The force required to maintain a polymeric fiber at a length L when its unstretched length is L₀ has been observed to be related to its temperature by

where _γ is a positive constant. The heat capacity of the fiber measured at the constant length L₀ is given by

where α and β 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 T₀ and an extension L₀ to its entropy at any other temperature T and extension L.

d. If the fiber at T = T_i and L = L_i is stretched slowly and adiabatically until it attains a length L_f , what is the fiber temperature at T_f ? 

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 F_U = (∂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.

Transcribed Image Text:

F = T(L-Lo)