(a) With reference to Figure 6.19, what is the relationship...

(a) With reference to Figure $6.19,$ what is the relationship between the number of nodes in an $s$ orbital and the value of the principal quantum number? (b) Identify the number of nodes; that is, identify places where the electron density is zero, in the 2$p_{x}$ orbital; in the 3 s orbital. (c) What information is obtained from the radial probability functions in Figure 6.19 (d) For the hydrogen atom, list the following orbitals in order of increasing energy: $3 s, 2 s, 2 p, 5 s, 4 d .$

Key Concepts

Principal Quantum Number

The principal quantum number (n) defines the primary energy level and size of an orbital in an atom. It not only indicates the energy of the orbital in a hydrogen-like atom but also determines the number of nodes present in the electron's wavefunction. In general, the total number of nodes in an orbital equals n–1, which, for s orbitals, translates entirely into radial nodes because they lack angular nodes.

Electron Nodes

Nodes are regions in space where the probability of finding an electron is zero. There are two types of nodes: radial nodes, which depend on the distance from the nucleus, and angular nodes, which depend on the direction in space. The count and types of nodes in an orbital provide important insights into the orbital’s shape and energy characteristics.

Radial Nodes

Radial nodes occur at specific distances from the nucleus where the radial part of the electron’s wavefunction goes to zero. For orbitals like the s orbital, which have spherical symmetry, all nodes are radial. Their number can be calculated as n–l–1, where l is the angular momentum quantum number. In s orbitals (l = 0), this simplifies to n–1.

Angular Nodes

Angular nodes are surfaces where the angular part of the wavefunction is zero, resulting in regions of zero electron density that are not dependent on distance from the nucleus. They occur in orbitals with nonzero angular momentum (l > 0), such as p, d, or f orbitals. For example, a p orbital (l = 1) inherently has one angular node, which gives it its characteristic dumbbell shape.

Radial Probability Function

The radial probability function describes how the likelihood of finding an electron varies as a function of distance from the nucleus. By indicating where the electron is most likely to be found, these functions provide important information about the size and shape of orbitals. Peaks in the radial probability distribution reveal the most probable distances at which an electron can be located, while nodes in the function correspond to distances where the electron is unlikely or forbidden to be found.

Orbital Energy Ordering

Orbital energy ordering refers to the ranking of orbitals based on their energy levels. In the hydrogen atom, energy levels depend solely on the principal quantum number, meaning that orbitals with the same n are degenerate, regardless of their shape. However, when comparing orbitals with different n values, the orbital with the lower principal quantum number generally has lower energy. This concept becomes more complex in multi-electron atoms due to electron–electron interactions and screening effects.

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