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
53 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, θand 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 dierent 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, En. 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
rst 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 con ned 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 de ne 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:
(∆x)(∆p)≥4πh
This inequality reveals that, when an electron moves in a small area with a correspondingly small uncer-tainty ∆x, the unceruncer-tainty in the momentum ∆p must be large. For ∆p to be large, the momentum must also be large, the electron must be moving with high speed, and so the kinetic energy must be high. (We won't need to use this inequality for calculations, but it is good to know that h is Planck's constant, 6.62×10-34 J·sec. We have previously seen Planck's constant in Einstein's equation for the energy of a photon.)
From the uncertainty principle we learn that the more compact an orbital is, the higher the kinetic energy will be for an electron in that orbital. If the electron's movement is con ned to a small region in space, its kinetic energy must be high. This extra kinetic energy is sometimes called the con nement energy, and it is comparable in size to the average potential energy of electron-nuclear attraction. Therefore, in general, an electron orbital provides an energy compromise, somewhat localizing the electron in regions of low potential energy but somewhat delocalizing it to lower its con nement energy.
What do these orbitals look like? In other words, other than the energy, what can we know about the motion of an electron from these orbitals? Quantum mechanics tell us that each electron orbital is given an identi cation, essentially a name, that consists of three integers, n, l, and m, often called quantum numbers.
The rst quantum number n tells us something about the size of the orbital. The larger the value of n, the more spread out the orbital is around the nucleus, and therefore the more space the electron has to move in.
n must be a positive integer (1, 2, 3, . . .), so the smallest possible n is 1. In a hydrogen atom, this quantum number n is the same one that tells us the energy of the electron in the orbital, En.
The second quantum number, l, tells us something about the shape of the orbital. There are only a
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54 CHAPTER 6. ELECTRON ORBITALS AND ELECTRON CONFIGURATIONS IN ATOMS handful of orbital shapes that we nd in atoms, and we'll only need to know two of these for now. l is a positive integer or 0, and it must be smaller than the value of n for the orbital. For example, if n is 2, l must be less than 2, so l can be either 0 or 1. In general, l must be an integer from the set (0, 1, 2, ... n-1).
Each value of l gives us a dierent orbital shape. If l = 0, the shape of the orbital is a sphere. Since the orbital tells us the probability for where the electron might be observed, a spherical orbital means that the electron is equally likely to be observed at any angle about the nucleus. There isn't a preferred direction.
Since there are only a few shapes of orbitals, each shape is given a one letter name to help us remember.
In the case of the l = 0 orbital with a spherical shape, this one letter name is s. (As an historical note,
s doesn't actually stand for sphere; it stands for strong. But that doesn't mean we can't use s as a way to remember that the s orbital is spherical.) Figure 6.5 is an illustration of the spherical shape of the s orbital.
Figure 6.5: Approximate probability distribution of the 1s orbital shown with axes to emphasize the 3-dimensional spherical shape.
For now, the only other shape we will worry about corresponds to l = 1. In this case, the orbital is not spherical at all. Instead, it consists of two clouds or lobes on opposite sides of the nucleus, as in Figure 6.6.
This type of orbital is given the one letter name p, which actually stands for principal. Looking at the p orbitals in Figure 6.6, it is reasonable to ask what direction the two lobes are pointing in. If it seems that the lobes could point in any of three directions, that is correct. There are three p orbitals for each value of n, each one pointed along a dierent axis, x, y, or z. The third quantum number m gives each orbital a name allowing us to distinguish between the three orbitals pointing perpendicularly to each other. In general, the
55 m quantum number must be an integer between l and +l. When l = 1, m can be -1, 0 or 1. This is why there are three p orbitals.
Figure 6.6: Approximate probability distributions of p orbitals shown along the x, y, and z axes.
Although we will not worry about the shape of the l = 2 orbitals for now, there are two things to know about them. First, these orbitals are given the single letter name d. Second there are ve d orbitals for each n value, since l = 2 and therefore m can be -2, -1, 0, 1, or 2.
Chemists describe each unique orbital with a name which tells us the n and l quantum numbers. For example, if n = 2 and l = 0, we call this a 2s orbital. If n = 2 and l = 1, we call this a 2p orbital. Remember though that there are three 2p orbitals since there are three m values (-1, 0, 1) possible.
The motion of an electron in a hydrogen atom is then easily described by telling the quantum numbers or name associated with the orbital it is in. In our studies, an electron can only be in one orbital at a time, but there are many orbitals it might be in. If we refer to a 2p electron, we mean that the electron is in an orbital described by the quantum numbers n = 2 and l = 1. The n = 2 value tells us how large the orbital is, and the l = 1 value tells us the shape of the orbital. Knowing the orbital the electron is in is for now everything that we can know about the motion of the electron around the nucleus.