Calculating Entropy Derivative of a Temperature-Dependent Spring

[painfive]

A spring has a temperature dependent k of aT²(L-L₀) and a heat capacity at constant length of C_L=bT, a, L₀, and b are constants. I need to calculate the derivative of entropy with respect to length at constant temperature. I can't figure out why the entropy should depend on the length. If dS=dQ/T, and when you stretch the spring you don't add any heat, why would the entropy rise? I'm not 100% sure I'm wrong, but based on the rest of the problem it doesn't seem likely that the answer actually is zero.

 

[painfive]

I really need help on this soon. I have to turn this in this afternoon.

 

[painfive]

Well I had to hand it in, and I just said there is no change in entropy, but I don't think I was right. Nobody has any ideas on this? It seems like a pretty basic thermodynamics question. I mean, I'm one to talk, but I'm just suprised nobody here has responded.

 

[JoAuSc]

I don't understand thermodynamics too well, but here's what I figure: A rubber band is made up of polymers, long chainlike molecules that can twist in a number of different directions at each bond. With three mer units, the molecule can't bend too much, but with tens of thousands of mer units the polymer's basically just like a piece of string. For each polymer, there's a finite number of states it can be in. There's much, much more crumpled states than organized states. An organized state would be one where the rubber band is stretched so that its polymers line up. The random vibrations of the polymers cause it to change states frequently, and since there are more crumpled states than aligned states, the polymers tend to a crumpled state and the rubber tends to be unstretched. When you stretch the rubber band, you're fighting its polymers' tendencies to remain in a crumpled state, so you feel a force. If you heat rubber (not hot enough to cause a chemical change), its polymers vibrate more strongly, and they have an even stronger tendency to return to a crumpled state, causing a higher restoring force for hot rubber than cooler rubber, the opposite it would be for metals.

If we're talking about a rubber spring, then the answer seems pretty clear: when you stretch it, you create order on a microscopic level, which I assume means the entropy of the system is lowered. There's probably a similar answer that applies to all elastic materials, but I don't know what it is.

 

[lightgrav]

Painfive:
Sorry I didn't see this Friday ...
When you stretch your spring you do Work on it
so its temperature rises as length increases.
they gave you c_v , you find c_p gas analogy.
Otherwise you can cool (Qout) as you stretch it
to keep it as constant T.