Question

1. (12 points) A cyclic process is given in the figure below. It starts at P1, Vi, goes along path A to P2, V2, and then returns to the start by going along path B. p is the pressure and V is the volume. This system is an ideal gas, which obeys the “ideal gas law": pV = NKT (this is sometimes called the equation of state). (a) Calculate the total work done. This will be a formula in terms of V1, V2, Pı and p2. (b) Calculate the efficiency n in terms of V1, V2, P1 and P2. This is given by n = Wout/Qin, where Wout is the work done by the system, and Qin is the heat input to the system. Use the formula: Q = Ufinal – Vinitial – W, where Q is the heat input to the system and W is the work done on the system. NOTE: you will need the formula for the internal energy of an ideal gas: U = {pV. Fig 3

Answer 1

solt: Given, Ideel, (P2 ) A2 (Para) 1. net PV=NKT еля! #ne/ expression ou UF3. PvPitan Also, Enpression for heat, 1Q=ufiner Unitial - W ] For Process 1-2. v=const. Boi , Vieva W= p. BV = 0 For Process 2-3 @ Vi=va V2 =V6 and P = Pa 1-2 o P=const. :: P = Pa work done during process 2-3, a This = P2 (V2-V.) Process 3-4, V=const. cV=V6 For We zo and For Processo TP P PCC ... 4. = P(V-V2)

Now, ow. Wnet. = Wout. I Wort West wyth & Want = 07 Pe(V2_W) +0+ Pily-vel = PzV2-P, VI +'P, V, - PVR = B (uz -V) - P(vz-V) I Wout. = (P2-P) (V2-vi) / Total work Now, during (b) from given, equation, Q = Uzinel - Vinitial - W For 1-20 procen Q = 3 Vi (P2 - P.) -0 L uintino 3 live For Proces 2-3 dene this cycle. urin 3 byl wca 19 = 34 (P2-P, 1-0 Q = 3 PV3- Z PV – P (V2-V1) = 3 P2(V3-V) – P2 (U2-4) 3 Pa (V2-V1] og tve 2-3 For 3-4 procees, Q = 3 P, V 2 - 3 P ₂ V2 - 0 For Procels 4-1, Q. = 3 Pivi - 3 P, V2 – P, (V, V2] Q / Pilvi-U2) - ve

From the expresion of an we can see, Q and O is positive and input heat is taken al positive so F = 34(62-203 76 Pz (v2=8) = 4(PzP.) +} Pelva-) - 3V, P2 - 3 Vipit PzV₂ -P₂4 = 2 P2 V, - 3 v P, + PZ VZ 2 Pz (1272 V, 5 – 3 PVi Gain = V (202 –58) Pava 2 So efficiency of cycle," n. Woul Qin (P2-P,) (V2_V, J x2 V(212-3p, the - Party. (P-P) (v2-V) V, (2P2 - 3 P.) + P2 V2