At low to moderate pressures the equilibrium state of the water—gas shift reaction CO + H₂O = CO₂ + H₂ is approximately described by the relation where T is the reactor temperature. K_e is the reaction equilibrium constant, and y, is the mole fraction of species I in the reactor contents at equilibrium.

The feed to a batch shift reactor contains 20.0 mole% CO. 10.0% CO₂. 40.0% water and the balance an inert gas. The reactor is maintained at T = 1123 K.

(a) Assume a basis of 1 mol feed and draw and label a flowchart. Carry out a degree-of-freedom analysis of the reactor based on extents of reaction and use it to prove that you have enough information to calculate the composition of the reaction mixture at equilibrium. Do no calculations.

(b) Calculate the total moles of gas in the reactor at equilibrium (if it takes you more than 5 seconds you’re missing the point) and then the equilibrium mole fraction of hydrogen in the product. (Suggestion.’ Begin by writing expressions for the moles of each species in the product gas in terms of the extent of reaction, and then write expressions for the species mole fractions.)

(c) Suppose a gas sample is drawn from the reactor and analyzed shortly after startup and the mole fraction of hydrogen is significantly different from the calculated value. Assuming that no calculation mistakes or measurement errors have been made, what is a likely explanation for the discrepancy between the calculated and measured hydrogen yields?

(d) Write a spreadsheet to take as input the reactor temperature and the feed component mole fractions X_CO, XH₂O, and X_CO2, (assume no hydrogen is fed) and to calculate the mole fraction YH2 in the product gas when equilibrium is reached. The spreadsheet column headings should be T x (C0) x (H₂0) x (C02) K_e . . .  y(H₂) Columns between Ke and y(H₂) may contain intermediate quantities in the calculation of y3. First test your program for the conditions of part (a) and verify that it is correct. Then try a variety of values of the input variables and draw conclusions about the conditions (reactor temperature and feed composition) that maximize the equilibrium yield of hydrogen.