For an elastic filament it is found that, at a finite range in temperature, a displacement x force J = ax - bT + cTx, where a, b, and c are constants. Furthermore, requires a its heat capacity at constant displacement is proportional to temperature, i.e. c_x = A(x)T. Use an appropriate Maxwell relation to calculate partial differential S/partial differential x| _T. Show that A has to in fact be independent of x, i.e. dA/dx = 0. Give the expression for S(T, x) assuming S(0, 0) = S_0. Calculate the heat capacity at constant tension, i.e. C_j = T partial differential S/partial differential T|, as a function of T and J.
