Methanol is formed from carbon monoxide and hydrogen in the gas-phase reaction The mole fractions of the reactive species at equilibrium satisfy the relation where P is the total pressure (atm). K_e the reaction equilibrium constant (atm^-2) and T the temperature (K) The equilibrium constant K_e equals 10.5 at 373 K. and 2.316 x 10^-4 at 573 K. A semi log plot of K_e (logarithmic scale) versus 1/ T (rectangular scale) is approximately linear between T = 300 K and T = 600 K.

(a) Derive a formula for K_e (T) and use it to show that K_e (450K) 0.0548 atm^-2.

(b) Write expressions for n_A. n_B. and n_C (gram-moles of each species), and then y_A, y_B and y_C, in terms of n_A0, n_B0 and n_C0 the molar extent of reaction. Then derive an equation involving only n_A0 n_B0, n_C0, P. T. and where (mol) is the value of the extent of reaction at equilibrium.

(c) Suppose you begin with equimolar quantities of CO and H₂ and no CH₃OH and the reaction proceeds to equilibrium at 423 K and 2.00 atm, Calculate the molar composition of the product (y_A. y_B and y_C) and the fractional conversion of CO.

(d) Write a set of equations for y_A. y_B, y_C and f_A (the fractional conversion of CO) in terms of y_AB, y_B0 T. and P (the reactor temperature and pressure at equilibrium). Enter the equations in an equation-solving program. Check the program by running it for the conditions of part (c) then use it to determine the effects on f_A (increase, decrease, or no effect) of separately increasing (I) the fraction of CO in the feed (ii) the fraction of CH3OH in the feed (in) temperature and (iv) pressure.

(e) Write a computer program to take as input y_A0. y_B0. T, and P (the reactor temperature and pressure at equilibrium) and to calculate and print out y_A, y_B, y_C, and f_A (the fractional conversion of CO) Test your program with the data of part (c). [Suggestion: Write the equilibrium relations derived in part (b) as a cubic equation in and use Newton’s rule—Appendix A.2—to obtain the solution.]