Question

Consider the following reaction at \(250^{\circ} \mathrm{C}\) : $$\mathrm{A}(s)+2 \mathrm{~B}(g) \rightleftharpoons \mathrm{C}(s)+2 \mathrm{D}(g)$$ (a) Write an equilibrium constant expression for the reaction. Call the equilibrium constant \(K_{1}\). (b) Write an equilibrium constant expression for the formation of one mole of \(\mathrm{B}(\mathrm{g})\) and call the equilibrium constant \(K_{2}\). (c) Relate \(K_{1}\) and \(K_{2}\).

Short answer

Based on the step-by-step solution, provide a short answer to the following questions: a) Write an equilibrium constant expression for the reaction (K1). The equilibrium constant expression for the reaction K1 is: $$K_{1} = \frac{[\mathrm{D}]^{2}}{[\mathrm{B}]^{2}}$$ b) Write an equilibrium constant expression for the formation of one mole of B(g) (K2). The equilibrium constant expression for the formation of one mole of B(g), denoted as K2, is: $$K_{2} = \frac{[\mathrm{D}]}{[\mathrm{B}]}$$ c) Relate K1 and K2. K1 and K2 are related as follows: $$K_{1} = (K_{2})^{2}$$

Step-by-step solution

Recall the equilibrium constant expression

For a general chemical reaction, the equilibrium constant expression is given by the formula: $$K = \frac{[\text{products}]^{\text{stoichiometric coefficients}}}{[\text{reactants}]^{\text{stoichiometric coefficients}}}$$ Where [products] and [reactants] are the molar concentrations of the products and reactants at equilibrium, and the stoichiometric coefficients are the coefficients in the balanced chemical equation.

Write the equilibrium constant expression for the given reaction (K1)

Using the given reaction and the general formula for the equilibrium constant expression, we can write the expression for K1 as: $$K_{1} = \frac{[\mathrm{C}]^{1}[\mathrm{D}]^{2}}{[\mathrm{A}]^{1}[\mathrm{B}]^{2}}$$ Since A and C are in solid form, their concentrations are constant and can be excluded from the expression. Therefore, the expression can be simplified to: $$K_{1} = \frac{[\mathrm{D}]^{2}}{[\mathrm{B}]^{2}}$$

Write the equilibrium constant expression for the formation of one mole of B(g) (K2)

First, we need to write an equation that represents the formation of one mole of B(g). We can achieve this by dividing the original equation by a factor of 2: $$\frac{1}{2}\mathrm{A}(s)+\mathrm{B}(g) \rightleftharpoons \frac{1}{2}\mathrm{C}(s)+\mathrm{D}(g)$$ Now, using the general formula for the equilibrium constant expression, we can write the expression for K2 as: $$K_{2} = \frac{[\frac{1}{2}\mathrm{C}][\mathrm{D}]}{[\frac{1}{2}\mathrm{A}][\mathrm{B}]}$$ Again, since A and C are in solid form, their concentrations are constant and can be excluded from the expression. The expression can then be simplified to: $$K_{2} = \frac{[\mathrm{D}]}{[\mathrm{B}]}$$

Relate K1 and K2

We are given the following expressions for K1 and K2: $$K_{1} = \frac{[\mathrm{D}]^{2}}{[\mathrm{B}]^{2}}$$ $$K_{2} = \frac{[\mathrm{D}]}{[\mathrm{B}]}$$ To relate K1 and K2, we can notice that the square of K2 is equal to K1: $$(K_{2})^{2} = \left(\frac{[\mathrm{D}]}{[\mathrm{B}]}\right)^{2} = \frac{[\mathrm{D}]^{2}}{[\mathrm{B}]^{2}} = K_{1}$$ Thus, the relation between K1 and K2 is given by: $$K_{1} = (K_{2})^{2}$$

Key concepts

Equilibrium Constant Expression

Understanding the equilibrium constant expression is pivotal in grasping how chemical reactions reach a state of balance. At the heart of this concept lies the equilibrium constant, denoted as K, which encapsulates the ratio of the concentration of products to reactants, each raised to the power of their stoichiometric coefficients, at equilibrium.

The equation for the equilibrium constant expression can be represented as:
\[K = \frac{[\text{products}]^{\text{stoichiometric coefficients}}}{[\text{reactants}]^{\text{stoichiometric coefficients}}}\]
This fundamental principle helps chemists predict the direction of the reaction and the extent to which reactants will convert into products. A significant point to note is that the equilibrium constant expression varies depending on the reaction phase. For instance, in the given example, solids do not appear in the K expression because their concentrations do not change during the reaction and are thus considered constant.

When dealing with gases, as is the case with B(g) and D(g) in the reaction, we use their molar concentrations. Solids and liquids, due to their constant concentration, are not included in the K expression. This leads us to an imperative insight: only the concentrations of gaseous and aqueous species are included in the equilibrium constant expression.

Stoichiometry

A fundamental cornerstone in chemistry, stoichiometry, relates to the quantitative aspect of chemical reactions. It involves the calculation of reactants and products using a balanced chemical equation. The ratio in which chemicals react is governed by their stoichiometric coefficients, which are the numbers placed before chemical formulas in a balanced equation.

In the context of equilibrium, stoichiometry helps dictate the equilibrium constant expression. As in the exercise, A(s) and 2B(g) react to form C(s) and 2D(g). The coefficients indicate the mole ratio at which reactants convert to products. For the reaction to reach equilibrium, the rate of the forward reaction (reactants forming products) must equal the rate of the reverse reaction (products reforming reactants).

Stoichiometry also extends to manipulating chemical equations, as we did when halving the coefficients to determine the expression for K2, pertaining to the formation of one mole of B(g). This alteration changes the K expression accordingly, showcasing the intrinsic connection between stoichiometry, equilibrium expressions, and the resultant equilibrium constants.

Equilibrium in Chemical Reactions

Every chemical reaction strives to reach a state of equilibrium, where the forward and reverse reactions occur at the same rate, resulting in constant concentrations of reactants and products. However, achieving equilibrium does not mean that the quantities of reactants and products are equal; it simply indicates that their rates of formation are matched.

An essential component we deduce from equilibrium is the equilibrium constant expression, which quantitatively describes this state. A high K value suggests that, at equilibrium, the reaction mixture is product-favored, meaning products are more abundant than reactants. Conversely, a low K value suggests that reactants predominate. This state is dynamic, meaning that even though we see constant concentrations, the reactions are still proceeding but do so in a balanced fashion.

The relationship between stoichiometry and equilibrium helps us understand the reversibility of chemical processes and calculate the concentrations of reactants and products under given conditions. When equilibrium is established, altering conditions such as concentration, temperature, or pressure can shift the equilibrium, a concept known as Le Chatelier's principle, which further illustrates the reactive nature of chemical equilibrium.