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fluidistic

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## Homework Statement

Hi guys, I'm trying to solve questions from past exams; here comes one.

There are 2 possible fundamental equations for the rubber bands:

##S=L_0\gamma \left ( \frac{aU}{L_0} \right ) ^{1/2} -L_0 \gamma \left [ \frac{1}{2} \left ( \frac{L}{L_0} \right ) ^2 + \frac{L_0}{L} -\frac{3}{2} \right ]## and ##S=L_0\gamma \exp \left ( \frac{aUn}{L_0} \right ) -L_0 \gamma \left [ \frac{1}{2} \left ( \frac{L}{L_0} \right ) ^2 + \frac{L_0}{L} -\frac{3}{2} \right ]## where ##L_0=nl_0##, a and gamma are constants. L is the length of the rubber band and U is the internal energy while ##l_0## is the molar length of the rubber band when the tension vanishes.

1)Which one of these equation is acceptable? Why?

2)From the right fundamental equation, determine the state equation ##f(T,L)## that relates the tension f with T and L.

3)We put 2 rubber bands in contact with the parameters ##T_0^i##, ##l_0^i##,##n_0^i## where i=1, 2. Assume that there's no flow of matter between the 2 bands nor heat flux around the system. Calculate T in function of the initial parameters.

## Homework Equations

##\left ( \frac{\partial S}{\partial L} \right ) _{U,n}=\frac{f(T,L)}{T}## (I believe).

## The Attempt at a Solution

1)The second expression for S isn't an extensive function because of the argument in the exponential. Basically if I double the system (n'=2n), I wouldn't get S'=2S, which isn't acceptable. On the other hand the first expression for S is an extensive function. So my answer would be the first expression is acceptable because the 2nd one isn't due to a lack of extensivity.

2)I used the expression given in "Relevant equations" and I reached that ##f(T,L)=T \gamma \left ( \frac{L_0^2}{L^2} - \frac{L}{L_0} \right )##. So when L=L_0 I get that the tension is worth 0 which seems good. When ##L>L_0## I get that the tension is negative... I have no idea if this is good. Intuitively I know that if I stretch the rubber band the tension would be like a restore force, but I don't know if a negative force means a force that goes against the stretching, by convention.

So I don't know here if my answer is correct. What do you think?

3) Will think about it.Edit:3) I know that the total entropy will be the sum of the entropy of each one of the rubber bands. Same for the internal energy. I was thinking about using the definition of temperature to get it. Namely that ##\left ( \frac{\partial S}{\partial U} \right ) _{L,n} =1/T##.

But this soon becomes a mess. I reached that ##1/T=\frac{\partial S ^{(1)}}{\partial U}+ \frac{\partial S ^{(2)}}{\partial U}## where ##U=U^{(1)}+U^{(2)}##. But I don't have ##S^{i}## in terms of U.

So I don't really know how to tackle this part. Any tip is appreciated!

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