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Entropy of Mixing

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I am trying to figure out how, for the entropy of mixing of two polymers, one gets from:

Change in entropy of mixing (delta S sub m) = k[(N1 + N2) ln (N1 + N2) - N1 ln N1 - N2 ln N2]

to the result:

change in entropy of mixing (delta S sub m) = -k(N1 ln v1 + N2 ln v2) where v1 and v2 are the volume fractions of the two polymers 1 and 2 respectively. What assumptions need to be made to go from the first equation to the second? How algebraically does one make the rearrangements necessary? Please be explicit.

Note that the first equation arises from Bolztmann's relationship:

delta S sub m (entropy of mixing) k ln (Omega) where omega is the number of possible arrangements in space that the molecules can assume. Omega ends up being equal to (No/(N1!N2!)) and then Stirling's approximation is applied to get the first equation I indicated above.

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Solution Summary

Applying the definition of partial volumes to the initial expression, with additional manipulation yields the required result.

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