In the Haber process for the synthesis of ammonia, N ₂( g ) + 3 H ₂( g ) 2 NH ₃( g ) , the equilibrium constant K _ p is 4.0 10⁻⁴ at 500 K . If the backward reaction rate constant k _ b at this temperature is 2.0 10⁻² , what is the value of the forward reaction rate constant k _ f ? [Given: R = 0.08 L atm K ⁻¹ mol ⁻¹ ]

Step-by-step solution

For the reaction N ₂( g ) + 3 H ₂( g ) 2 NH ₃( g ) , the change in the number of moles of gaseous species is: n _ g = 2 - (1 + 3) = -2 The relationship between K _ p and K _ c is: K _ p = K _ c ( RT )^ n _ g Rearranging to solve for K _ c : K _ c = K _ p ( RT )^ n _ g = K _ p ( RT )^2 Calculate RT : RT = 0.08 500 = 40 Now, substitute the values to find K _ c : K _ c = (4.0 10⁻⁴) (40)^2 = (4.0 10⁻⁴) 1600 = 0.64 The equilibrium constant K _ c is also given by the ratio of the forward and backward rate constants: K _