Consider an elastic rubber band of length L being stretched by a temperature-dep
ID: 159742 • Letter: C
Question
Consider an elastic rubber band of length L being stretched by a temperature-dependent external force f(T), where T is temperature. (a) Write down the fundamental thermodynamic relation for this system relating the change in internal energy dU to the infinitesimal change in length dL and the heat supplied TdS, where U is the internal energy and S is the thermodynamic entropy (it is assumed here, of course, that thermodynamic processes in this system are reversible). (b) In one experiment the length of the band was fixed to L=1 m and the temperature of the band T =300 K is raised by delta T=3 K. This caused the force needed to maintain the band's length to increase by delta f=1.2 N. In another experiment, the band was stretched from L to L + delta L at constant temperature T=300 K. As a result the band exchanges heat with the environment. What is the amount of heat exchanged for delta L=2 cm? Is the heat released or absorbed by the band? [Hint to part (b): Use the Helmholtz function and the associated Maxwell relation involving S, L, f and T. Also, use finite differences to compute partial derivatives where necessary, that is, for the arbitrary thermodynamic variables X, Y, and Z, the partial derivative (partial X/ partial Y)z can be approximated as the ratio of the finite difference in X to the finite difference in Y given by (delta X/ delta Y)z. ]Explanation / Answer
Consider an elastic rubber band of length L being stretched by a temperature-dep