$W=Z^N,Z=4\frac{k_{B}T}{fa}sinh(\frac{fa}{k_{B}T})$

A$)$ use the definition of the Gibbs free energy, ignore

the enthalpy term, and show that the average

polymer length is given by:

$($:$L$:$)=-\frac{\text{d}\text{e}\text{l}\text{G}}{\text{d}\text{e}\text{l}\text{f}}=Na[coth(\frac{fa}{k_{B}T})-\frac{k_{B}T}{fa}]$

B$)$ Linearize the result $($use a Taylor series

approximation$)$ for the small force limit small to obtain a

linear relationship as below. Write the equation for the

effective spring stiffness in this linearized case? How is

this different from the result for a $1$D random walk

polymer derived in class?

$f$~~$k_{\text{e}\text{f}\text{f}}($:$L$:$),\frac{fa}{k_{B}T}1$