2012 Semifinal Exam Part A 6 Question A2 An ideal(but not necessarily perfect monatomic)gas undergoes the following cycle. The gas starts at pressure Po,volume Vo and temperature To The gas is heated at constant volume to a pressure aPo,where a>1. The gas is then allowed to expand adiabatically (no heat is transferred to or from the gas)to pressure Po The gas is cooled at constant pressure back to the original state. The adiabatic constant y is defined in terms of the specific heat at constant pressure Cp and the specific heat at constant volume Co by the ratio y=Cp/Co. a.Determine the efficiency of this cycle in terms of a and the adiabatic constant y.As a reminder,efficiency is defined as the ratio of work out divided by heat in. b.A lab worker makes measurements of the temperature and pressure of the gas during the adiabatic process.The results,in terms of To and Po are Pressure units of Po1.211.411.591.732.14 Temperature units of T62.112.212.282.342.49 Plot an appropriate graph from this data that can be used to determine the adiabatic constant. c.What is y for this gas? Solution a.Label the end points as 0,1,and 2.A quick application of PV =nRT requires that Ti=aTo It takes more work to do the process 1-2;it is acceptable to simply state the adiabatic law of PVY=constant;if you don't know this,you will need to derive it. In the case that you know the adiabatic process law, PiVi=PV=aPiV, so that =(a)号 Another quick application of PV =nRT requires that T2=(a)To. Heat enters the gas during isochoric process 0-1,so Qin nCuAT=nCu(a-1)To Heat exits the system during process2一→0，so Qout nCpAT nCp(a/-1)To Copyright C2012 American Association of Physics Teachers2012 Semifinal Exam Part A 6 Question A2 An ideal (but not necessarily perfect monatomic) gas undergoes the following cycle. • The gas starts at pressure P0, volume V0 and temperature T0. • The gas is heated at constant volume to a pressure αP0, where α > 1. • The gas is then allowed to expand adiabatically (no heat is transferred to or from the gas) to pressure P0 • The gas is cooled at constant pressure back to the original state. The adiabatic constant γ is defined in terms of the specific heat at constant pressure Cp and the specific heat at constant volume Cv by the ratio γ = Cp/Cv. a. Determine the efficiency of this cycle in terms of α and the adiabatic constant γ. As a reminder, efficiency is defined as the ratio of work out divided by heat in. b. A lab worker makes measurements of the temperature and pressure of the gas during the adiabatic process. The results, in terms of T0 and P0 are Pressure units of P0 1.21 1.41 1.59 1.73 2.14 Temperature units of T0 2.11 2.21 2.28 2.34 2.49 Plot an appropriate graph from this data that can be used to determine the adiabatic constant. c. What is γ for this gas? Solution a. Label the end points as 0, 1, and 2. A quick application of P V = nRT requires that T1 = αT0. It takes more work to do the process 1 → 2; it is acceptable to simply state the adiabatic law of P V γ = constant; if you don’t know this, you will need to derive it. In the case that you know the adiabatic process law, P1V γ 1 = P2V γ 2 = αP1V γ 2 , so that V2 = V1 (α) 1 γ . Another quick application of P V = nRT requires that T2 = (α) 1 γ T0. Heat enters the gas during isochoric process 0 → 1, so Qin = nCv∆T = nCv(α − 1)T0 Heat exits the system during process 2 → 0, so Qout = nCp∆T = nCp(α 1/γ − 1)T0 Copyright c 2012 American Association of Physics Teachers