0.5 Electron orbitals and electron configurations in atoms  (Page 5/10)

$\text{(∆}x\text{)(∆}p\text{)≥}\frac{h}{\mathrm{4\pi }}$

This inequality reveals that, when an electron moves in a small area with a correspondingly small uncertainty ∆x , the uncertainty 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 confined to a small region in space, its kinetic energy must be high. This extra kinetic energy is sometimes called the “confinement 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 confinement 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 identification, essentially a name, that consists of three integers, n, l, and m, often called “quantum numbers.” The first 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, E n .

The second quantum number, l , tells us something about the shape of the orbital. There are only a handful of orbital shapes that we find 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 different 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.) [link] is an illustration of the spherical shape of the s orbital.