In this example we see that the lowest probabilitythe

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Unformatted text preview: e molecules are ) = (¼ two not zero. So, we can o er Perspectivelogic applies to H..,532*0if #1: the right-hand chamber; the probability of of 3 × (⅛) for Cases 3 and 4. is results in on of spontaneity by saying ButOne same le and two in the right: 4. the in the indistinguishable we have a probability nding ess moves from states of bothSuppose wein the le -the behavior)–( is (½)(½) =in¼). e probability of ndmolecules examine hand chamber AB ) ( C of molecules ( a two-chamber apparatus, tes of high probability. In this example, we see that the lowest probabilitythe histogram shown to the right for three indistinguishable molecules. shown molecule 4.11. ing onein Figurethein the le chamber )–( AC )again, we note the pattern: (a) eis are n + 1 = 3 + 1 = 4 possible states and one molecule in the right chamber re ( Once that are relatively ordered, i.e. a large fraction of gas molecules can B ½. and the states of high probability are(or)–( BC urations), and (b) the higher of two ( A con g ) he right or the left of the container; us, we construct our histogram accordingly. Notice that for our case probability cases are those for which there Chapter 4 The Case for 2 Molecules e the system is relatively disordered. identical molecules, there are three (n + 1) possible con gurations and that there L5/6763.308 Determination of Probability at the Molecular Level 4.9 L5/6763.308 L5/6763.308 L5/6763.308 is the most balanced distribution between the two chambers. But while each con g nding the molecules evenly has the same probability is a higher probability ofuration of the three moleculesdistributed between the two Spontaneity and Probability if we distinguish the individual molecules usion that we draw fromchambers. progression is of great prac- ((½)(½)(½) = ⅛), if the molecules are Molecules this simple The Case for 10 The Case oceed to consider theimportant10 molecules in a illustrated for 3 Molecules ndistinguishable we have probability we of cal) importance. It is icase for because we haveFigure 4.13,that3 × (⅛) for Cases 3 and 4. is results in counting becomes rapidly moreparticular(or tedious!),between indistinguishable molecules. probFIGURE 4.12 c initial case of two but FIGURE f .13 As we continue to of high the histogram heult to the right for the f individual molecules seeking a di Tshowndistributionmol- three o4low probability to states add mes spontaneous process based on a purely sso we will keep add- molecules to n + system, counting ossible states more pronounced:eculeshistogram we note the pattern: (a) ere are the 1 = 3 + 1 = 4 p the states e is straightforward, in the middle, corOnce again, peakstatistical treatment ability. ) is a ing molecules the system and ndchambers, out how molecules being distributedinvolvetobetween the energy, the en- assigning a probability to each arrange-there (or con gurations), con guration changes.and no point did we discuss or equallyof eachand (b)two higher probability cases are those for which the probability the i...
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This note was uploaded on 11/08/2012 for the course CHEMISTRY 109 taught by Professor Flemingcrim during the Fall '10 term at University of Wisconsin.

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