In this chapter, we will explore the quantum mechanical model for where electrons reside in an atom. We will examine the electronic transitions, the events when an electron moves from one energy level to another. If the transition is to a higher energy level, energy is absorbed, and the energy change has a positive value. To obtain the amount of energy necessary for the transition to a higher energy level, a photon is absorbed by the atom. A transition to a lower energy level involves a release of energy, and the energy change is negative. This process is accompanied by emission of a photon by the atom.

The quantum mechanical model for where electrons reside in an atom can be used to look at electronic transitions, the events when an electron moves from one energy level to another. If the transition is to a higher energy level, energy is absorbed, and the energy change has a positive value. To obtain the amount of energy necessary for the transition to a higher energy level, a photon is absorbed by the atom. A transition to a lower energy level involves a release of energy, and the energy change is negative. This process is accompanied by emission of a photon by the atom.

The principal quantum number is one of three quantum numbers used to characterize an orbital. An atomic orbital is a general region in an atom within which an electron is most probable to reside. The quantum mechanical model specifies the probability of finding an electron in the three-dimensional space around the nucleus and is based on solutions of the Schrödinger equation. In addition, the principal quantum number defines the energy of an electron in a hydrogen or hydrogen-like atom or an ion and the general region in which discrete energy levels of electrons in a multi-electron atom and ions are located.

Another quantum number is the secondary quantum number, which is an integer that may take the values, l = 0, 1, 2, etc. This means that an orbital with n = 1 can have only one value of l = 0, whereas n = 2 permits l = 0 and l = 1. Whereas the principal quantum number, n, defines the general size and energy of the orbital, the secondary quantum number specifies the shape of the orbital. Orbitals with the same value of l have a spherical distribution, while those with different values of l have a dumbbell shape.

There are certain distances from the nucleus at which the probability density of finding an electron located at a particular orbital is zero. In other words, the value of the wavefunction ψ(r) is zero at this distance for this orbital. Such a distance is called a radial node. The number of radial nodes in an orbital is n … 1.

Consider the examples in Figure 10.5, which show the probability density of finding an electron for the s, p, d, and f orbitals as a function of distance from the nucleus. It can be seen from the graphs that there are 1 … 0 … 1 = 0 places where the density is zero for the s orbitals and 1 … 0 … 1 = 1 node for the p orbitals.

The s subshell electron density distribution is spherical, and the d subshell has a dumbbell shape. The d and f orbitals are more complex. These shapes represent the three-dimensional regions within which the electron is likely to be found.

In Figure 10.6, we see the shapes of the s, p, d, and f orbitals. The s subshell can only have one value of l = 0, while the p subshell can have three values of l = 1, 2, or 3. Similarly, the d subshell can have five values of l = 0, 1, 2, 3, or 4, and the f subshell can have seven values of l = -1, 0, 1, 2, 3, or 4.

The Pauli exclusion principle states that no two electrons in an atom can have the same set of quantum numbers. This means that each electron in an atom must occupy a unique orbital, and the probability density of finding an electron in a particular orbital is zero outside of that orbital.

In summary, the quantum mechanical model for electrons in atoms provides valuable insights into the behavior of electrons in atoms. The principal quantum number defines the size and energy of an orbital, while the secondary quantum number specifies its shape. The Pauli exclusion principle ensures that each electron occupies a unique orbital, and the probability density of finding an electron in a particular orbital is zero outside of that orbital.