, and orientation in space respectively.

Atomic orbitals come in various shapes, such as spherically symmetric s-orbitals and dumbbell-shaped p-orbitals. The higher the principal quantum number, the more complex the orbital shapes become, including d and f orbitals. For a given value of , there are ^2 atomic orbitals, revealing the multi-faceted nature of electron arrangement within an atom. It's crucial to visualize orbitals as dynamic entities of an atom's electron configuration rather than fixed paths.

Electron Configuration

Electron configuration is the roadmap that outlines how electrons are distributed in an atom's orbitals. The arrangement of electrons across various orbitals follows a set of principles such as the Aufbau principle, Pauli exclusion principle, and Hund's rule to minimize the energy of the atom.

Electron configuration is often notated with the help of spectroscopic notation, which combines the principal quantum number with a letter representing the type of orbital (s, p, d, f), and a superscript number showing the amount of electrons in those orbitals. For example, the notation 2p^4 indicates that there are four electrons in the p orbital of the second energy level. An understanding of electron configuration isn't just about writing out electrons' addresses within an atom; it's about grasping how electrons fill these orbitals in an energy-efficient manner to give atoms their unique properties.