Question

What are the differences between the \(2 s\) orbital and the \(1 s\) orbital of hydrogen? How are they similar?

Short answer

The \(1s\) and \(2s\) orbitals of hydrogen differ in size, energy level, and the number of nodes. The \(2s\) orbital is larger and has a higher energy level than the \(1s\) orbital due to its higher principal quantum number (\(n=2\)), while the \(1s\) orbital has no nodes and the \(2s\) orbital has one radial node. The similarity between them is that both orbitals have a spherical shape, which is characteristic of \(s\) orbitals.

Step-by-step solution

Define Orbitals

Orbitals are regions of space around an atom's nucleus where electrons are most likely to be found. In the hydrogen atom, the simples model of orbitals is based on the Schrödinger equation, which describes the distribution of electron density around the nucleus. There are different types of orbitals, such as \(s\), \(p\), \(d\), and \(f\), each with a specific shape and energy level. In this exercise, we will focus on the \(s\) orbitals, specifically the \(1s\) and \(2s\) orbitals of hydrogen.

Shape of \(1s\) and \(2s\) Orbitals

The \(s\) orbitals have a spherical shape. Their electron density is distributed uniformly around the nucleus, so they appear as a sphere when portrayed in a 3D view. Both the \(1s\) and \(2s\) orbitals share this spherical shape, which is a similarity between them.

Size of \(1s\) and \(2s\) Orbitals

The size of an orbital is determined by its principal quantum number (\(n\)). A higher principal quantum number corresponds to a larger orbital. The \(1s\) orbital has a principal quantum number of \(n=1\), while the \(2s\) orbital has a principal quantum number of \(n=2\). Therefore, the \(2s\) orbital is larger in size than the \(1s\) orbital. This is a difference between the two orbitals.

Energy Level of \(1s\) and \(2s\) Orbitals

The energy level of an orbital is also determined by its principal quantum number (\(n\)). A higher principal quantum number corresponds to a higher energy level. The \(1s\) has a principal quantum number of \(n=1\), which means it has a lower energy than the \(2s\) orbital with a principal quantum number of \(n=2\). This is another difference between the two orbitals.

Node in \(1s\) and \(2s\) Orbitals

A node is a region where the probability of finding an electron is zero. The number of nodes in an orbital depends on its principal quantum number (\(n\)), and is given by \(n-1\). In this case, the \(1s\) orbital has no nodes, while the \(2s\) orbital has one radial node. This is a difference between the two orbitals.

Summary of Differences and Similarities

Differences: 1. The \(2s\) orbital is larger than the \(1s\) orbital due to its higher principal quantum number. 2. The \(2s\) orbital has a higher energy level than the \(1s\) orbital. 3. The \(1s\) orbital has no nodes, while the \(2s\) orbital has one radial node. Similarities: 1. Both the \(1s\) and \(2s\) orbitals possess a spherical shape, characteristic of \(s\) orbitals.