(a) What are the similarities of 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? ( \(\mathbf{d}\) ) For the hydrogen atom, list the following orbitals in order of increasing energy (that is, most stable ones first): \(4 f, 6 s, 3 d, 1 s, 2 p\)

The 1s and 2s orbitals are both spherically symmetric, meaning that they have the same general shape. Both orbitals have a radial distribution function that depends only on the distance from the nucleus. However, there are some differences between the 1s and 2s orbitals. The 2s orbital has a higher energy than the 1s orbital, which means that an electron in the 2s orbital is, on average, farther from the nucleus compared to an electron in the 1s orbital. Additionally, the 2s orbital has a radial node, while the 1s orbital does not have any nodes.

The 2p orbitals have directional character because their electron density distribution is not spherically symmetric. Instead, the distribution is concentrated along specific axes. For the px orbital, the electron density is primarily along the x-axis; similarly, for the py and pz orbitals, the electron density is concentrated along the y-axis and z-axis, respectively. The directional characteristic of the px orbital is similar to the directional characteristic of the dx^2-y^2 orbital, as both orbitals show electron densities concentrated along specific axes. However, the dx^2-y^2 orbital has electron density concentrated along four axes instead of one, with the electron density being distributed along the x^2-y^2 plane.

When comparing the average distance of electrons in a 2s orbital to those in a 3s orbital, the electrons in the 3s orbital are, on average, farther from the nucleus. This is because the 3s orbital has a higher energy, and an electron in a higher energy orbital spends more time farther from the nucleus, on average, than an electron in a lower energy orbital.

The energy of an orbital in a hydrogen atom depends only on the principal quantum number (n). The order of the orbitals in terms of increasing energy (most stable first) is as follows: 1s (n=1) < 2p (n=2) < 3d (n=3) < 4f (n=4) < 6s (n=6)