Question

A \(3 s\) orbital is illustrated here. Using this as a reference to show the relative size of the other four orbitals, answer the following questions.(a) Which orbital has the greatest value of \(n ?\) (b) How many orbitals have a value of \(\ell=1 ?(\mathrm{c})\) How many other orbitals with the same value of \(n\) would have the same general shape as orbital (b)?

Short answer

(a) n = 5, (b) 12 orbitals, (c) 3 orbitals.

Step-by-step solution

Understanding Quantum Numbers

Quantum numbers describe the size, shape, and orientation of orbitals. The principal quantum number, \(n\), determines the energy level and relative size of the orbital. The azimuthal quantum number, \(\ell\), determines the shape of the orbital. For example, \(\ell = 0\) corresponds to "s" orbitals, \(\ell = 1\) corresponds to "p" orbitals, and so on.

Identifying Orbital with Greatest n

The principal quantum number \(n\) is associated with the overall size and energy of the orbital. Among the given orbitals, the one with the greatest value of \(n\) is the largest orbital with the highest energy. Here, the question hints that there are other orbitals compared to a \(3s\) orbital, typically including higher \(n\) values like \(4s\), \(4p\), or \(5s\). The largest will have the greatest \(n\), which is \(5s\), thus \(n = 5\).

Determining Orbitals with \(\ell = 1\)

The azimuthal quantum number \(\ell = 1\) indicates "p" orbitals. For each value of \(n\) greater than or equal to 2, there are three p orbitals (for \(m_\ell = -1, 0, 1\)). Considering the typical sequence of orbitals up to \(n = 5\), orbitals with \(\ell = 1\) can be found in the \(2p\), \(3p\), \(4p\), and \(5p\) sets. Thus, there are 12 orbitals in total with \(\ell = 1\).

Counting Similar Orbital Shapes at Same n

With the given \(3s\) orbital, observe that it's an "s" orbital (\(\ell = 0\)). For \(3p\) orbitals (\(\ell = 1\)), there are three \(3p\) orbitals having the same value of \(n = 3\). Since the question relates to this specific \(n\), there are three orbitals of the same shape (p orbitals) when \(n = 3\), having \(\ell = 1\).

Key concepts

Principal Quantum Number

The principal quantum number, denoted as \(n\), is a fundamental number in quantum mechanics that helps describe the overall energy and size of an atom's orbital. It's essentially like the floor level in a building. If you imagine electrons as residents, the principal quantum number defines how high their floor is within the "atom building".
Let's break this down:

• The value of \(n\) can be any positive integer (1, 2, 3, etc.).
• Each increase in \(n\) represents an increase in energy level and usually corresponds to a larger orbital size.
• For example, a \(1s\) orbital is closer to the nucleus compared to a \(2s\) or \(3s\) orbital.

When comparing orbitals, the one with the greatest \(n\) will generally be the largest and have the highest energy. So, in our case, if we're asked which orbital has the largest \(n\) beyond a given \(3s\) orbital, the largest \(n\) would be something like \(5s\), indicating it's the orbital with the greatest energy and relative size.

Azimuthal Quantum Number

The azimuthal quantum number, denoted by \(\ell\), plays a crucial role in determining the shape of an orbital. Think of this as the architectural design of each floor in our "atom building" analogy. The value of \(\ell\) defines the symmetry and form of the orbitals on each level.
Here’s a quick guide:

• \(\ell = 0\) represents "s" orbitals, which are spherical.
• \(\ell = 1\) represents "p" orbitals, which have a dumbbell shape.
• \(\ell = 2\) represents "d" orbitals, which are more complex.
• \(\ell = 3\) represents "f" orbitals, which are even more intricate.

For orbitals with \(\ell = 1\), known as "p" orbitals, these exist from \(n = 2\) onward. At each energy level, there are three different p orbitals due to the possible values of \(m_\ell\), the magnetic quantum number. So, at energy level \(n = 2\), for instance, the \(2p\) orbitals include \(2p_x\), \(2p_y\), and \(2p_z\) orbitals, and this pattern repeats for higher levels.

Orbital Shapes

Exploring orbital shapes is like examining the unique designs that make up the structure of atoms. The shapes of these orbitals define not only the space where electrons can be found but also influence how atoms interact with each other through bonds.
Here’s a detailed look at common orbital shapes corresponding to different azimuthal quantum numbers:

• **"s" Orbitals (\(\ell = 0\))**: Spherical in shape, these orbitals are non-directional and simply look like a ball around the nucleus. This shape means electrons are equally likely to be found in any direction relative to the nucleus.
• **"p" Orbitals (\(\ell = 1\))**: Have a dumbbell shape, with lobes on opposite sides of the nucleus. These orbitals are directional, meaning they have specific orientations (like \(p_x, p_y, p_z\)) along the x, y, or z axis.
• **"d" Orbitals (\(\ell = 2\))**: Generally more complex, these orbitals can resemble clover leaves or other lobe patterns, often extending between the axes rather than along them.
• **"f" Orbitals (\(\ell = 3\))**: Involve even more lobes and unique shapes, contributing to the complexity of this orbital, especially in larger atoms.

These shapes influence an atom's chemical properties and reactivity. Understanding them allows chemists to predict molecular behavior and bonding patterns in different chemical reactions.