CountingDistinguishableArrangements

CountingDistinguishableArrangements - Counting...

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Counting distinguishable arrangements Notes on General Chemistry http://quantum.bu.edu/notes/GeneralChemistry/CountingDistinguishableArrangements.pdf Last updated Tuesday, March 14, 2006 22:15:46-05:00 Copyright © 2006 Dan Dill ([email protected]) Department of Chemistry, Boston University, Boston MA 02215 Spontaneous change proceeds in the direction that increases the number of distinguishable arrangements. We consider two different kinds of arrangements. The first is the ways molecules can be distributed in space. The second is the ways energy can be distributed among molecules. à Molecules distributed in space The number of distinguishable ways to distribute n molecules of one type and m molecules of another type is W molecules H n , m L = H n + m L ! ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅ n ! m ! . For example, the number of ways of arranging 3 ink molecules and 3 water molecules is H 3 + 3 L ! ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅ 3 ! 3 ! = 6 μ 5 μ 4 μ 3 ! ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅ 3 ! H 3 μ 2 μ 1 L = 6 μ 5 μ 4 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅ 3 μ 2 μ 1 = 2 μ 5 μ 2 = 20. Here is a way to derive W molecules H n , m L . The number of arrangements of n + m molecules, ignoring whether the arrangements are distinguishable is H n + m L ! , since the first molecule can be in any of n + m places, the second can be in any of the remaining n + m - 1 places, and so on to the last molecule which can be in the single remaining space. Now, the number of arrangements H n + m L ! must be the product of (1) the number of unique arrangements, (2) the number of ways n ! that a particular arrangement of the n molecules can arise, and (3) the number of ways m ! that a particular arrangement of the m molecules can arise: W molecules H n , m L μ n m != H n + m L ! . Solving this expression for W molecules H n , m L , we get the expression above. Note that in this expression n m ! is the number of ways that a particular one of the W molecules H n , m L distinguishable arrangements can occur.
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Questions How many distinguishable ways can n different objects be arranged: n , n 2 , n n or n ! ? Answer: n ! How many indistinguishable ways can w identical objects be arranged: w , w 2 , w n or w ! ? Answer: w ! How many ways can w different objects and i different objects (a total of w + i different objects) be arranged: w + i , H w + i L 2 , H w + i L w + i , or H w + i L ! ? Answer: H w + i L ! What is true about the number, W p H w , i L , of distinquishable ways can w identical objects on one kind and i identical objects of another kind (a total of w + i different objects) be arranged: W p H w , i L w ! i != H w + i L ! , W p H w , i L = w ! i ! , or W p H w , i L = H w + i L ? Answer:
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This note was uploaded on 05/01/2010 for the course CHEM Chem617 taught by Professor Otto during the Spring '09 term at Algoma University.

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CountingDistinguishableArrangements - Counting...

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