The following two reactions take place in a gas-phase reactor: If the system comes to equilibrium at 3000 K and 1 atm, the product gas mole fractions satisfy the relations

(a) Let n_A0 . . . n_E0 be the initial number of gram-moles of each species and ?₁ and ?_2?be the extents of reactions 1 and 2 respectively, at equilibrium (see Equation 4.6-6). Derive expressions for the mole fractions y_A, y_B . . . ,y_E in terms of n_A0, n_B0 . . . n_E0, ?₁ and ?₂._?Then substitute in the equilibrium relations to derive two simultaneous equations for the two extents of reaction.

(b) One-third of a gram-mole each of CO₂, O₂, and N₂ are charged into a batch reactor and the reactor contents equilibrate at 3000 K and 1 atm. Without doing any calculations, prove that you have enough information to calculate the component mole fractions of the reactor contents at equilibrium.

(c) Perform the calculation of part (b), using either (i) an equation solving program or (ii) a spreadsheet that implements the Newton-Raphson method outlined in Section A.2i of Appendix A. If you use the spreadsheet, guess initial values of 0.1 for both ?_1?and ?₂ and iterate until successive estimates of these values differ by less than 0.1%.

(d) Write a computer program to implement the Newton?Raphson procedure of part (c) for an arbitrary starting composition. The program should take input values of n_A0, n_B0, n_C0, n_D0, and n_E0 and calculate the total moles and mole fractions of each species at equilibrium, stopping when the values of and each change by less than 0.001% from one iteration to the next. Run the program for the following feed mixtures:

MAD 1/3 0 1/2 1/5 2 CO (A) O + N (C) (D) = 0.1071 0 1/3 0 1/5 nco 1/3 1/3 0 1/5 2CO + O (B) (C) 2NO (E) YE YCYD = 0.01493 nDo 1/3 1/3 0 1/5 NEO 0 0 1/2 1/5