Question

Discuss the similarities and differences between a \(1 s\) and a \(2 s\) orbital.

Short answer

The \(1s\) and \(2s\) orbitals are similar in shape and capacity but differ in energy, size, and presence of nodes.

Step-by-step solution

Define the Orbitals

An orbital is a region in an atom where there is a high probability of finding an electron. The designation "1s" refers to a spherical orbital at the first energy level closest to the nucleus, while "2s" is a spherical orbital at the second energy level slightly farther from the nucleus.

Similarities Between \(1s\) and \(2s\) Orbitals

Both the \(1s\) and \(2s\) orbitals are spherical in shape, meaning they resemble a three-dimensional circle around the nucleus. They are both part of the s-subshell category and can hold a maximum of 2 electrons.

Differences in Energy Levels

The main difference between the \(1s\) and \(2s\) orbitals is their energy levels. The \(1s\) orbital is at the lowest energy level, closest to the nucleus, while the \(2s\) orbital is at a higher energy level, indicating it is further away from the nucleus and requires more energy.

Differences in Size and Node Presence

The \(2s\) orbital is larger in size compared to the \(1s\) orbital because it is in the second energy level, further out from the nucleus. Moreover, the \(2s\) orbital has a radial node, a region where the probability of finding an electron is zero, whereas the \(1s\) orbital does not have any nodes.

Key concepts

s-orbital

The s-orbital is one of the fundamental types of atomic orbitals in quantum chemistry. This orbital has a characteristic spherical shape, meaning it encloses the nucleus in a perfect three-dimensional sphere. As with all orbitals, the s-orbital represents a region in space where electrons are most likely found.
The s-orbitals are part of the s-subshell and they include the following properties:

• Spherical shape: because of their shape, s-orbitals have no angular nodes unlike the p, d, or f orbitals which have more complex shapes.
• Capacity: Every s-orbital can hold up to two electrons.
• Identification: The subscript number indicates the principal energy level, for instance, 1s or 2s, signifying their distance from the nucleus.

Understanding the s-orbital is crucial as it forms the basic building block upon which more complex orbitals are conceptualized.

energy levels

Energy levels in an atom refer to the fixed distances from the nucleus where electrons can orbit. These levels are quantized, meaning electrons can only exist at particular energy states, and not in between them. In the context of orbitals:

• The 1s orbital is the closest to the nucleus and possesses the lowest energy level since its electrons are tightly bound.
• The 2s orbital is further from the nucleus, representing a higher energy level. Electrons here have more potential energy compared to those in a 1s orbital.

Energy levels are denoted by the principal quantum number, labeled as 'n'. For example, n=1 for the 1s orbital and n=2 for the 2s orbital.
This principle helps understand why electrons fill orbitals closest to the nucleus first, obeying the "lowest energy first" rule known as the Aufbau principle.

electron configuration

Electron configuration is a way of arranging electrons in an atom's available orbitals. It's like planning a seating arrangement, where electrons fill available lower energy seats first. For example, hydrogen is expressed as 1s¹, indicating that it has a single electron in the 1s orbital.

• The s-orbitals are filled before other types of orbitals, due to their lower energy.
• Each orbital follows Pauli Exclusion Principle, allowing only two electrons with opposite spins.
• Configuration follows Hund's rule, where each electron will occupy lower energy orbital singly before pairing up.

Understanding electron configuration is instrumental in predicting the chemical behavior and reactivity of elements on the periodic table.

radial nodes

Radial nodes are specific to the three-dimensional shapes of orbitals and represent areas where there is zero probability of finding an electron. In simple terms, they are like "gaps" or "holes" in an orbital.
For s-orbitals:

• 1s orbital has no radial nodes because it is the first energy level.
• 2s orbital has one radial node due to its higher energy level (n=2), reducing the probability of electron presence in certain regions.

Radial nodes are crucial because they help define the size and energy of the orbital. The number of radial nodes is calculated by the formula: \[\text{number of radial nodes} = n - l - 1\]where \(n\) is the principal quantum number and \(l\) is the azimuthal quantum number. Understanding nodes are essential for visualizing quantum mechanical properties of electrons.