Consider a general, single-step reaction of the type A + B ∆ C. Show that the equilibrium constant is equal to the ratio of the rate constants for the forward and reverse reactions, Kc = kf>kr.
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Identify the forward and reverse reactions: In the given reaction A + B ⇄ C, the forward reaction is A + B → C, and the reverse reaction is C → A + B.
Define the rate constants: Let kf be the rate constant for the forward reaction and kr for the reverse reaction.
Write the rate expressions for both reactions: The rate of the forward reaction can be expressed as rate_forward = kf[A][B], and the rate of the reverse reaction as rate_reverse = kr[C].
Set up the equilibrium condition: At equilibrium, the rate of the forward reaction equals the rate of the reverse reaction. Therefore, kf[A][B] = kr[C].
Solve for the equilibrium constant, Kc: Rearrange the equilibrium condition to find Kc = \frac{kf}{kr}. This shows that the equilibrium constant for the reaction is indeed the ratio of the rate constants of the forward and reverse reactions.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Equilibrium Constant (Kc)
The equilibrium constant (Kc) is a dimensionless value that expresses the ratio of the concentrations of products to reactants at equilibrium for a given reaction at a specific temperature. For the reaction A + B ⇌ C, Kc is defined as [C]/([A][B]), indicating the extent to which the reaction favors products or reactants at equilibrium.
Rate constants (kf for the forward reaction and kr for the reverse reaction) quantify the speed of a reaction. They are specific to each reaction and depend on factors such as temperature and activation energy. The relationship between these constants is crucial for understanding how quickly reactants convert to products and vice versa.
In a dynamic equilibrium, the rates of the forward and reverse reactions are equal, leading to constant concentrations of reactants and products. This balance allows us to relate the equilibrium constant to the rate constants, as Kc = kf/kr, demonstrating that the position of equilibrium is influenced by the relative speeds of the forward and reverse processes.
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