Question

(a) What are the similarities of and differences between the 1s 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 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

(a) Both 1s and 2s orbitals are spherically symmetric s orbitals, but the 1s orbital has a lower energy level, no radial node, and is closer to the nucleus. (b) 2p orbitals have a directional character as their electron probability density is concentrated along specific axes, such as px along the x-axis. In comparison, px has electron density along the x-axis while dx²-y² is concentrated between x and y axes. (c) The average distance of electrons in a 3s orbital is greater than that in a 2s orbital. (d) The order of increasing energy for given orbitals is: 1s, 2p, 3d, 4f, 6s.

Step-by-step solution

(a) Similarities and differences between 1s and 2s orbitals

The 1s and 2s orbitals are both s orbitals, which means that they have a spherical shape and are symmetric around the nucleus. They share the same principal quantum number n, which represents the energy level of the orbitals. However, there are some differences between these orbitals: 1. The 1s orbital has a lower energy level and is closer to the nucleus than the 2s orbital, as it has a smaller principal quantum number, n=1, whereas the 2s orbital has n=2. 2. The 2s orbital has a radial node, where the electron's probability density goes to zero, while the 1s orbital does not have any radial node. 3. Due to their difference in energy levels, the 1s orbital is more stable and has a higher probability of containing an electron than the 2s orbital.

(b) Directional character of 2p orbitals and comparison with px and dx²-y² orbitals

The 2p orbitals have directional character because their electron probability density is concentrated along specific axes in space. There are three 2p orbitals: px, py, and pz, each aligned along the x, y, and z-axis, respectively. In contrast, the s orbitals are spherically symmetric and have no directional character. Now let's compare the directional characteristics of the px and dx²-y² orbitals. The px orbital is a p orbital, so it has electron density concentrated along the x-axis. On the other hand, the dx²-y² orbital is a d orbital, and its electron density is concentrated in regions between the x and y axes. Thus, while both orbitals exhibit directional character, the px orbital has its electron density concentrated along a single axis (x-axis), while the dx²-y² orbital has its electron density concentrated between the x and y axes.

(c) Comparing the average distance from the nucleus for electrons in 2s and 3s orbitals

The average distance of an electron from the nucleus can be determined by the principal quantum number (n). As n increases, the electron's distance from the nucleus also increases. Since the 2s orbital has n=2 and the 3s orbital has n=3, the average distance of an electron in a 3s orbital is greater than that in a 2s orbital.

(d) Ordering the orbitals in terms of increasing energy

For hydrogen atom, the energy of an orbital depends solely on the principal quantum number, n. Higher values of n represent higher energy levels. We can arrange the orbitals in this exercise in increasing order of their principal quantum numbers: 1. 1s (n=1) 2. 2p (n=2) 3. 3d (n=3) 4. 4f (n=4) 5. 6s (n=6) So, the order of increasing energy is: 1s, 2p, 3d, 4f, 6s.

Key concepts

Spherical and Directional Orbitals

In the exploration of hydrogen atom orbitals, it's fundamental to distinguish between spherical and directional orbitals. Spherical orbitals, or s orbitals, present a characteristic round shape centered around the nucleus. This symmetry means that the probability of finding an electron at a given distance is equal in all directions. For instance, both the 1s and 2s orbitals are spherical; however, the 2s orbital is larger and encompasses a radial node – a spherical shell where the probability of finding an electron is zero.

Directional orbitals, such as the p and d orbitals, are elongated in nature, displaying regions of high electron probability density along certain spatial axes. The p orbitals (subdivided into p_x, p_y, and p_z) concentrate electron density along the x, y, or z axes. Comparatively, d orbitals, including the d_x²-y² type, exhibit a clover-shaped pattern, with electron density between the axes. This allows us to visualize the directional nature of these orbitals, illustrating the non-uniform distribution of electron probability in a defined direction or region of space.

Quantum Numbers and Energy Levels

Quantum numbers are like an address system for electrons, indicating their location and energy status within an atom. Each electron in an atom is described by a set of four quantum numbers: the principal (n), angular momentum (l), magnetic (m_l), and spin (m_s) quantum numbers.

The principal quantum number, n, is particularly important when discussing energy levels; it determines the electron's energy level and distance from the nucleus. A smaller n means the electron is closer to the nucleus and in a lower energy state, while a larger n implies a higher energy level and typically a greater average distance from the nucleus. Using this framework, we deduced that, in the hydrogen atom, the 1s orbital is at the lowest energy level, followed orderly by the 2p, 3d, 4f, and 6s orbitals based on their principal quantum numbers.

Electron Probability Density

Electron probability density is a concept central to quantum mechanics, describing the likelihood of locating an electron in a specific region around the nucleus. It's often represented by complex shapes called orbitals, which are illustrations of where one is most likely to find an electron. These shapes are not definitive paths but rather 'clouds' where the presence of an electron is probable. For instance, in comparing the average distances of electrons in different orbitals, such as 2s and 3s, we notice that the electron in a 3s orbital has a higher average distance from the nucleus due to its greater principal quantum number.

This characteristic hints at the probabilistic nature of quantum mechanics, where higher energy levels mean electrons can be located further from the nucleus, but the precise position at any given moment isn't fixed. It's this probability density that dictates electron behavior and chemical bonding properties, essential for understanding atomic and molecular structures.