The equilibrium constant for the ethane dehydrogenation reaction,

is defined as

where P(atm) is the total pressure and yi is the mole fraction of the ith substance in an equilibrium mixture. The equilibrium constant has been found experimentally to vary with temperature according to the formula

The heat of reaction at 1273K is +145:6 kJ, and the heat capacities of the reactive species may be approximated by the formulas

Suppose pure ethane is fed to a continuous constant-pressure adiabatic reactor at 1273K and pressure P(atm), the products emerge at T_f(K) and P(atm), and the residence time of the reaction mixture in the reactor is large enough for the outlet stream to be considered an equilibrium mixture of ethane, ethylene, and hydrogen.

(a) Prove that the fractional conversion of ethane in the reactor is

(b) Write an energy balance on the reactor, and use it to prove that

Where

Finally, substitute for ΔHr and the heat capacities in Equation 4 to derive an explicit expression for ϕ(T_f).

(c) We now have two expressions for the fractional conversion f : Equation 2 and Equation 3. If these expressions are equated, Kp is replaced by the expression of Equation 1, and ϕ(T_f) is replaced by the expression derived in Part (b), the result is one equation in one unknown, T_f. Derive this equation, and transpose the right side to obtain an expression of the form

Ψ(T_f) = 0 …………… (5)

(d) Prepare a spreadsheet to take P as input, solve Equation 5 for Tf (use Goal Seek or Solver), and determine the final fractional conversion, f . (Suggestion: Set up columns for P, T_f , f , K_p, ϕ, and ψ.)

Run the program for P(atm) = 0:01, 0.05, 0.10, 0.50, 1.0, 5.0, and 10.0. Plot T_f versus P and f versus P, using a logarithmic coordinate scale for P.

Transcribed Image Text:

C,H6(g)= C,H4(g) + H2(g)