b. Develop an equation that relates the change in inter- nal energy of the fiber to changes in its temperature and length. c. Develop an equation that relates the entropy of the fiber at a temperature To and an extension Lo to its entropy at any other temperature T and exten- sion L. d. If the fiber at T = T: and L = L; is stretched slowly and adiabatically until it attains a length If, what is the fiber temperature at If? e. In polymer science it is common to attribute the force necessary to stretch a fiber to energetic and entropic effects. The energetic force (i.e., that part of the force that, on an isothermal exten- sion of the fiber, increases its internal energy) is Fu = (OU/OL)r, and the entropic force is Fs = -T(OS/OL)T. Evaluate Fu and Fs for the fiber being considered here.6.26 The force required to, maintain a polymeric ber at a length L when its unstretched length is L0 has been observed to be related to its temperature by F = 'YT(L L0) where 7 is a positive constant. The heat capacity of the ber measured at the constant length L0 is given by CL=0z+T where or and [3 are parameters that depend on the ber length. a. Develop an equation that relates the change in en tropy of the ber to changes in its temperature and length, and evaluate the derivatives (8S/8L)T and (88/81?) L that appear in this equation