0.5 Electron orbitals and electron configurations in atoms  (Page 4/10)

Observation 2: electron orbitals in hydrogen atoms

Quantum mechanics tells us that the motion of the electron in a hydrogen atom is described by a function, often called the “wave function” or the “electron orbital” and typically designated by the symbol Ψ. The electron orbital is the best information we can get about the motion of the electron about the nucleus. For a particular position (x,y,z) in the space about the nucleus, quantum mechanics tells us that |Ψ| ² is the probability of observing the electron at the location (x,y,z). The uncertainty principle we worked out above tells us that the probability distribution is the most we can know about the electron’s motion. In a hydrogen atom, it is most common to describe the position of the electron not with (x,y,z) but rather with coordinates that tell us how far the electron is from the nucleus, r , and what the two angles which locate the electron, θ and φ. We won’t worry much about these angles, but it will be valuable to look at the probability for the distance of the electron from the nucleus, r .

There isn’t just one electron orbital for the electron in a hydrogen atom. Instead, quantum mechanics tells us that there are a number of different ways for the electron to move, each one described by its own electron orbital, Ψ. Each electron orbital has an associated constant value of the energy of the electron, E _n . This agrees perfectly with our earlier conclusions in the previous Concept Development Study. In fact, quantum mechanics exactly predicts the energy levels and the hydrogen atom spectrum we observe.

The energy of an electron in an orbital is determined primarily by two characteristics of the orbital. The first characteristic determines the average strength of the attraction of the electron to the nucleus, which is given by the potential energy in Coulomb’s law. An orbital which has a high probability for the electron to have a low potential energy will have a low total energy. This makes sense. For example, as we shall see shortly, the lowest energy orbital for the electron in a hydrogen atom has most of its probability near the nucleus. By Coulomb’s law, the potential energy for the attraction of the electron to the nucleus is lower when the electron is nearer the nucleus. In atoms with more than one electron, these electrons will also repel each other according to Coulomb’s law. This electron-electron repulsion also adds to the potential energy, since Coulomb’s law tells us that the potential energy is higher when like charges repel each other.

The second orbital characteristic determines the contribution of kinetic energy to the total energy. This contribution is more subtle than the potential energy and Coulomb’s law. As a consequence of the uncertainty principle, quantum mechanics predicts that, the more confined an electron is to a smaller region of space, the higher its average kinetic energy must be. Remember that we cannot measure the position of electron precisely, and we define the uncertainty in the measurement as ∆ x . This means that the position of the electron within a range of positions, and the width of that range is ∆ x . Quantum mechanics also tells us that we cannot measure the momentum of an electron precisely either, so there is an uncertainty ∆ p in the momentum. In mathematical detail, the uncertainty principle states that these uncertainties are related by an inequality: