Thermodynamics of an elastic band

[bon]

Homework Statement

The Gibbs free energy of an elastic band satisfied dG = -SdT - LdF where F is the tension and L the string's length.

At fixed temp the entropy is given by S = So - a(L-Lo)^2 + b(L-Lo)^2

Where Lo is the length of the elastic at zero tension and a and b are positive constants.

Show that the thermal expansion coefficient at zero tension is negative.

Homework Equations

The Attempt at a Solution

So I'm trying to show that partial L wrt partial T at constant F < 0.

Using a maxwell-type relation we can say dL/Dt)f = dS/dF)T

But I'm not sure where to go from here..the expression for S doesn't contain F and I can't see how to solve..
thanks!

 

[Mapes]

At fixed temp the entropy is given by S = So - a(L-Lo)^2 + b(L-Lo)^2

Is this part correct? It seems like an odd way of writing it; I suspect there's a typo or another condition in there somewhere.

 

[bon]

It is correct yes! And no typo..this was on a past exam paper.

 

[Mapes]

It is correct yes! And no typo..this was on a past exam paper.

It doesn't seem odd that the entropy expression could be simplified to S = So + c(L-Lo)^2, where c = b-a? I don't see a way to solve it unless it's also assumed that a > b.