The work done on a rubber band when it is stretched can be written as dL. where

Question

The work done on a rubber band when it is stretched can be written as dL. where r is the tension and L is the length of the rubber band. Neglecting p. V terms, set up an expression for dE for a rubber band. By a Legendre transformation, obtain an expression for dA. where A Ls the Helmholtz free energy. Experiment shows that a rubber band heats up when it is stretched adiabatically. From this fact, determine whether a rubber band will contract or expand when it is cooled at constant tension. The same amount of heat flows into two identical rubber bands, but one is held at constant tension and the other at constant length. Which has the largest increase in temperature?

Explanation / Answer

dE = TdS + dL A = E - TS dA = -SdT + dL b) dT/dL|S > 0 (dT/dL|S)*(dL/dS|T)*(dS/dT|L) =-1            Triple Product Rule dS/dL|T = d/dT|L      MaxwellRelation from Helmholtz Free Energy => dL/dS|T = dT/d|L (dT/dL|S)*(dT/d|L)*(dS/dT|L) = -1 d/dT|L *(dT/dL|)*(dL/d)|T = -1 => (dT/dL|)*(dL/d|T) = -(dT/d|L) (dT/dL|S)*(dT/dL|)*(dL/d|T)*(dS/dT|L) = 1 dL/dT| = (dT/dL|S)*(dL/d|T)*(dS/dT|L) dT/dL|S > 0 (given) dS/dT|L =CL/T         where CL = heat capacity at constant length andCL > 0 dL/d|T > 0    As you increase thetension, the length should increase So dL/dT| = (dT/dL|S)*(dL/d|T)*(dS/dT|L) =positive*positive*positive dL/dT| > 0 so it should strectch when T is increased (heated) at fixed. c) The one at constant tension, because then its energy increases whenwork is done on the rubber band (From part b, you know that Lincreases when T increases at constant , so work is done) No work can be done for the one at constant length.

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