Forces
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Fritz London
London Forces: The glue that permits most things in the Universe to condense at sufficiently low temperatures is the attractive intermolecular force. Naturally, there are repulsive forces as well; try to crush solids to higher than their normal density, and they resist you because their electron charge clouds are overlapping, and similar charges repel. (That's the same force that keeps two cations apart, for example; Coulombic repulsion.) But those same electron clouds can instead and do lead to attraction in a subtle way first recognized by Fritz London.
The trick is that electrons, being nearly 2000 times lighter than the smallest nucleus, move much, much faster than the nuclei. The electrons that are least strongly bound to their nucleus (viz., the valence electrons) are often free to roam the entire molecule (molecular orbital), and even those well-bound to their nucleus can at least wash from one side of it to the other. All this wandering implies that the center of mass of the electronic charges will not always coincide with the center of mass of the protons!
That sounds like a recipe for a dipole. But it's not a permanent dipole since on average those centers of charge do overlap exactly for non-polar molecules. And that's what we're discussing first: non-polar molecules.
So even non-polar molecules have dipoles instantaneously, as their electrons fidget and squirm. What London proposed is that one molecule's instantaneous dipole induces its neighbors to fidget their own electrons into its opposite. Then opposites attract. But an attosecond later, the electrons are somewhere else. No matter. Whatever new instantaneous dipole has arisen is mirrored (in reverse) by its neighbors, and again there's an intermolecular attraction. In fact, the only time there's no attraction is when a molecule's electrons happen to configure themselves to zero out the dipole instantaneously.
Johannes van der Waals
Bear in mind that bigger molecules let their electrons roam further, increasing the instantaneous dipoles and concomitant attractions. And more electropositive (less electronegative) atoms hold their electrons less strongly, permitting them to wander more easily under the influence of induced dipoles or any external electrical field. So London forces increase down the Periodic Table (valence electrons further from the nucleus) and decrease to the right (just as electronegativity rises that way).
To see an animation of London forces at work, click the "Many Kinds"
marginal title to the left.
Intermolecular
Forces
Hydrogen Bonds: And the strongest kind of non-ionic attractions are dipoles of hydrogen bonded to extremely electronegative atoms like N, O, and F. They binding here can be about 10% as strong as even valence binding; so "hydrogen bonds" (not H-H but rather X-H...H-X where X is very electronegative) are often spoken of in the same breath as chemical bonding. Indeed, your DNA is held together with hydrogen bonds, a fact which explains how easily the cell can unzip the double helix to replicate.
The best and most ubiquitous example of hydrogen bonding, of course, is in liquid water. In fact, by all rights, a molecule as small as H2O ought to be a gas; after all, H2S is gaseous. But while S is electronegative (cS=2.5), it isn't that much more electronegative than H (cH=2.1). But oxygen (cO=3.5) over triples the electronegativity difference with hydrogen compared to sulfur; so H2S makes lousy hydrogen bonds compared to water's.
You could say that Life As We Know It owes its existence to the hydrogen bond both for the replication of DNA and the simple existence of liquid water at Earth-like temperatures!
MS Encarta 99
MS Encarta 99
While diamond represents a single-molecule solid, carbon's most common form, graphite is a solid bond into infinite planes of hexagons via sp2 hybridization. But the planes themselves are loosely aligned (a fact you prove each time you write with a pencil) by London forces.
For a comparison of the solid structures arising from different kinds of bindings, click on Purdue's Categories of Solids webpage.
Since these electrons in the metal solid can be anywhere, the metal conducts electricity. Since the MO overlaps are all bonding, the nuclei are content to be a solid.
For a description of the many ways the metal atoms can nestle next to one another, try this link to Purdue's Structure of Metals webpage.
The regularity gives reinforcement to the pattern but the position of the pattern gives atomic dimensions in the crystal due to the fact that Xray wavelengths are comparable to interatomic and intermolecular distances.
The story they tell is of regular repeating units called unit cells. The organization of the relevant units (atoms or molecules) in those cells give rise to symmetrical structures best understood by motion pictures. If you have configured RASMOL for your computer, link to Case Western Reserve's Solids webpage. (Configure to display "spacefilled" with your right mouse click.) If your computer is already configured for MPEG movies, you'll find Cornell's Cubic Crystal Structures interesting to watch. (Click the stop button before the movie ends so that you can replay it as many times as you like. If you let it end, the viewer goes away.)
Indices
We catalog the planes by how they carve through a unit cell. The picture at left is a small part of a slide on this subject at the University of Binghampton. You can go there by clicking on the picture and follow their slide presentation by clicking their forward arrows. (You'll have to click not their back arrow but your BACK button a few times to return here.) Many sites have a good introduction to Miller Indices including this one that (while terse) has an animated image that flips through many different crystal planes.
In the picture at left, we are looking along the planes that appear as diagonal lines. The unit cell's dimensions are a (vertical), b (horizontal), and c (out of the screen). The planes shown intersect EVERY unit cell along a but cut through each cell at ½b. Since the planes are parallel to c, they don't advance (up d, the interplane direction) from cell to cell that way. So the three numbers of interest for these planes are 1, ½, and infinity. The Miller Indices are their reciprocals: 1, 2, and 0. So these planes are referred to as the 120 planes. This triple has a generic name: hkl and here h=1, k=2, and l=0.
A bit of trigonometry (or Pythagorean geometry) will convince you that
Of course, those values of d are the same as in the Bragg Equation; so they govern at which angle the X-rays congregate. And this site shows how X-ray spectra determine the ordering of dh,k,l values. Note that backward slanting planes intersect a, b, or c with negative Miller Indices, denoted by a number with a bar over the top. (Our typeface here has no way of showing "k bar" for example.)
The geologists are, of course, keenly interested in Miller Indices and X-ray crystallography since it helps identify minerals for them. A web textbook on crystallography for amateur rock hounds by Mike and Darcy Howard, includes the use of Miller Indices in describing the macroscopic crystals.