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Entropy derivation

It is possible to treat quantitatively the entropy change on extending a polymer chain:

If one end of the chain is at (0,0,0), then the probability of the other end being at point (x,y,z) is:

where

a = bond length
n = number of links

If we stretch the chain, so that the end is at a new location (x',y',z') such that (x'² + y'² + z'²) > (x² + y² + z²), then p(x,y,z) will decrease, leading to a decrease in entropy. The entropy is given by

S = klnΩ

where Ω = total number of possible conformations leading to the same end position. Now

On stretching a chain, so that the initial end point (x,y,z) changes to (x',y',z') where

x' = λ_xx
y' = λ_yy
z' = λ_zz

the associated change in entropy is given by

In the unstressed state, with overall length r, we expect no preferential direction, so:

So, on average:

From random walk theory,

The entropy of a single chain segment can be multiplied by N (the no of chain segments) to give the total entropy change:

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