The van der Waals equation of state (Equation 5.3-6) is to be used to estimate the specific molar volume V (L/mol) of air at specified values of T (K) and P(atm). The van der Waals constants for air are a = 1.33 atm ? L²/mol² and b = 0.0366 L/mol.

(a) Show why the van der Waals equation is classified as a cubic equation of state by expressing it in the form f(V) = c₃V^3?+ c₂V^2?+ c₁V + c₀ = 0 where the coefficients c₃, c₂, c₁, and c₀ involve P. R, T, a, and b. Calculate the values of these coefficients for air at 223 K and 50.0 atm. (Include the proper units when giving the values.)

(b) What would the value of V be if the ideal gas equation of state were used for the calculation? Use this value as an initial estimate of V for air at 223 K and 50.0 atm and solve the van der Waals equation by trial and error to obtain a better estimate. What percentage error results from the use of the ideal gas equation of state, taking the van der Waals estimate to be correct?

(c) Set up a spreadsheet to carry out the calculations of part (b) for air at 223 K and several pressures. The spreadsheet should appear as follows:

The polynomial expression for V (f = c₃V³ + c₂V²+ ???) should be entered in the f(V)column, and the value in the V column should be varied until f(V) is essentially zero. Use goal-seeking if the spreadsheet program includes this feature.

(d) Do the calculation for 223 K and 50.0 atm using the Newton-Raphson method (Appendix A.2).

Transcribed Image Text:

T(K) P(atm) 223 223 223 223 223 1.0 10.0 50.0 100.0 200.0 c3 c2 cl ... ... ..E *** ... ⠀⠀ *** : ... ... c0 V(ideal) (L/mol) ... ... ... ... V (L/mol) ... f(V) ... ... ... % error