Problem 13 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

For a reversible reaction, the rate constants for the forward and backward reactions are \(0.16\) and \(4 \times 10^{4}\), respectively. What is the value of equilibrium constant of the reaction? (a) \(0.25 \times 10^{6}\) (b) \(2.5 \times 10^{5}\) (c) \(4 \times 10^{-6}\) (d) \(4 \times 10^{-4}\)

Step-by-Step Solution

Answer

The value of the equilibrium constant of the reaction is (c) 4 x 10^{-6}.

Step 1 - Understanding the Relationship Between Rate Constants and Equilibrium Constant

The equilibrium constant, K, for a reversible reaction is related to the rate constants of the forward (k_f) and reverse (k_r) reactions by the equation K = k_f / k_r. This relationship is derived from the equilibrium condition where the rate of the forward reaction equals the rate of the reverse reaction.

Step 2 - Substituting the Given Values

Given the rate constants for the forward and backward reactions are 0.16 and 4 x 10^4 respectively, substitute these values into the equilibrium constant equation: K = 0.16 / (4 x 10^4).

Step 3 - Calculating the Equilibrium Constant

Perform the division to find the equilibrium constant. So, K = 0.16 / (4 x 10^4) = 4 x 10^-6.

Step 4 - Compare with Given Options

Match the calculated value of the equilibrium constant with the given options. The correct answer is (c) 4 x 10^-6.

Key Concepts

Reversible Reaction

Understanding reversible reactions is crucial when studying chemical kinetics and equilibrium. A reversible reaction is one where the conversion of reactants to products and the conversion of products back to reactants occur simultaneously. This means that these reactions can proceed in both directions - forward and reverse.

For instance, if we have a reaction as follows: \( A + B \rightleftharpoons C + D \), both the formation of \( C + D \) from \( A + B \) (forward reaction) and the reverse process (reverse reaction) are happening at the same time. What's interesting about these reactions is that they can reach a state where the rates of the forward and reverse reactions are equal, leading to what we know as chemical equilibrium. This balance is dynamic, as the reactions continue to occur, but the concentrations of the reactants and products remain constant over time.

In the context of the given exercise, we have rate constants provided for both the forward and backward reactions. These constants are simply the speeds at which the forward and reverse reactions proceed. Knowing these allows us to determine the overall behavior of the system and calculate the equilibrium constant, which quantifies the position of equilibrium.

Rate Constants

Rate constants are the numerators in the algebra of chemical reactions. They quantitatively describe how quickly a chemical reaction proceeds. For any chemical reaction, the rate constant for the forward reaction (\(k_f\)) and the rate constant for the reverse reaction (\(k_r\)) are unique and determined experimentally. These values vary with temperature and are independent of the concentration of reactants and products.

Importance of Rate Constants

Knowing the rate constants for a reversible reaction provides the ability to understand how likely a reaction is to go forward or reverse under given conditions. For example, a high forward rate constant compared to the reverse rate constant means the reaction strongly favors the formation of products. Conversely, if the reverse rate constant is larger, the reaction prefers to stay as reactants or to re-form reactants if the products are formed.

In practical terms, calculating the equilibrium constant from rate constants allows chemists to predict the extent to which a reaction will proceed which greatly aids in process optimization and control within chemical industries.

Chemical Equilibrium

Chemical equilibrium is a fundamental concept in chemistry that describes a state where the rates of the forward and reverse reactions are equal, resulting in no net change in the concentrations of reactants and products over time. This does not imply that the reactions have stopped, but rather that they are proceeding at an equal rate in both directions.

The equilibrium constant (\(K\)) is a ratio that quantifies the state of equilibrium. It's derived from the rate constants of the forward and reverse reactions (\(K = k_f / k_r\)), making it clear how these kinetic parameters directly influence the position of equilibrium. For a simple reaction such as \(aA + bB \rightleftharpoons cC + dD\), the equilibrium constant is given by \(K = [C]^c [D]^d / [A]^a [B]^b\), where the square brackets indicate the concentration of each species.

Significance of the Equilibrium Constant

Knowing the value of the equilibrium constant gives insight into the extent of a reaction: a larger value of \(K\) suggests that, at equilibrium, the concentration of products is high relative to the concentration of reactants. This is crucial when designing chemical processes or synthesizing compounds - chemists must know whether the reaction will yield enough product to be practical. In the exercise provided, by calculating the equilibrium constant from the given rate constants, the position of equilibrium can be determined, answering the crucial question of whether the reaction favors the products or the reactants.