How Does Temperature Affect the Length of a Rubber Band Under Constant Tension?

[derrickb]

Homework Statement

For the rubber band model, calculate the fractional change in (L-L₀) that results from an increase δT in temperature, at constant tension. Express the result in terms of the length and temperature.

Homework Equations

U=cL₀T
τ=bT((L-L₀)/(L₁-L₀)); τ=tension, L₁=elastic limit
d/dL(1/T)=d/dU(-τ/T)

The Attempt at a Solution

I'm sort of at a loss on this one. I've tried subbing in all sorts of equations, but can't seem to make any real progress.

 

[rude man]

Homework Statement

For the rubber band model, calculate the fractional change in (L-L₀) that results from an increase δT in temperature, at constant tension. Express the result in terms of the length and temperature.

Homework Equations

U=cL₀T
τ=bT((L-L₀)/(L₁-L₀)); τ=tension, L₁=elastic limit
d/dL(1/T)=d/dU(-τ/T)

The Attempt at a Solution

I'm sort of at a loss on this one. I've tried subbing in all sorts of equations, but can't seem to make any real progress.

EDIT:

By using Maxwell's 4th equation you can show that T dS = C_τ dT if τ is constant.

You can also rewrite the 1st law as dU = T dS + τ dL.

Just thinking - if we can assume an "ideal rubber band" analogously to an ideal gas, such that U is a function of T only, then dU = C_L dT similar to dU = C_V dT for an ideal gas.