Question

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

Short answer

The total number of electrons is \( 2n^2 \) for a given principal quantum number \( n \).

Step-by-step solution

Understanding Principal Quantum Number

The principal quantum number, denoted by \( n \), indicates the energy level or shell of an electron in an atom. An energy level with principal quantum number \( n \) can hold different types of orbitals (s, p, d, f, etc.), depending on the value of \( n \).

Understanding Orbital Capacity

Each type of orbital can hold a specific number of electrons: one s orbital can hold 2 electrons, three p orbitals can hold a total of 6 electrons, five d orbitals can hold a total of 10 electrons, and seven f orbitals can hold a total of 14 electrons.

Determining the Number of Orbitals for Given n

For a given principal quantum number \( n \), the types of orbitals and their corresponding numbers are as follows: n=1 (1s), n=2 (2s, 2p), n=3 (3s, 3p, 3d), and n=4 (4s, 4p, 4d, 4f) and so on. The total number of orbitals in a shell is given by \( n^2 \).

Calculating Total Electron Capacity

The total number of electrons that can be held in orbitals with the same principal quantum number \( n \) is given by the formula \( 2n^2 \). This is because each orbital can hold 2 electrons, so with \( n^2 \) orbitals, the total capacity is \( 2n^2 \).

Application of Formula

If we need the total number of electrons for a specific \( n \), simply apply the formula \( 2n^2 \). For example, if \( n = 3 \), \( 2 \times 3^2 = 18 \) electrons can be held.

Key concepts

Electron Configuration

Electron configuration refers to the arrangement of electrons in an atom's orbitals. It offers insights into the distribution of electrons across various energy levels and sub-levels within an atom. Understanding the electron configuration of an element is crucial as it accounts for the chemical properties and reactivity of an element.

Each electron in an atom is described by a set of quantum numbers, with the principal quantum number ( ) being one of them. Electron configurations are often written in a notational form that begins with the principal quantum number, followed by the type of orbital (e.g., s, p, d, f). For instance, the electron configuration of helium is written as 1s², indicating that it has two electrons in the first energy level, specifically in the s orbital. This organized notation helps chemists and physicists to predict how atoms will interact in chemical reactions.

Understanding the electron configuration not only assists in identifying the placement of electrons but also makes it easier to determine the total number of electrons that an atom can hold within a certain principal quantum number.

Orbital Capacity

Orbital capacity is key to understanding how electrons fill the various orbitals in an atom. Each type of orbital - s, p, d, and f - can hold a different maximum number of electrons:

• The s orbital can contain up to 2 electrons.
• The p orbitals can contain a combined maximum of 6 electrons (3 orbitals, each holding 2 electrons).
• The d orbitals can accommodate a total of 10 electrons (5 orbitals, each with 2 electrons).
• The f orbitals can house up to 14 electrons (7 orbitals, each capable of supporting 2 electrons).

Knowing these capacities is essential when determining how many electrons can fit within a specific principal quantum number. For example, when the principal quantum number ( ) equals 2, the orbitals include one s orbital (2 electrons) and three p orbitals (6 electrons), totaling 8 electrons.

Energy Levels

Energy levels in atoms, denoted by the principal quantum number ( ), are significant in understanding how electrons are distributed. The energy levels represent distinct shells where electrons reside. These shells correspond to increasing values of the principal quantum number: 1, 2, 3, etc. At each level, specific sub-levels or orbitals can occur, all with differing energy states.

The number of orbitals in an energy level equals ², so each shell can hold a maximum of 2n² electrons. For instance, for =3, the third energy level harbors 9 orbitals (3² = 9) that can accommodate up to 18 electrons in total (2 x 3² = 18).

The pattern of filling, dictated by "Aufbau principle", states that electrons fill lower-energy orbitals first before advancing to higher energy ones. Comprehending these energy level arrangements helps to unravel why atoms engage in their particular chemical behavior.

Quantum Mechanics

Quantum mechanics forms the foundational science that articulates the behavior of electrons in atoms. It delineates the probabilistic nature of electron positions within orbitals, rather than fixed paths. The cornerstone of this theory involves quantum numbers - principal, azimuthal, magnetic, and spin - which together describe each electron's unique state in an atom.

The principal quantum number ( ) primarily determines the energy level of electrons, while the azimuthal quantum number specifies the shape of the orbital ( s, p, d, or f). In contrast, the magnetic quantum number identifies a particular orbital's orientation in space, and the spin quantum number distinguishes the electron's spin direction.

Understanding quantum mechanics allows insight into not only the structure of the atom but also provides the framework for predicting an atom's interaction in different chemical processes. Harnessing these concepts supports the foundational arrangement of the periodic table while clarifying bond formation and electron sharing among atoms.