Chapter 5: Isothermal Reactor Design: Conversion

Deriving The Equilibrium Constant (K_C) and Equilibrium Conversion(X_e) for a Constant Volume System:

You are given the reversible reaction:

\( 2A \leftrightharpoons B \)

which takes place in a constant volume batch reactor. The equilibrium constant, K_C, for this reaction is:

\( K_C = \frac{C_{Be}}{C_{Ae}^2} \)

where C_Ae and C_Be are:

\( C_{Ae} = C_{A0}(1 - X_e) \)

\( C_{Be} = C_{A0}\left(\frac{1}{2}X_e\right) \)

Substituting for C_Ae and C_Be gives us:

\( K_C = \frac{C_{A0}\left(\frac{1}{2}X_e\right)}{C_{A0}^2(1 - X_e)^2} \)

\( K_C = \frac{X_e}{2C_{A0}(1 - X_e)^2} \)

Substituting known values (C_A0 = 0.2 mol/dm³ and K_C = 100 dm³/mol):

\( 100 = \frac{X_e}{2(0.2)(1 - X_e)^2} \)

Solving for the equilibrium conversion, X_e, yields:

\( X_e = 0.83 \)