1) A polymer is a molecule built out of smaller units called monomers that are covalently linked together to form the la

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1 A Polymer Is A Molecule Built Out Of Smaller Units Called Monomers That Are Covalently Linked Together To Form The La 1 (83.43 KiB) Viewed 28 times

1) A polymer is a molecule built out of smaller units called monomers that are covalently linked together to form the larger molecule. Many polymers are linear chains in which each monomer is linked to at most two monomers (the "first" and "last" monomers have only one link each.) A linear polymer consisting of N monomers each of length b has an end-to-end length R when fully extended of R = Nb. (You can picture each monomer as a cylinder of length b and negligible diameter. Although the covalent bond between neighboring monomers is hard to break at room temperature - it has an energy of -1 ev, which is -100 x the energy of thermal fluctuations at room temperature - each such bond can easily change its spatial orientation, with random "kicks" from the surrounding solvent molecules providing enough energy to decorrelate bond orientations along the chain. This results in the chain adopting contorted configurations, which fluctuate continuously. Consequently, the actual measured end- to-end extension R is less than its maximum possible value of Nb. (You can think of Ras the distance a bullet would travel if a marksman sitting on the first monomer were to aim for the last monomer. For contorted configurations, the two ends would be closer together than the maximum allowed value, so R < Nb.) The following is an expression for the free energy of a single polymer chain in "good" solvent: F(TR) 3R2 vN2 KAT 2Nb2+ R3 As before, R is the measured end-to-end extension of the chain, b is the length of a single monomer, and N is the number of monomers in the chain. The quantity v = v(T) is called the excluded volume parameter and can be positive, zero, or negative. Positive values of v correspond to solvents in which each monomer prefers solvent molecules - has a net attractive interaction with solvent molecules - over the other monomers on its chain. Solvents whose molecules have a net attractive interaction with the monomers of a chain are called good solvents. Positive values of v measure the strength of the attraction between monomer and solvent molecules or equivalently, the strength of the net repulsion between two monomers (not necessarily neighbors) on the chain. As the notation suggests, v, in general, depends on temperature but the exact dependence of von T does not concern us here. (a) Show that at room temperature of T = 300 K, 1 koT = 0.01 ev. (b) State the version of the second law relevant to conditions of constant temperature (constant T). i.e. what quantity is minimized with respect to the other extensive quantities (X2,..., X.) when T is held fixed? (c) By applying the second law to the free energy expression above, find an expression for R that would be observed at equilibrium when the temperature is held fixed at T. 11 Page You will obtain an expression giving R as a function of N (and implicitly, through the excluded volume parameter.)