(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)

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4. Note that at the critical point, the three roots of the van der Waals equation (pv³ - (pb + RT)v² + av ab = o) are equal (i.e., (v - vc)³ = 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/27b². (Solution) Compare Pcv³ - (pcb + RTc) v² + a v - ab = 0 and Pcv³ – (3pcvc) v² + (3pcvc²) v − PcVc³ = 0. a = = 3pcvc²; ab = pcvc³ ⇒ 3pcvc²b = pcvc³ ⇒ 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/(3vc²)= a/27b². (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+ ³ Por² (1-16) V Pc P + 3 (v/vc)² 3 ( ²² + ²³² ) ( ^_-) ) = - RT V RT Pc Vc 1 3 8 = RT RT a - 36 276² 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 لطفيله 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.) 4. Note that at the critical point, the three roots of the van der Waals equation (pv³ - (pb + RT)v² + av ab = o) are equal (i.e., (v - vc)³ = 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/27b². (Solution) Compare Pcv³ - (pcb + RTc) v² + a v - ab = 0 and Pcv³ – (3pcvc) v² + (3pcvc²) v − PcVc³ = 0. a = = 3pcvc²; ab = pcvc³ ⇒ 3pcvc²b = pcvc³ ⇒ 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/(3vc²)= a/27b². (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+ ³ Por² (1-16) V Pc P + 3 (v/vc)² 3 ( ²² + ²³² ) ( ^_-) ) = - RT V RT Pc Vc 1 3 8 = RT RT a - 36 276² 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 لطفيله 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.)

Answer rating: 100% (QA)

a compare pcv3pcbRTcv2avab0 and pcv33pcvcv23pcvc2pcvc30 a3pcvc2 abp... View the full answer