27 For a real fluid, the internal energy u, and in general, C = (au/at), depend on both v and T. (a) In this part, we will derive how G, depends on v given a fixed T. To do so, we must obtain an expression for (8c,/aut. We will start with the expression for that we derived in class: C = T(as/aT"), and switch the order in which derivatives with respect to v and T are taken (in the spirit of Maxwell relationships), to show that: ac, a as = T- au at Then, using the appropriate Maxwell relationship, show that: ace a2p au ar This expression highlights that the dependence of c, on v (at constant T') is dictated entirely by the EOS. T () (6)]. , (). -"). T (b) Using the above expression, determine whether c, depends on v for a fluid that obeys the: (i) ideal gas EOS, (ii) van der Waals EOS, (iii) Berthelot EOS. (c) For a fluid that obeys the Berthelot EOS and for which was a function of T in the ideal gas limit (v +00) is given, e.g., CLC (T) = A + BT, integrate the expression you derived in part (a) to show that: 2a c(0,T) = (T) + 0,1 T2y IG