15-2 (a) For a system of localized distinguishable oscillators, Boltzmann statistics applies. Show that the entropy S is given by N In N (b) Substitute the Boltzmann distribution in the previous result to show that U S = T + Nk in Z. (c) Using the expressions derived in the text for U and T, prove that 8/T S = Nk - 8/T 1 = where hvik. Examine the behavior of S as T approaches zero.