Given the Arrhenius equation, k = Ae-Ea>RT, and the rela- tion between the equilibrium constant and the forward and reverse rate constants, Kc = kf>kr, explain why Kc for an exothermic reaction decreases with increasing temperature.
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Step 1: Understand the Arrhenius equation. The Arrhenius equation, k = Ae^(-Ea/RT), describes how the rate constant (k) of a reaction depends on the temperature (T). Here, A is the pre-exponential factor, Ea is the activation energy, R is the gas constant, and T is the temperature. The equation shows that as temperature increases, the rate constant also increases, but this increase is moderated by the activation energy.
Step 2: Understand the relationship between equilibrium constant and rate constants. The equilibrium constant (Kc) is the ratio of the forward rate constant (kf) to the reverse rate constant (kr), as given by Kc = kf/kr. This means that the equilibrium constant depends on the rates of the forward and reverse reactions.
Step 3: Consider an exothermic reaction. In an exothermic reaction, the forward reaction releases energy (negative Ea), while the reverse reaction requires energy (positive Ea). Therefore, according to the Arrhenius equation, an increase in temperature will have a greater effect on the rate constant of the reverse reaction (kr) than on the forward reaction (kf).
Step 4: Apply the relationship between equilibrium constant and rate constants. As temperature increases, both kf and kr increase, but kr increases more. Therefore, the ratio kf/kr (which is Kc) decreases.
Step 5: Therefore, for an exothermic reaction, the equilibrium constant Kc decreases with increasing temperature. This is because the increase in temperature favors the endothermic reverse reaction more than the exothermic forward reaction, shifting the equilibrium towards the reactants and decreasing Kc.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Arrhenius Equation
The Arrhenius equation describes how the rate constant (k) of a chemical reaction depends on temperature (T) and activation energy (Ea). It states that k = Ae^(-Ea/RT), where A is the pre-exponential factor. As temperature increases, the exponential term becomes larger, leading to an increase in k, which generally accelerates the reaction rate.
The equilibrium constant (Kc) quantifies the ratio of the concentrations of products to reactants at equilibrium for a reversible reaction. For a reaction with forward rate constant (kf) and reverse rate constant (kr), Kc = kf/kr. Changes in temperature can affect kf and kr differently, influencing the value of Kc.
Le Chatelier's Principle states that if a system at equilibrium is subjected to a change in temperature, pressure, or concentration, the system will adjust to counteract that change and restore a new equilibrium. For exothermic reactions, increasing temperature shifts the equilibrium position towards the reactants, resulting in a decrease in Kc.
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