For a reversible gas-phase reaction, X(g) + Y(g) Z(g) the rate constant for the forward reaction is 82 times the rate constant for the backward reaction at 500 K. The value of equilibrium constant K _ p for the reaction at 500 K is: [Given : R = 0.082 ~L~atm~mol⁻¹~K⁻¹ ]

Step-by-step solution

The equilibrium constant in terms of concentration, K_ c , is given by the ratio of the forward and backward rate constants: K_ c = k_ f k_ b Given that k_ f = 82 k_ b , we have: K_ c = 82 For the reaction X(g) + Y(g) Z(g) , the change in the number of moles of gas is: n_ g = moles of gaseous products - moles of gaseous reactants = 1 - (1 + 1) = -1 The relationship between K_ p and K_ c is: K_ p = K_ c (RT)^ n_ g Substituting the given values ( R = 0.082 ~L~atm~mol⁻¹~K⁻¹ and T = 500 ~K ): K_ p = 82 (0.082 500)⁻¹ K_