Consider a system of a single macromolecule (polymeric molecule) as a chain of M atoms connected by (M?1) bonds. The molecule sits in a bath of a liquid solvent of N particles with associated chemical potential ?, has length L and force at its ends, F. At fixed length, volume, number of solvent molecules, and entropy, the insertion of a new polymeric atom into the system changes internal energy by ??. At fixed number of polymer atoms M, volume, number of solvent molecules, and entropy, the work done on the system for marginal increase of length is FdL.
(a) What are the additional intensive?extensive conjugate variables of this system, in addition to the usual conjugate variables: P?V; S?T; N??. Fill in the other variables in the fundamental relation U(S,V,N,.....) and S(U,V,N,.....).
(b) Write the differential form of the energy and entropy representations of the fundamental equation. Identify the new intensive variables and how they relate to partial derivatives of the fundamental relations.
(c) Suppose you are performing a stretching experiment on the macromolecule. You are controlling the temperature, pressure, and length, and both the number of solvent and polymer molecules in the system are held fixed. What is the thermodynamic potential that is minimized under these conditions? If it is not one of the two fundamental ones (energy or entropy representation of fundamental equation), then identify it by the Legendre Transformation that defines it.
(d) You decide to repeat the experiment, this time switching to controlling the force on the molecule. What is the thermodynamic potential that is minimized, as defined by it Legendre Transformation? What are its natural variables? Write the differential form of this thermodynamic potential.