8. A rubber band is suspended vertically such that the upper end is attached rigidly to...

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8. A rubber band is suspended vertically such that the upper end is attached rigidly to a point on the ceiling and the lower end is attached to a mass of weight W which hangs freely from the lower end. The rubber band is modelled as a polymer chain of N links joined end to end. Each link has the same length d and can assume two possible orientations l, one pointing down towards the floor in which case it contributes l = +d to the total length and one pointing up towards the ceiling in which case it contributes / = - d to the total length. It follows that the maximum length of the rubber band is Lmax = Nd if all the links are pointing down and the minimum length is Lmin = 0 if half of the links point up and half point down. Because the weight acts to stretch the rubber band, the downward orientation is energetically favoured while the upward orientation is energetically unfavourable. If the energy of a link is E = -Wl, then Edown -Wd and Eup = +Wd, which satisfies above requirement since Edown < Eup. Neglect the kinetic energies of the links and the weights of the links themselves and any interaction between the links. L W Y+d (a) Write down an expression for the probability Pup that a single link points up and a similar expression for the probability Pdown that a single link points down, assuming that the rubber band is in thermal equilibrium at temperature T. (b) Calculate the mean length I of a single link and hence show that mean total length of the rubber band is = Nd tanh = Wd 8. A rubber band is suspended vertically such that the upper end is attached rigidly to a point on the ceiling and the lower end is attached to a mass of weight W which hangs freely from the lower end. The rubber band is modelled as a polymer chain of N links joined end to end. Each link has the same length d and can assume two possible orientations l, one pointing down towards the floor in which case it contributes l = +d to the total length and one pointing up towards the ceiling in which case it contributes / = - d to the total length. It follows that the maximum length of the rubber band is Lmax = Nd if all the links are pointing down and the minimum length is Lmin = 0 if half of the links point up and half point down. Because the weight acts to stretch the rubber band, the downward orientation is energetically favoured while the upward orientation is energetically unfavourable. If the energy of a link is E = -Wl, then Edown -Wd and Eup = +Wd, which satisfies above requirement since Edown < Eup. Neglect the kinetic energies of the links and the weights of the links themselves and any interaction between the links. L W Y+d (a) Write down an expression for the probability Pup that a single link points up and a similar expression for the probability Pdown that a single link points down, assuming that the rubber band is in thermal equilibrium at temperature T. (b) Calculate the mean length I of a single link and hence show that mean total length of the rubber band is = Nd tanh = Wd