Question

(a) What are the similarities and differences between the \(1 s\) and \(2 s\) orbitals of the hydrogen atom? (b) In what sense does a \(2 p\) orbital have directional character? Compare the "directional" characteristics of the \(p_{x}\) and \(d_{x^{2}-y^{2}}\) orbitals (that is, in what direction or region of space is the electron density concentrated?). (c) What can you say about the average distance from the nucleus of an electron in a \(2 s\) orbital as compared with a \(3 s\) orbital? (d) For the hydrogen atom, list the following orbitals in order of increasing energy (that is, most stable ones first): \(4 f, 6 s, 3 d, 1 s, 2 p\).

Short answer

\(1 s\) and \(2 s\) orbitals are both spherically symmetric, but the \(2 s\) orbital is larger with a radial node. The \(2 p\) orbital has a directional character with electron density distribution concentrated in lobes along the axes. The \(p_{x}\) orbital has two lobes along the x-axis, while the \(d_{x^2-y^2}\) orbital has four lobes along the x and y axes. An electron in a \(3 s\) orbital will be on average further from the nucleus compared to a \(2 s\) orbital. In a hydrogen atom, the energy order is: \(1 s < 2 p < 3 d < 4 f < 6 s\).

Step-by-step solution

(a) Similarities and Differences between 1s and 2s Orbitals

Both 1s and 2s orbitals are spherically symmetric, meaning their electron density distribution does not have any directional preferences. They both belong to the s-orbital family as they have the same angular momentum quantum number (l = 0). The differences between 1s and 2s orbitals lie in their basic quantum numbers and their shapes. The principal quantum number (n) for a 1s orbital is 1, while for the 2s orbital, it is 2. The 2s orbital is larger than the 1s orbital, indicating that electrons in the 2s orbital are on average further from the nucleus compared to the 1s orbital. Additionally, unlike the 1s orbital, the 2s orbital has a radial node. This means there is a region of space where the electron density is zero (no electron can be found there).

(b) Directional Character of 2p Orbital

A 2p orbital has directional character because its electron density distribution is not spherically symmetric, unlike s orbitals. The 2p orbitals consist of three different types: \(p_x\), \(p_y\), and \(p_z\), which are aligned along the x, y, and z axes, respectively. The electron density distribution in a 2p orbital is concentrated at two opposite lobes along the respective axis. To compare the directional characteristics of the \(p_x\) and \(d_{x^2-y^2}\) orbitals, remember that the \(p_x\) orbital has electron density concentrated along the x-axis, while the \(d_{x^2-y^2}\) orbital has electron density concentrated along the x and y axes. The difference is that the \(d_{x^2-y^2}\) orbital has a cloverleaf-like shape with four lobes along the x and y axes, whereas the \(p_x\) orbital has a dumbbell shape with two lobes along x-axis.

(c) Average Distance from the Nucleus of an Electron in a 2s Orbital vs a 3s Orbital

The average distance of an electron from the nucleus in an s orbital is related to the principal quantum number (n). As n increases, the average distance from the nucleus also increases. Thus, an electron in a 3s orbital will be on average further from the nucleus compared to an electron in a 2s orbital.

(d) Ranking Orbitals by Energy

In a hydrogen atom, the energy of an orbital depends only on its principal quantum number n. The energy order of the orbitals will be based on their respective n values. Given orbitals: \(4f, 6s, 3d, 1s, 2p\) Order of principal quantum number (n): \(1s < 2p < 3d < 4f < 6s\) Therefore, the increasing energy order is: \(1s < 2p < 3d < 4f < 6s\)

Key concepts

Quantum Numbers

In the dazzling world of quantum mechanics, quantum numbers are critical for understanding the electron configurations of atoms. They are like the addresses that pinpoint the probable location of electrons within an atom. There are four quantum numbers: the principal quantum number (), the angular momentum quantum number (), the magnetic quantum number (), and the spin quantum number ().

Think of the principal quantum number, '', as the city where an electron resides—it tells us the energy level of an electron in an atom. The angular momentum quantum number, '', can be compared to the neighborhood in that city, indicating the shape of the orbital. The magnetic quantum number, '', pinpoints the exact street, giving the orientation of the orbital in space, and the spin quantum number, '', differentiates between two electrons in the same orbital, akin to specifying the apartment unit of each electron. Insight into quantum numbers is quintessential for students to grasp the complex ballet of electrons within an atom.

Electron Density Distribution

Imagine you're throwing a party, and you've scattered snacks in various rooms. Guests are most likely to be found where you've placed the snacks, right? In atomic terms, the electron density distribution is somewhat similar—it describes where electrons are likely to 'hang out' around the nucleus of an atom. This distribution is depicted by orbitals, the regions in space where there's a high probability of finding an electron.

For instance, the 1s orbital is like a balanced party room, with snacks evenly distributed, offering no favored direction—eletrons could be anywhere within a spherical region. But as we move to other orbitals, say the 2p or d orbitals, we get more 'themed rooms' with distinct shapes, like a dumbbell or a cloverleaf, indicating a more complex distribution of electron probability. Understanding this concept is like unraveling where the party crowd—in our case, electrons—prefers to congregate in the atom's vast space.

Angular Momentum Quantum Number

Dive into the atomic realm, and you'll encounter the angular momentum quantum number (l), a defining aspect of electron behavior. Think of it like the different dance styles at a club—the angular momentum quantum number gives us the type or the 'style' of orbital, which in turn dictates the electron's motion pattern. For s-orbitals, 'l' is 0, indicating a simple, non-directional (spherical) dance that electrons follow. P-orbitals have an 'l' of 1, with a more complex, two-lobed 'dance floor.'

As 'l' increments, we encounter d and f orbitals, where electrons exhibit even more intricate 'dance moves.' These moves represent the orbital's shape, from dumbbells (p-orbitals) to cloverleaves (d-orbitals) and beyond. Grasping this quantum number is essential for students to visualize the electron’s choreography within an atom.

Radial Node

If we're plotting a journey through an atom's electron cloud, we'd encounter regions called radial nodes, the no-go zones for electrons. These are spherical regions within an atom where the probability of finding an electron is exactly zero—it's as if electrons have taken a solemn oath to avoid these spaces. Each time the principal quantum number 'n' increases, more radial nodes appear in the orbital's landscape, much like increasing the number of barriers in a maze game.

For example, while the 1s orbital is node-free, a 2s orbital incorporates one radial node—a spherical shell where an electron will never be found. Understanding radial nodes helps students visualize the internal structure of an atom's orbitals and the layered complexity of electron distribution.

Directional Character of Orbitals

Not all atomic orbitals are born equal—some have distinct directional character. This is the property that describes how an orbital's shape and electron density distribution give it a preferred direction in three-dimensional space. Consider the s orbitals as the communism of electron density—equal in all directions. In contrast, p and d orbitals are the capitalists, with wealth (electron density) amassed in specific directional 'lobes.'

A 2p orbital, for example, has a dumbbell shape extending along an axis, showcasing its directional preference. The lobes of a orbital aim along the x-axis, while those of a orbital span the x and y axes. This characteristic is pivotal in bonding, as atoms connect via these 'high-density' avenues. When students understand this concept, they unlock the secrets of how atoms interact to form the molecular structures that make up our world.