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), Ke the reaction equilibrium constant (atm^-2), and T the temperature (K). The equilibrium constant Ke equals 10.5 at 373 K, and 2.316x10^-4 at 573 K. A semilog 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(450 K) = 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; n_C0, and ξ, the extent of reaction. Then derive an equation involving only n_A0; n_B0; n_C0, P, T, and ξ_e , where ξ_e is 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) The conversion of CO and H₂ can be enhanced by removing methanol from the reactor while leaving unreacted CO and H₂ in the vessel. Review the equations you derived in solving Part (c) and determine any physical constraints on ξ_e associated with n_A0 = nB0 = 1 mol. Now suppose that 90% of the methanol is removed from the reactor as it is produced; in other words, only 10% of the methanol formed remains in the reactor. Estimate the fractional conversion of CO and the total gram moles of methanol produced in the modified operation. 

(e) Repeat Part (d), but now assume that n_B0 = 2 mol. Explain the significant increase in fractional conversion of CO. 

(f) Write a set of equations for y_A; y_B ; y_C , and f A (the fractional conversion of CO) in terms of y_A0; 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 CH₃OH in the feed, (ii) temperature, and (iii) pressure.

Transcribed Image Text:

CO + 2H2 = CH3OH (C) (A) (B)