Compute partition function Z and Free Energy. This question was too tough for me

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Compute partition function Z and Free Energy.

This question was too tough for me to handle- -and it was last year's final question.

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I. A one-dimensional polymer can be modeled as consisting of N elementary links of length i each, attached to each other to form an link chain. Each link can point left or right at random: in the absence of an external force stretching the polymer, there is no energy difference to links pointing left or right. Associate a discrete variable s 1, i N, with each link and declare that if the k-th link points to the right sk 1, while if it points to the left, s -1. Define the total "length" of the chain, L Si, as the distance from the beginning of the first link to the end of the N-th one. When the polymer is subjected to an external force f its energy is E fL, whose sign ensures that the chain aligns itself with the applied force. 1. Compute the partition function Z and the free energy F of the chain subject to an external force f 2. Find the average length C (L) as a function of the temperature and external force. Argue that in an appropriate temperature regime, a linear relation between the length and applied force, f kC holds. This relation is known as Hooke's law. Find the dependence of k on the temperature for this polymer 3. In this thermodynamic system, the partition function and free energy are functions of the external parameters (N,T, f). Show that C m. Then, referring to the known relation between F N,T and the entropy, argue that the fixed N thermodynamic identity for the polymer is dF -SdT-Cdf

Explanation / Answer

Ans:- Given one dimension polymer N elementary link. the polymer length l. each link can point left or right at random.

            total length of polymer chain L = l si external force stretching the polymer. when the polymer external force applied f . so the energy E=-f L

(a). the partition function of Z is related for N number of link chain. so that  we are using binomial formula to calculate the N number of chain link of f external force applied to calculated the Z as a partition function. and individual N elementary link

so that our formula is

b(Z; f, N) = fCz * Nz * (1 - N)f - Z
or
b(Z; f, N) = { f! / [ Z! (f - Z)! ] } * Nz * (1 - N)f - z

so we can calculated the value . if it is numerical based .

(b).  Now we getting the average length of the polymer due to applied the external force as well as temperature. as we are using to the Hooke's law we know that force(f) applied on polymer so that the extend the polymer link.

Averaged length of ln tn / ni                           ( l = number of length , t = each polymer temp , n = number of polymer )

We know if increase the temperature so that stiffened is loss .it mean if increase the temperature so reduce the need of the external force applied .

for example:-the applied the temp for polymer to displacement of k link in polymer and using the work conjugate & Cauchy stress are applied on the stress–strain relation so we derived from a temperature strain energy density denoted as (U),

So we prove the dependence k on the temperature for this polymer.

(c). the thermodynamic system the force applied in polymer . the relation between the force (F) & Entropy (s). the both are vice versa because of entropy measure of the molecular randomness, of a system . if we increase the temperature of polymer so that increase the entropy. it mean polymer length deformation or change permanently.

now as per given parameter (N,T,f) L =- (F / f) NT  

Fixed - N thermodynamic identical for the polymer dF= -SdT - Ldf   that equation is the average length increase the polymer in base on force as well as entropy increase when the temprature increase so that both are - ve sign in thas equation so that id will total effect on the polymer length in behafl of the external forces.

L=- (F / f) NT   so that its very clear the only number of polymer in link (N) and Temperature(T) to change the value of the External forces applied in the polymer.

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