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Fa08-MT1-SunRev1-Soln

Fa08-MT1-SunRev1-Soln - Sterling’s approximation N is...

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4. There are N /2 particles each of substances A and B . First, we want to ask how many ways are there of putting these N total particles into the N possible slots of the alloy. First, let’s look at the A particles. We want to put N /2 objects into N slots. There are N - choose- N /2 ways to do this. Remembering a little probability theory, this equals N N / 2 " # $ % & = N ! ( N / 2)!( N / 2)! Now, we have N/2 empty spaces left, and exactly N /2 particles of B to fill them! There’s only one way of doing this, so the total number of microstates for the binary allow is: " = N ! ( N / 2)! [ ] 2 The entropy is S = k B ln Ω . The logarithm of the number of states is found using
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Unformatted text preview: Sterling’s approximation ( N is very large!): ln " = ln N ! # 2ln( N /2)! $ N ln N # 2 N 2 ln N 2 = N ln N # ln N /2 ( ) = N ln2 Therefore, the entropy of the binary alloy, with N = 10 23 is: S = Nk B ln2 = 7 " 10 22 k B Another way of seeing this is that there are 2 choices for each of the N slots! b) There is no work, heat transfer, or change in temperature, so the change in internal energy should be 0! c) If, on the other hand, A and B were not distinct from eachother, there would only be 1 way of putting all the particles into the N slots! So, S = 0....
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