Entropy and Probability(A statistical view)Entropy ~ a measure of the disorder of a system.A state of high order = low probabilityA state of low order = high probabilityIn an irreversible process, theuniverse moves from a state of lowprobability to a state of higherprobability.We will illustrate the concepts byconsidering the free expansion of a gasfrom volume Vi to volume Vf.The gas always expands to fill theavailable space. It never spontaneouslycompresses itself back into the originalvolume.First, two definitions:Microstate: a description of a system thatspecifies the properties (positionand/or momentum, etc.) of eachindividual particle.Macrostate: a more generalizeddescription of the system; it can be interms of macroscopic quantities, suchas P and V, or it can be in terms of thenumber of particles whose propertiesfall within a given range.In general, each macrostate contains alarge number of microstates.An example: Imagine a gas consisting ofjust 2 molecules. We want to considerwhether the molecules are in the left orright half of the container.L RThere are 3 macrostates: both moleculeson the left, both on the right, and oneon each side.There are 4 microstates:LL, RR, LR, RL.How about 3 molecules? Now we have:LLL, (LLR, LRL, RLL), (LRR, RLR, RRL), RRR(all L) (2 L, 1 R) (2 R, 1 L) (all R)i.e. 8 microstates, 4 macrostatesHow about 4 molecules? Now there are16 microstates and 5 macrostates(all L) (3L, 1R) (2L, 2R) (1L, 3R) (all R) 1 4 6 4 1 number of microstates12In general: N W M 1 1 1 2 2 1 2 1 2 4 3 1 3 3 1 3 8 4 1 4 6 4 1 4 16 5 1 5 10 10 5 1 5 32 6 1 6 15 20 15 6 1 6 64 7 1 7 21 35 35 21 7 1 7 128 81 8 28 56 70 56 28 8 1 8 256 9 2N N+1 This table was generated using the formulafor # of permutations for picking n itemsfrom N total: WN,n = i.e. W6,2 = = 15 N!N! (N-n)! 6!2! 4!“multiplicity”Fundamental Assumption of StatisticalMechanics: All microstates are equallyprobable.Thus, we can calculate the likelihood offinding a given arrangement of moleculesin the container.E.g. for 10 molecules:Conclusion: Events such as thespontaneous compression of a gas (orspontaneous conduction of heat from acold body to a hot body) are notimpossible, but they are so improbablethat they never occur.We can relate the # of microstates W ofa system to its entropy S by consideringthe probability of a gas to spontaneouslycompress itself into a smaller volume.If the original volume is Vi, then theprobability of finding N molecules in asmaller volume Vf isProbability = Wf/Wi = (Vf/Vi)N• ln(Wf/Wi) = N ln(Vf/Vi) = n NA ln(Vf/Vi)We have seen for a free expansion thatDS = n R ln(Vf/Vi) ,so DS = (R/NA) ln(Wf/Wi) = k ln(Wf/Wi)orSf - Si = k ln(Wf) - k ln(Wi)Thus, we arrive at an equation, firstdeduced by Ludwig Boltzmann, relating theentropy of a system to the number ofmicrostates:S = k ln(W)He was so pleased with this relation thathe asked for it to be engraved on
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