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Lecture 24sf - A crystalline solid holds atoms molecules...

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A crystalline solid holds atoms, molecules, ions together in a regular long-range pattern called a lattice. The smallest repeating unit in this lattice is called a unit cell . An analysis of the crystal structure and the size of the unit cell gives valuable information relating density and molar mass of the substance. X-ray diffraction data gives us such information about the crystal structure and size of the unit cell. The distance between atoms in a lattice is roughly the same as X-ray wavelengths, resulting in a special scattering pattern called diffraction. When the reflected and incident rays reinforce each other, a diffraction pattern emerges. The Bragg equation can be used to determine the distance d between atomic layers in a lattice. n λ = 2d sin θ where λ = wavelength of the incoming X-rays, d = distance between layers θ = the angle of incidence between the incoming X-rays and the line of atoms in the crystal. n = the order of the diffraction, usually taken as 1. Example: X-rays having wavelength of 154 pm produce a diffraction pattern when aimed at a layer in the lattice at an angle of 19.3. Calculate d, the distance between the layers. Assuming first order diffraction pattern (n = 1): d = By changing the incident angle, we aim the X-rays at different layers, and can infer the crystal structure. In practice, this can be a complex process, not as easy as just using one simple equation such as the Bragg equation shown above. Some common unit cells are:
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Simple cubic: atom in each corner of a cube Face-centered cube: atom in each corner of a cube and an atom in the center of each face. Body-centered cube: atom in each corner of a cube, and an atom in the center of the cube. A diagram of these unit cells are shown below.
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A good recommended website is: http://www.molsci.ucla.edu/pub/explorations.html Then take the link to crystalline solids, and then to download setup program. The downloaded program will give excellent three-dimensional views of different unit cells and crystal structures. Students should learn to do problems relating: Dimensions of unit cell Atomic radius Number of atoms in each cell Molar mass Avogadro’s Number Density Several examples follow. Note that in each case, dimensional analysis is an extremely helpful tool. Calcium crystallizes in a face-centered cubic unit cell. The side of the unit cell is 556 pm (determined by X-ray diffraction data). The density of calcium is 1.54 3 cm g . The molar mass of calcium is 40.08 . Calculate the value of Avogadro’s number based on the above data.
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