Question

**The relation between force and extension for a random walk “macromolecule” in** **three**

** dimensions**

**(a)** In class, we have derived the relation between force and extension for a one-dimensional random walk “macromolecule”. Derive the analogous relation between force and extension for a three-dimensional random walk “macromolecule”, in which each monomer is permitted to point in one of six directions. Make a plot of the resulting function. *Hint: find the average length of one segment in the direction of the force, <L*_{0}*>, and then the average length for the entire molecule is N times as large, N<L*_{0}*>.*

**(b)** In the small-force limit, the force-extension curve is linear; that is, in this regime, the polymer behaves like an ideal Hookean spring with a stiffness constant k ∝k_{B}T/L_{tot}a

Demonstrate this claim and deduce the numerical factor that replaces the proportionality with a strict equality.

#### Top Answer

Great question... View the full answer