Stretch your imagination: A rubber band can be considered a thermodynamic

system where the $-PdV$ mechanical work is replaced by the $+fdL$ work of

extension. Here, $f$ is the force $($tension$)$ along the rubber band and $L$ is its length.

Consider this system $($with no such $-PdV$ term$).$

a$.$ Account for the work of stretching the rubber band and the chemical

potential work. Combining the first and second laws, give an expression

for the internal energy change $dU,$ assuming a reversible process.

delW$=$

delQ$=$

$dU=$

b$.$ Write $dH,dF,$ and $dG$ as Legendre transforms from the potential from part

a$.$ Note that, by definition, $H-=U-FL,F-=U-TS,$ and $G-=H-TS.$

$dH=$

$dF=$

$dG=$

c$.$ For a polymer, it is known that the entropy decreases as its length is

increased, at constant temperature. Show that this information is

sufficient to prove that the tension along a rubber band increases with

temperature, at constant length. Hint: consider second derivatives of $F.$

d$.$ Assume that the internal energy is given by $U(S,L,n)=S^2\frac{L}{n^2}$ where $$

is a constant. Does this equation of state satisfy the third law of

thermodynamics? Show any and all calculations to make your case.