(c) Show that Zc = pcvc/RTc = 3/8 = 0.375. Compare this value from the value seen from the above generalized compressibility chart. Which one is larger? (You may discuss the accuracy.) ( Only this part is needed to be solved)

4. Note that at the critical point, the three roots of the van der Waals equation (pv - (pb + RT)v + av ab = o) are equal (i.e., (v - vc) = 0). (Upper bars are omitted for the sake of convenience.) (a) Show that the critical values of the specific volume, temperature, and pressure are given by vc = 3b, Tc = 8a/27Rb, and pc = a/27b. (Solution) Compare Pcv - (pcb + RTc) v + a v - ab = 0 and Pcv (3pcvc) v + (3pcvc) v PcVc = 0. a = = 3pcvc; ab = pcvc 3pcvcb = pcvc b = vc/3 or vc = 3b. (a = 3pcvc(3b) pcvc = a/9b) pcb + RTc = 3pcvc RTc = 3pcvc - pcb = 3pcvc - pcvc/3 = (3 - 1/3) pcvc = (8/3) pcvc = (8/3) (a/9b) = 8a/27b Tc = 8a/27Rb pc=a/(3vc)= a/27b. (b) Show that in terms of the reduced quantities v = v/vc,T =T/Tc,p = p/ pc, the van der Waals equation becomes + VR V (Solution) a (p++/ ) (v-b) = RT 3 p+ 1+ Por (1-16) V Pc P + 3 (v/vc) 3 ( + ) ( ^_-) ) = - RT V RT Pc Vc 1 3 8 = RT RT a - 36 276 T a 9Rb T 3 8a 8 27 Rb 8 T 3 3 Tc 8/3 ==T RT Z = PV 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.5 Co+7404 8078 1.0 x Au ex TR = 2.00 TR= 1.00 0 D X TR= 1.10 X X TR = 1.20 1.5 2.0 2.5 TR = 1.50 TR = 1.30 X X 08 * Methane o Ethylene A Ethane Propane n-Butane 3.0 3.5 4.0 4.5 Reduced pressure PR A Legend Isopentane en-Heptane 6 Nitrogen 5.0 5.5 Average curve based on data on hydrocarbons Carbon dioxide Water 6.0 B 6.5 7.0 (c) Show that Zc = pcvc/RTc = 3/8 = 0.375. Compare this value from the value seen from the above generalized compressibility chart. Which one is larger? (You may discuss the accuracy.)