Equation of State for a Polymer Chain

The development for a single chain follows the same line of attack as for the ensemble, starting with the first law of thermodynamics 

In the case of our model chain, which has no energy barrier hindering the rotation of the segments, the internal energy of the chain is the same for all conformations. Thus the first term on the right side of equation (d) is zero, and 

To calculate the entropy for our Gaussian chain, we use the Boltzmann s relation from statistical mechanics 

Inserting equation (h) into equation (f) and differentiating according to equation (e), we obtain the equation of state for a single polymer chain  

In equation (i), f is a vector aligned with the vector r. Thus if the ends of a chain are held a fixed distance r = r apart by a tensile force, then this force is directly proportional to r, and to absolute temperature. It is also inversely proportional to the mean square length of the chain, since b2 =3/2r2 [equation 

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