(Solved): 6.26 The force required to maintain a polymeric fiber at a length L when its unstretched length is ...

$$6.26$$ The force required to maintain a polymeric fiber at a length $$L$$ when its unstretched length is $$L_0$$ has been observed to be related to its temperature by $$F=?T\left(L?L_0\right)$$ where $$?$$ is a positive constant. The heat capacity of the fiber measured at the constant length $$L_0$$ is given by $$C_{\mathrm{L}}=?+?T$$ where $$?$$ and $$?$$ are parameters that depend on the fiber length. a. Develop an equation that relates the change in entropy of the fiber to changes in its temperature and length, and evaluate the derivatives $$(?S/?L{)}_T$$ and $$(?S/?T{)}_L$$ that appear in this equation. b. Develop an equation that relates the change in internal 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 $$T_0$$ and an extension $$L_0$$ to its entropy at any other temperature $$T$$ and extension $$L$$. d. If the fiber at $$T=T_f$$ and $$L=L_j$$ is stretched slowly and adiabatically until it attains a length $$L_f$$, what is the fiber temperature at $$TR$$ ? 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 extension of the fiber, increases its internal energy) is $$FU=(?U/?L{)}_T$$, and the entropic force is $$F_S=?T(?S/?L{)}_T$$. Evaluate $$F_U$$ and $$F_S$$ for the fiber being considered here.