Question

Make a chart of all allowable orbitals in the first four principal energy levels of the hydrogen atom. Designate each by type (e.g., \(s, p\) ), and indicate how many orbitals of each type there are.

Short answer

The first four principal energy levels have 1s, 2s/2p, 3s/3p/3d, and 4s/4p/4d/4f orbitals with 1, 1&3, 1&3&5, and 1&3&5&7 orbitals, respectively.

Step-by-step solution

Understand Principal Energy Levels

A principal energy level, denoted by the quantum number \( n \), specifies the energy level that the electron can occupy around the nucleus of an atom. In the hydrogen atom, each principal energy level can have various types of orbitals associated with it.

Identify Orbital Types in Each Energy Level

Each principal energy level \( n \) can contain different types of orbitals: - For \( n = 1 \), the type is \( 1s \)- For \( n = 2 \), the types are \( 2s, 2p \)- For \( n = 3 \), the types are \( 3s, 3p, 3d \)- For \( n = 4 \), the types are \( 4s, 4p, 4d, 4f \)

Determine the Number of Orbitals of Each Type

Each type of orbital can have a certain number of orbitals based on the geometric orientation:- The \( s \) orbital has 1 orientation.- The \( p \) orbitals have 3 orientations.- The \( d \) orbitals have 5 orientations.- The \( f \) orbitals have 7 orientations.

Compile into a Chart

Now, create a chart that specifies the number and type of orbitals for \( n = 1 \) to \( n = 4 \):- \( n = 1 \): - 1s: 1 orbital- \( n = 2 \): - 2s: 1 orbital - 2p: 3 orbitals- \( n = 3 \): - 3s: 1 orbital - 3p: 3 orbitals - 3d: 5 orbitals- \( n = 4 \): - 4s: 1 orbital - 4p: 3 orbitals - 4d: 5 orbitals - 4f: 7 orbitals

Key concepts

Principal Energy Levels

In the world of atomic structure, a principal energy level is like the ground floor, first floor, and so on, in a building. Scientists refer to these as shells or levels, and they mark them by a quantum number, \( n \). This number essentially tells us the energy level an electron resides in within an atom. For the hydrogen atom, these levels start from \( n = 1 \) and go upwards.

The principal energy level determines many things: the size of the electron cloud, its energy, and the number of sublevels within a shell. The higher the \( n \), the more energy the electron has.

Each increase in \( n \) signifies an additional level further from the nucleus, and with each step up, more types of orbitals become available for electrons to occupy.

Quantum Numbers

Quantum numbers are like the unique identification number for electrons in an atom. Imagine them as the address telling where exactly within the atom an electron is likely to be found. The principal quantum number \( n \) is one of four key quantum numbers, each serving a different function.

• \( n \): It tells us the shell number or the energy level.
• \( l \): The azimuthal quantum number indicates the type of orbital (\(s, p, d, \) or \( f \)).
• \( m_l \): The magnetic quantum number describes the specific sub-orbital within a type of orbital.
• \( m_s \): The spin quantum number specifies the spin direction of the electron (either up or down).

Each quantum number adds more detail, narrowing down the position and energy characteristics of the electron, from the size of its path to the shape of its orbital.

Types of Orbitals

As we delve into the atomic structure, understanding the types of orbitals is crucial. Orbitals are regions in space where the probability of finding an electron is high. Different types of orbitals have distinct shapes and energy levels.

• \( s \) Orbital: Spherical in shape, it has a single orientation and can hold a maximum of two electrons.
• \( p \) Orbitals: Dumbbell-shaped, these have three orientations, labeled as \( p_x, p_y, \) and \( p_z \), accommodating six electrons in total.
• \( d \) Orbitals: More complex in shape, these orbitals have five orientations and can hold ten electrons.
• \( f \) Orbitals: Even more intricate, with seven orientations, they have a capacity for fourteen electrons.

Understanding these shapes and orientations helps explain the distribution of electrons in an atom as well as their chemical properties.

Orbital Orientations

While each type of orbital has a distinct shape, the orientations within these shapes offer a deeper level of detail. Each orbital's unique orientation plays a role in determining how these electrons interact and bond with atoms.

• \( s \) Orbital: With its simple spherical shape, the \( s \) orbital has only one possible orientation, making it the simplest type.
• \( p \) Orbitals: These three orientations—\( p_x, p_y, \) and \( p_z \)—are key in forming molecular geometries that dictate how atoms bond with one another.
• \( d \) Orbitals: Five orientations provide a complex landscape for electron flow, impacting the shapes of compounds they form.
• \( f \) Orbitals: With seven orientations, these are less frequently discussed but are pivotal in heavier elements and complex bonding situations.

These orientations allow electrons to occupy the most stable and lowest energy positions possible, influencing everything from color to the behavior of materials.