Question

Statistical Mechanics Problem Counting Configurations ?

A certain protein molecule consists of a one-dimensional chain of six molecular sub-units, or monomers. Each monomer can take a short or a long structural form, with lengths L and 2L, respectively, but with the same energy in each case. A particular configuration of the protein with total length 8L is il- lustrated below, with a short monomer represented using a dot and a long monomer represented as a dash.

1. How many configurations of the protein are there in total? -> I calculated this one as 15, using the combinatorics formula used in the hint for the problem: the number of ways in which NS indistinguishable objects can be placed on N sites is given by N!/(NS!NL!) where N = NS + NL.]

2. Macrostates of the protein can be labelled by its total length Ltot, taking
   values between 6L and 12L. Calculate the multiplicity of configurations,
   or microstates, available to the protein in each of the possible macrostates. ....Here I figured that Ns + 2Nl should be equal to the energies values between 6LE and 12LE but how can I further proceed? Thank you!

3. Calculate the microcanonical average length of the protein, stating any assumptions you make.

Answer 1

When the total length is 8L, the only possible configuration is (4S + 2L) [S: small; L: large]. In that case, the number of possible configurations is as you have calculated.

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You can represent a macrostate using L(tot) of that particular macrostate. But as you can see, this corresponds to a microcanonical ensemble where total energy of each macrostate possible is the same and here it has the value 6E (assuming E is the energy associated with individual monomers) irrespective of the total length of the polymer.

Macrostate (L(tot))	Possible configuration	Number of microstates
6L	6S	1
7L	5S + 1L	6
8L	4S + 2L	15
9L	3S + 3L	20
10L	2S + 4L	15
11L	1S + 5L	6
12L	6L	1

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Basic assumption of microcaonical ensemble is that all the macrostates are equaly probable (i.e. probability of the system being in any of the macrostates is equal). Here the probability is (1/7)

  

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Hope this clarifies your doubt.

All the best...