Question

An amount of 1 mole each of \(\mathrm{A}\) and \(\mathrm{D}\) is introduced in \(1 \mathrm{~L}\) container. Simultaneously the following two equilibria are established: \(\mathrm{A} \rightleftharpoons \mathrm{B}+\mathrm{C} ; K_{\mathrm{C}}=10^{6} \mathrm{M}^{2}\) and \(\mathrm{B}+\mathrm{D} \rightleftharpoons \mathrm{A} ; K_{\mathrm{C}}=10^{-6} \mathrm{M}^{-1}\) The equilibrium concentration of \(\mathrm{A}\) will be (a) \(10^{-6} \mathrm{M}\) (b) \(10^{-3} \mathrm{M}\) (c) \(10^{-12} \mathrm{M}\) (d) \(10^{-4} \mathrm{M}\)

Short answer

The equilibrium concentration of A is close to 1 M, so none of the given options are correct. The correct answer would likely be slightly less than 1 M due to the small amount of A converted into B and C.

Step-by-step solution

Identify Given Information

We are given a reaction mixture with starting amounts of 1 mole each for reactants A and D in a 1 L container. We have two equilibrium reactions: Reaction 1 is A ⇌ B + C with an equilibrium constant, Kc = 10^6 M^2. Reaction 2 is B + D ⇌ A with an equilibrium constant, Kc = 10^-6 M^-1.

Expression for Equilibrium Constants

Write the expression for the equilibrium constants for both reactions. For Reaction 1, Kc1 = [B][C]/[A], and for Reaction 2, Kc2 = [A]/[B][D].

Let’s Assume Change in Concentration

Since A ⇌ B + C, let x be the equilibrium concentration of B and C formed from A. So, [A] = [A]initial - x, [B] = x, [C] = x. For reaction B + D ⇌ A, assume y moles A is formed back. [A] = [A]initial - x + y, [B] = x - y, and [D] = [D]initial - y.

Apply Stoichiometry

Stoichiometry of both reactions suggests for every mole of A consumed, one mole each of B and C is produced, and that when one mole of B reacts with one mole of D, one mole of A is produced. This implies that at equilibrium, x = y. Thus, [A] at equilibrium remains 1 - x, [B] = x - x = 0, and [D] remains at 1 because no B is available to react with D due to B's complete consumption in forward reaction.

Substitute Equilibrium Concentrations into Kc Expressions

Since [B] = 0 at equilibrium, Reaction 2 does not proceed and does not affect the equilibrium for Reaction 1. Therefore, we have Kc1 = 10^6 = [C][B]/[A]. With [B] and [C] equal to x and [A] equal to 1 - x, Kc1 = 10^6 = x^2 / (1 - x).

Solve for x assuming (1 - x) is approximately 1

Since Kc1 is large, we can assume that virtually all A converts to B and C, making (1 - x) approximately equal to 1. Thus, Kc1 = 10^6 = x^2, which gives x = 10^-3 M. Since x represents the equilibrium concentration of B and C, and A is initially 1 M, the change x is negligible and [A] remains approximately 1 M at equilibrium.

Assess the Validity of the Assumption

Since we assumed (1 - x) to be approximately 1, we need to check if x is indeed much smaller than 1. x = 10^-3 is much smaller than 1, so our assumption is valid, and the equilibrium concentration of A is not significantly changed by the reaction, remaining close to 1 M.

Key concepts

Equilibrium Constant

Understanding the equilibrium constant (Kc) is crucial when analyzing chemical reactions at equilibrium. It's a snapshot of the relative concentrations of the reactants and products at a specific point when the rates of the forward and reverse reactions are equal. For a generic reaction, where reactants A and B form products C and D, the Kc is expressed as: \[\begin{equation}K_{c} = \frac{[C][D]}{[A][B]}\text{,}\end{equation}\]where the concentrations are measured in molarity (M), and stoichiometric coefficients become exponents in the expression.
In the exercise presented, two Kc expressions are provided for two different equilibria. The large Kc value for the first reaction indicates a strong tendency to form products, while the small Kc for the second reaction implies a preference for reactants. By writing out the mathematical expressions for Kc, you can visualize the relationship between reactants and products. This fundament helps in understanding why, at equilibrium, certain reactants or products dominate the mixture, which, in this case, is why the concentration of A remains approximately 1M.

Reaction Stoichiometry

Reaction stoichiometry involves the quantitative relationship between reactants and products in a chemical reaction. For every stoichiometrically balanced equation, the moles of each substance are related by whole numbers. This concept guides our understanding of how molecules interact and combine to form new products. For instance, in the exercise, for each mole of A consumed, one mole each of B and C is produced, illustrating a 1:1:1 stoichiometry in the first reaction.
Likewise, the second reaction's stoichiometry shows that one mole of B reacts with one mole of D to form one mole of A. However, due to the equilibrium dynamics and stoichiometry, we find that in the given scenario, B is entirely consumed in the forward reaction, leaving none available to react with D. This highlights the importance of considering both the stoichiometry and the directionality implied by the equilibrium constants when predicting the outcome of simultaneous equilibria. Such understanding is necessary for determining the final concentrations of reactants and products.

Equilibrium Concentration

Equilibrium concentration refers to the concentration of reactants and products in a reaction mixture when the reaction has reached a state of balance and no further change in concentrations occurs. The detailed exercise explores the calculation of the equilibrium concentration of a substance, A, when two equilibria involving A are established.
Initially, by setting up an assumption for the change in concentration (denoted by x and y), one can express the final equilibrium concentrations in terms of these changes. This approach allows simplification of potentially complex systems by relating the changes in concentration back to initial conditions and equilibrium constants. In the given exercise, using the assumptions and the value of Kc, we see that the equilibrium concentration of A remains close to its initial concentration. The correct solution, therefore, relies on a solid understanding of how the initial amount, the stoichiometry of the reactions, and the values of the equilibrium constants influence those final concentrations. It illuminates how some reactions may lead to a significant conversion of reactants to products, while others may not, guiding students towards a clearer grasp of chemical equilibrium concepts.