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Problem: The following virial equation of state (originally developed by Beattie and Bridgeman, Proc. Am. Acad. Arts Sci., 63, 229-308 (1928)) was proposed by Deming and Shupe, J.Am. Chem. Soc., 52, 1382-1389 (1930), for the prediction of the thermal properties of Nitrogen in the temperature range [-70℃,400°C] and in the pressure  range [0atm, 400atm]

where {a, A₀, b, B₀, c} are numerical constants that were determined from the experimental data. The values reported by Deming and Shupe are the following when P the pressure,is in atm, V the specific volume,is in g/cm³and R the Universal Gas constant,in cm³atm/(g·K)

Table 1. VirialEquation Constants

Later in 1952, Lunbeck, Michels and Wolkers, Appl. Sci. Res., A3:197(1952),reported Z the compressibility factor of Nitrogen at very high pressures. The data are shown in the table below:

The following modification of the Deming-Shupe virial equation is proposed to include the new high-pressure data:

where d is a new numerical constant to be determined by the high-pressure data.

Calculations to be performed:

1.Develop a methodology to estimate the new constant d from the high-pressure experimental by minimizing the following objective function:

Hint:Recast the virial equation of state in the form.

where Z is the compressibility factor,p is the molar[or mass]density of nitrogen and  {β,y,8} are coefficients that depend on temperature and on the constants {a,A,b₃B,c,@.For the estimation of the new constant dkeep the values of the rest of the constants {a,A,b,B₀,c}the same as given in the Table 1 above.Prepare two parity plots,the first with the value of the constant d set equal to zero and the second with the optimal value calculated by the nonlinear regression.

2.Calculate the thermal properties of Nitrogen{β,Cp,Cv,μπ,Dsomd}at temperature

T=-60C and pressure P=800atm using the modified virial equation.Express your answers in SI units.Compare your results with the values obtained using the(RKS)Redlich- Kwong-Soave Equation of State and the values reported in the NIST Webbook.

3.Calculate the power required to compress Nitrogen reversibly and adiabatically from

T=-60℃ and P=800atm to P=1000 atm.Express your result in kJ/kg and compare it with the one obtained using the(RKS)Redlich-Kwong-Soave Equation of State.

4.Verify whether the Beattie-Bridgeman Equation of State is consistent at the critical point of  nitrogen.If based on the deviations from consistency,it is not,make a recommendation how to rectify the inconsistency.