The Hampson-Linde cycle is the key step in most gas liquefaction processes, invented by William Hampson and Carl von Linde in 1895. The cycle uses successive pressurization and expansion through a Joule-Thomson orifice (essentially a throttling valve) to achieve cooling until the gas is cold enough to condense. In this problem, we will study a simplified version of the process. Argon gas at 290K and 1bar in a steady flow of 1mol/s is first compressed isothermally to 70 bar, and then passed through a throttling valve until the pressure goes back to 1 bar. The throttling valve can be assumed to be adiabatic. The ideal gas assumption does not work in our case, because it fails to produce the cooling effect when a real gas is passed through a valve (called the Joule-Thomson effect). Instead of using the ideal gas assumption, we will use the Pitzer B correlation for argon: Z=RTPV=1+B0TrPr where: B0=0.083Tr1.60.422 Here, Tr=T/Tc is the reduced temperature and Pr=P/Pc is the reduced pressure. Note that for argon, the acentric factor =0, so we can leave out the B1 term. The critical temperature is Tc= 150.9K and the critical pressure is Pc=48.98 bar. (a) Prove that the residual molar enthalpy and residual molar entropy are given by the following expressions for this equation of state: HR(Tr,Pr)=RTcPr(B0TrdTrdB0)SR(Tr,Pr)=RPr(dTrdB0) (b) Assuming the compression step is reversible, calculate the heat rate and the shaft power of the compressor, using the residual property formulae above. (c) Calculate the final temperature of the argon gas after passing through the valve. For argon gas CPig=2.5R and can be assumed to be independent of temperature