A simple model of a polymer (a long-chained molecule as in rubber), assumes it f

Question

A simple model of a polymer (a long-chained molecule as in rubber), assumes it forms a one dimensional chain made up of N identical elements, each of length L The angle between successive elements can be either 0 degree or 180 degree, but there is no difference in internal energy between these two possibilities. For the sake of counting, one can think of each element as either pointing to the right (+, if oriented at 0 degree from the previous element) or to the left (-, if oriented at 180 degree from the previous element). Then one has N = n_+ + n_- for the total number of elements and L = l(n_+ - n_-) for the length of the chain to the right of the first element. Assume N, n_+ and n_- are all very large. Find the entropy, S, as a function of N, n_+ and k. The relationship between entropy and the tension, F, on the chain is given by the first law of thermodynamics: 0 = TdS + FdL. Use this to find an explicit expression for the tension as a function of N, n_+, T, k and l Using your result in b), show that in the high temperature limit (for which L

Explanation / Answer

a) In the situation presented,entropy can be defined as the number of ways the system can be arranged.

Given, the number of elements in the chain = N=( n+ )+( n- )

Number of + -ve charges = N/2

Number of -ve charges = N/2

Entropy= C(N,N/2)= combination of (N,N/2) =

N!/((N-N/2)!(N/2)!)

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