How Does Stretching a Polymer Relate to Hooke's Law?

[CottonHill]

The question tells me that some experiments have shown that the force needed to stretch a polymer molecule by an amount (Δl) is given approximately by:

F= (k_BT/P)*((1/4)*(L_o²/(L-Δl)²-(1/4)+(Δl/L))

Yea, I know, nasty looking. Anywho:

F = Force
T = Temp.
k_B = Boltzmann's Constant
P = Persistence Length
L_o = Contour Length
Δl = Amount of Stretch of Polymer Molecule

After giving all of that lovely info, the question says to show that if Δl is small relative to L_o, it approximates Hooke's Law. Then it asks to find the effective spring constant of the molecule.


So far I have taken that equation and said if Δl approaches zero and L₀ approaches ∞ then we get down to:

F=(k_B*T)/P

I can explain that Hooke's Law says that a force (F) equals a constant relating to energy of a spring (k) times the distance (x)...
And that relates to the above equation by having a force (F) equals a constant relating to energy of stretching (k_B) times a a distance (P). Ignoring T.

Is my logic correct and how would I find the spring constant of this molecule? Thanks in advance.

 

[BvU]

Don't understand F is a constant. Does that mean F is non-zero even when Δl is zero ? You say

So far I have taken that equation and said if Δl approaches zero and L₀ approaches ∞ then we get down to:
F=(kB*T)/P

This is true, but it's not the same as taking the limit ${\Delta {\rm l}\over {\rm L}}\, \rightarrow 0$

(Note you don't want L $\rightarrow \infty$)