Methanol is formed from carbon monoxide and hydrogen in the
gas-phase reaction

CO + 2H₂ <-> CH₃OH

(A) (B) (C)

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 semilog plot of Ke (logarithmic
scale) versus 1/T (rectangular scale) is approximatel ylinear
between T = 300K and T = 600K.

a) Derive a formula for K_e(T), and use it to show
that K_e(450) = 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 , the
extent of reaction. Then dervice an equation involving only
n_A0, n_B0, n_C0, P, T, and
_e where
_e is the
extent of reaction at equilibrium

c) Suppose you begin with equimole 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 fraction conversion of CO.