# The force required to maintain a polymeric fiber at a length L when its unstretched length is

## Question:

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

where _{γ} is a positive constant. The heat capacity of the fiber measured at the constant length L_{0} 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 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_{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.

## Step by Step Answer:

**Related Book For**

## Chemical Biochemical And Engineering Thermodynamics

**ISBN:** 9780470504796

5th Edition

**Authors:** Stanley I. Sandler