1: Filament: For an elastic filament it is found that, at a finite range in temperature, a displacement x requires a force J= ax -bT + cTx where a, b, and c are constants. Furthermore, its heat capacity at constant displacement is proportional to temperature, i.e. Cx = A(x)T. (a) Use an appropriate Maxwell relation to calculate S/Ox/r (b) Show that A has to in fact be independent of x, i.e. dA/dx=0. (c) Give the expression for S(T, x) assuming S(0, 0) = So. (d) Calculate the heat capacity at constant tension, i.e. Cu= TS/oTy, as a function of T and J. Q2: Photon gas Carnot cycle: The aim of this problem is to obtain the blackbody radiation relation, E(T, V ) & V T*, starting from the equation of state, by performing an infinitesimal Carnot cycle on the photon gas. (a) Express the work done, W, in the above cycle, in terms of dV and dP. (b) Express the heat absorbed, Q, in expanding the gas along an isotherm, in terms of P, dV , and an appropriate derivative of E(T, V ). (c) Using the efficiency of the Carnot cycle, relate the above expressions for W and O to T and dT. (d) Observations indicate that the pressure of the photon gas is given by P= AT*, where A = T7kp$ /45 (hc)3 is a constant. Use this information to obtain E(T, V ), assuming E(T, 0) = 0. (e) Find the relation describing the adiabatic paths in the above cycle. 03: Irreversible Processes: (a) Consider two substances, initially at temperature T, and T2, coming to equilibrium at a final temperature Tr through heat exchange. By relating the direction of heat flow to the temperature difference, show that the change in the total entropy, which can be written as must be positive. This is an example of the more general condition that "in a closed system, equilibrium is characterized by the maximum value of entropy S.' (b) Now consider a gas with adjustable volume V , and diathermal walls, embedded in a heat bath of constant temperature T, and fixed pressure P The change in the entropy of the bath is given by A@bath ASbath AQgaS 1 7 - (AEgas + PAVas). By considering the change in entropy of the combined system establish that "the equilibrium of a gas at fixed T and P is characterized by the minimum of the Gibbs free energy G=E + P V - TS