ps5 - Chem120B Problem Set 5 Due 1 Problems 19-7 19-13...

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Chem120B Problem Set 5 Due: February 29, 2008 1. Problems 19-7, 19-13, 19-14, and 19-34 in Simon and McQuarrie’s Physical Chemistry . For problem 19-34, notice that the heat capacity C p = parenleftbigg ∂H ∂T parenrightbigg p,N is very large at the transitions between different phases (solid liquid and liquid gas). Explain this fact in terms of fluctuations in energy and/or volume. 2. (i) In the previous assignment you explored how the Gibbs entropy of a probability distribution de- pends on its width. Here you will show that a uniform distribution actually has the largest possible entropy. To do this, you need to find the combination of microstate probabilities p ν that maximize S = k B ν p ν ln p ν . [We have changed notation slightly from the usual P ( ν ) to p ν , which is more convenient for this calculation.] If we think of each p ν as an independent variable, then the maximum would be specified by parenleftbigg ∂S ∂p ν parenrightbigg p ν negationslash = ν = 0 for each microstate ν . These probabilities, however, cannot actually vary independently, since they must all sum up to 1. One way to deal with this is to single out one microstate, say ν = 1 , and write its probability in terms of all the others: p 1 = 1 M summationdisplay ν =2 p ν , where M is the total number of microstates. Now we can differentiate S = k B bracketleftBigg p 1 ( p 2 , p 3 , . . . , p M ) ln p 1 ( p 2 , p 3 , . . . , p M ) + M
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