Question

How is the (a) energy and (b) size of an \(n s\) atomic orbital affected by an increase in \(n ?\)

Short answer

Increasing \(n\) increases both the energy and size of \(n s\) orbitals.

Step-by-step solution

Understanding Quantum Number n

The principal quantum number, denoted as \(n\), determines the energy level or shell of an electron in an atom. The larger the value of \(n\), the higher the energy level the electron occupies.

Relation between Energy and n

As \(n\) increases, the energy of the \(n s\) orbital increases. Electrons in higher energy levels are farther from the nucleus due to reduced effective nuclear charge and increased distance.

Relation between Size and n

The size of the \(n s\) orbital is determined by the most probable distance from the nucleus. As \(n\) increases, the size of the orbital also increases, meaning the orbital extends further from the nucleus.

Summarizing the Effects

In conclusion, an increase in the principal quantum number \(n\) leads to both an increase in the energy and size of an \(n s\) atomic orbital.

Key concepts

Principal Quantum Number

The principal quantum number, symbolized as \( n \), plays a vital role in quantum mechanics by defining the main energy level or shell that an electron occupies within an atom. This number can take any positive integer value (e.g., 1, 2, 3, ...). Each increase in \( n \) denotes a higher energy level, meaning that the electron is located in a shell farther from the nucleus. This is why electrons with higher principal quantum numbers often have more energy and play a significant role in determining the chemical properties of an element.

Together with other quantum numbers, the principal quantum number helps categorize electrons in an atom and predict how they will interact with each other and external forces. When \( n \) increases, not only does the energy level go up, but the separation between energy levels becomes more prominent. This characteristic is essential for understanding electron transitions and emission spectrums.

Atomic Orbitals

Atomic orbitals are regions around an atom's nucleus where electrons are most likely to be found. Each orbital can hold a maximum of two electrons with opposite spins, thanks to the Pauli exclusion principle. These orbitals come in various shapes (such as s, p, d, f), with the simplest being the s orbital, which is spherical in shape.

The size and shape of atomic orbitals are influenced by the quantum numbers, especially \( n \). As \( n \) increases, the atom's orbitals become larger because the most probable location of the electron moves further away from the nucleus. In simple terms, higher \( n \) levels "stretch" the atom, allowing the electrons to exist in a larger space. This fact also explains why elements with larger atomic numbers tend to have more complex atomic structures.

Electron Energy Levels

Electron energy levels, also known as electron shells, describe the particular energy states in which electrons reside around an atom's nucleus. An increase in the principal quantum number \( n \) indicates a higher energy level and therefore higher energy electrons. Electrons in these energy levels are less tightly bound to the nucleus, allowing them to be more easily influenced or removed by external forces.

This directly impacts an atom's chemical reactivity; for example, elements with higher energy level electrons become more reactive. The configuration of these levels is elemental in understanding the emission or absorption spectra of atoms. When an atom absorbs energy, electrons can be "excited" to higher energy levels. Later, when they return to a lower energy state, they release energy in the form of light. The arrangement and interaction of electron energy levels underpin much of atomic behavior and chemical bonding.