Question

For each of the following pairs of hydrogen orbitals indicate which is higher in energy: (a) \(1 s, 2 s ;\) (b) \(2 p, 3 p\); (d) \(3 s, 3 d\); (c) \(3 d_{x y}, 3 d_{y z}\) (e) \(4 f, 5 s\).

Short answer

(a) 2s; (b) 3p; (c) 3d; (d) Equal; (e) 5s.

Step-by-step solution

Assign Energy Levels

In hydrogen atoms, the energy of an orbital depends only on the principal quantum number \(n\). Orbitals with a higher principal quantum number \(n\) have higher energy.

Compare Orbital Energies for Part (a)

For (a) \(1s\) vs \(2s\): The \(1s\) orbital has \(n = 1\) and the \(2s\) orbital has \(n = 2\). Therefore, \(2s\) is higher in energy than \(1s\).

Compare Orbital Energies for Part (b)

For (b) \(2p\) vs \(3p\): The \(2p\) orbital has \(n = 2\), and the \(3p\) orbital has \(n = 3\). Therefore, \(3p\) is higher in energy than \(2p\).

Compare Orbital Energies for Part (c)

For (c) \(3s\) vs \(3d\): Both the \(3s\) and the \(3d\) orbitals have \(n = 3\). However, within the same principal quantum level, \(s < p < d\) in energy. Thus, \(3d\) is higher in energy than \(3s\).

Compare Orbital Energies for Part (d)

For (d) \(3d_{xy}\) vs \(3d_{yz}\): Both orbitals are \(d\) orbitals within the same principal quantum number \(n = 3\). Therefore, they are of equal energy.

Compare Orbital Energies for Part (e)

For (e) \(4f\) vs \(5s\): The \(4f\) orbital has \(n = 4\), and the \(5s\) orbital has \(n = 5\). Therefore, \(5s\) is higher in energy than \(4f\).

Key concepts

Principal Quantum Number

The principal quantum number, denoted as \( n \), plays a crucial role in the world of atomic physics. It is an integer that signifies the energy level of an electron within an atom. In hydrogen atoms, the principal quantum number determines the energy and size of an orbital. The higher the \( n \), the higher the energy level and the larger the orbital.

In the context of the exercise given, the principal quantum number is used to compare different hydrogen orbitals. For example, if you are comparing a \( 1s \) orbital to a \( 2s \) orbital, the \( 2s \) has a higher principal quantum number \( (n = 2) \) compared to \( (n = 1) \) of \( 1s \), thus it is higher in energy. Understanding the principal quantum number is essential for grasping how electrons are arranged in atoms and how they interact with their environment.

Orbital Energy Comparison

To compare the energy of different orbitals, especially in hydrogen, the principal quantum number is the primary factor. In simple terms, hydrogen orbital energies depend solely on \( n \).

Here's a guideline for comparing energy levels:

• Higher \( n \) value means higher energy.
• For the same \( n \), different types of orbitals (s, p, d, f) follow the general trend: \( s < p < d < f \) in terms of energy.

For hydrogen, an important rule to remember is that orbitals belonging to the same principal quantum level, regardless of their type (e.g., \( 3s \) vs \( 3p \)), follow the energy sequence mentioned above. This approach is fundamental to solving problems involving hydrogen atom orbitals.

Atomic Orbitals

Atomic orbitals are regions within an atom where electrons are most likely to be found. Each atomic orbital is defined by a set of quantum numbers, with the principal quantum number \( n \) being one of them.

For hydrogen, orbitals are typically categorized by letters:

• \( s \) orbitals – spherical in shape
• \( p \) orbitals – dumbbell-shaped
• \( d \) orbitals – more complex shapes

Every type of orbital can hold a different maximum number of electrons:

• \( s \) orbitals can hold 2 electrons.
• \( p \) orbitals can hold 6 electrons.
• \( d \) orbitals can hold 10 electrons.

Understanding atomic orbitals is crucial for recognizing how electrons are organized in an atom, which in turn influences chemical bonding and the properties of elements.

Energy Levels

Energy levels, also known as electron shells, define the general energy distribution of electrons in an atom. These levels are broad categories that contain specific orbitals.
In the hydrogen atom, each energy level corresponds to a specific principal quantum number \( n \). The rule here is straightforward: the higher the energy level, the higher the energy. This means electrons in higher energy levels are further away from the nucleus.

Electrons tend to occupy the lowest available energy levels first (ground state) and can jump to higher levels (excited state) when they absorb energy. For instance, in scenarios like what was discussed in the exercise, electrons in the \( 3p \) orbital are at a higher energy level than those in the \( 2p \) orbital.

Orbital Subshells

Within each principal energy level, orbitals are grouped into subshells – namely, \( s \), \( p \), \( d \), and \( f \). These subshells have different shapes and energies.

The order of filling subshells with electrons, especially for hydrogen, follows specific rules:

• \( s \) subshells fill before \( p \) subshells of the same principal energy level.
• \( p \) fill before \( d \) in higher levels.

For example, a problem comparing \( 3s \) with \( 3d \) involves recognizing that \( 3d \) will have a higher energy compared to \( 3s \) even though they share the same principal quantum number. Understanding subshells is key to predicting how various elements might behave chemically.