In crystals of the salt cesium chloride, cesium ions Cs^+ from the eight corners

Question

In crystals of the salt cesium chloride, cesium ions Cs^+ from the eight corners of a cube and a chlorine ion Cl^- is at the cube's center (see the figure). The edge length of the cube is L = 0.49 nm. The Cs^+ ions are each deficient by one electron (and thus each has a charge of +e), and the Cl^- ion has one excess electron (and thus has a charge of -e). (a) what is the magnitude of the net electrostatic force exerted on the Cl^- ion by the eight Cs^+ ions at the corners of the cube? (b) If one of the Cs+ ions is missing, the crystal is said to have a defect; what is the magnitude of the net electrostatic force exerted on the Cl^- ion by the seven remaining Cs+ ions?

Explanation / Answer

PART A

The cesium ion at a corner of the cube exerts a force of the same magnitude on the chlorine ion at center. each force is a force of attraction and is directed toward the cesium ion that exerts it , along the body diagonal of the cube . Since the ions in such a pair exert forces that have the same magnitude but are oppositely directed, two forces sum zero and , since every cesium ion can be paired total force on the chlorine ion is zero.

b)   The length of a body diagonal of a cube is 3a and the diistance from the center is d = 3/2 * a

F = k ( -e) ( -e)/d2

    = ke2/d2

   = 9*109 * ( 1.6 * 10-19C)2/ 3/4 *( 0.49 * 10-9)2

    =1.28 * 10-9

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