Question

At $800\mathrm{K},$ the equilibrium constant for the reaction ${\mathrm{A}}_2(g)\rightleftharpoons 2\mathrm{A}(g)$ is $K_c=3.1\times {10}^{-4}$ . (a) Assuming both forward and reverse reactions are elementary reactions, which rate constant do you expect to be larger, $k_f$ or $k_r$ ? (b) If the value of $k_f=0.27{\mathrm{s}}^{-1},$ what is the value of $k_r$ at 800 $\mathrm{K}?(\mathrm{c})$ Based on the nature of the reaction, do you expect the forward reaction to be endothermic or exothermic? (d) If the temperature is raised to $1000\mathrm{K},$ will the reverse rate constant $k_r$ increase or decrease? Will the change in $k_r$ be larger or smaller than the change in $k_f?$

Short answer

(a) The reverse rate constant, $k_r$, is larger than the forward rate constant, $k_f$. (b) The value of the reverse rate constant, $k_r$, is approximately $870.97{\mathrm{s}}^{-1}$. (c) The forward reaction is endothermic. (d) At 1000 K, the reverse rate constant, $k_r$, will decrease, and the change in $k_r$ will be smaller than the change in $k_f$.

Step-by-step solution

Equilibrium Constant Relationship

Equilibrium constants can be related to the forward and reverse reaction rate constants with the following formula: $$K_c=\frac{k_f}{k_r}$$ Here, Kc is the equilibrium constant, kf is the forward rate constant and kr is the reverse rate constant. Step 2: Determine the larger rate constant

Comparing kf and kr

To determine which rate constant is larger, we can analyze their respective values when the reaction is at equilibrium. Since Kc is 3.1 x 10^(-4) and this value is less than 1, it indicates that the denominator (kr) is larger than the numerator (kf). Thus, the reverse rate constant, kr, is larger than the forward rate constant, kf. Step 3: Calculate the reverse rate constant (kr)

Calculating kr

Given the value of the forward rate constant, kf = 0.27 s^(-1), we can find the reverse rate constant, kr, using the formula for the equilibrium constant as seen in Step 1. $$kr=\frac{k_f}{K_c}$$ $$kr=\frac{0.27{\mathrm{s}}^{-1}}{3.1\times {10}^{-4}}$$ Now, calculate the value of kr: $$kr\approx 870.97{\mathrm{s}}^{-1}$$ Step 4: Determine the endothermic or exothermic nature of the forward reaction

Endothermic or Exothermic

Since the value of Kc increases with temperature (given in part d of the problem), it indicates that the forward reaction is endothermic. The equilibrium shifts for an endothermic reaction to favor the products when the temperature is increased. Step 5: Analyzing how rate constants change with temperature

Changes in Rate Constants with Temperature

If the temperature is raised to 1000 K: (a) The reverse rate constant kr will decrease because the equilibrium shifts to favor the products (due to the endothermic nature of the forward reaction). (b) The change in kr will be smaller than the change in kf. The reaction becomes more product-favored at higher temperatures, which indicates that the forward reaction rate will increase by a larger amount compared to the reduction in the reverse reaction rate.

Key concepts

Rate Constants

Understanding rate constants is crucial when studying chemical reactions. Rate constants, denoted as $k_f$ for the forward reaction and $k_r$ for the reverse, dictate the speed at which reactants transform into products and vice versa. The forward rate constant $k_f$ relates to how fast the reaction proceeds from left to right, converting ${\mathrm{A}}_2(g)$ to $2\mathrm{A}(g)$. Similarly, $k_r$ represents the speed of the reverse process.In a chemical equilibrium, the equilibrium constant $K_c$ is given by $K_c=\frac{k_f}{k_r}$. This ratio helps determine which rate constant is larger based on the equilibrium scenario. When $K_c$ is less than 1, as in this problem, it indicates a larger $k_r$ compared to $k_f$, meaning the reaction favors the formation of reactants over products at equilibrium. Adjustments in rate constants by changing conditions like temperature can significantly affect equilibrium positions, highlighting the dynamic nature of chemical reactions.

Equilibrium Constant

The equilibrium constant, represented by $K_c$, gives us vital information about the composition of a chemical system at equilibrium. It is calculated as the ratio of the rate constants for the forward and reverse reactions as $K_c=\frac{k_f}{k_r}$. In simpler terms, $K_c$ expresses how much of the reactants have converted into products at a given temperature.A small $K_c$ value, like 3.1 x 10^{-4}, means that the concentration of reactants is much higher than that of products when equilibrium is reached, implying that the reverse reaction rate constant $k_r$ is greater than the forward rate constant $k_f$. When the temperature changes, the equilibrium constant can shift, favoring either the reactants or products more. As such, it becomes a crucial parameter to predict how shifts in conditions such as pressure and temperature affect the balance between reactants and products.

Endothermic and Exothermic Reactions

In the world of chemical reactions, the terms endothermic and exothermic describe how energy is exchanged. An endothermic reaction absorbs energy from its surroundings, resulting in a positive enthalpy change. Conversely, an exothermic reaction releases energy, reflected by a negative enthalpy change.The problem hints at an endothermic forward reaction because the equilibrium constant $K_c$ increases with temperature. According to Le Chatelier's principle, if a reaction is endothermic, increasing the temperature will shift the equilibrium towards the products, signifying that more energy is needed to reach an equilibrium state favoring the forward reaction.Interestingly, higher temperatures will reduce the reverse rate constant $k_r$ but increase the forward rate constant $k_f$ more significantly. This reflects how systems can adjust to balance reaction energies, aligning with our understanding of thermochemical behavior.