Problem 19 - Chemistry and Chemical Reactivity

Question

Which of the following correctly relates the equilibrium constants for the two reactions shown? \(A+B \rightleftarrows 2 C \quad K_{1}\) \(2 \mathrm{A}+2 \mathrm{B} \rightleftharpoons 4 \mathrm{C} \quad K_{2}\) (a) \(K_{2}=2 K_{1} \quad\) (c) \(K_{2}=1 / K_{1}\) (b) \(K_{2}=K_{1}^{2}\) (d) \(K_{2}=1 / K_{1}^{2}\)

Step-by-Step Solution

Answer

The correct relation is (b) \(K_2 = K_1^2\).

Step 1: Identify Reaction Relationships

We need to relate two given reactions. The first reaction is given as \(A+B \rightleftharpoons 2C\) with equilibrium constant \(K_1\), and the second one as \(2A+2B \rightleftharpoons 4C\) with equilibrium constant \(K_2\). Notice that the second reaction is just the first one multiplied by 2.

Step 2: Equilibrium Constant Concept

The equilibrium constant \(K\) for a reaction is defined in terms of the concentrations of products and reactants raised to the power of their coefficients in the balanced equation. For example, for the reaction \(aA + bB \rightleftharpoons cC\), the equilibrium constant \(K\) is expressed as \(K = \frac{[C]^c}{[A]^a[B]^b}\).

Step 3: Express Equilibrium Constants for Given Reactions

For the reaction \(A + B \rightleftharpoons 2C\), the equilibrium constant \(K_1\) is given by \(K_1 = \frac{[C]^2}{[A][B]}\). For the reaction \(2A + 2B \rightleftharpoons 4C\), the equilibrium constant \(K_2\) is given by \(K_2 = \frac{[C]^4}{[A]^2[B]^2}\).

Step 4: Relate the Constants

The second reaction is essentially the first reaction times 2. According to the rules of equilibrium constant calculations, when a reaction is multiplied by a factor, the equilibrium constant is raised to the power of that factor. Therefore, \(K_2 = K_1^2\).

Step 5: Conclusion

Comparing this outcome with the options given, we find that option (b) \(K_2 = K_1^2\) correctly describes the relationship between the equilibrium constants of the two reactions.

Key Concepts

Chemical Equilibrium

Chemical equilibrium refers to a state in a chemical reaction where the concentrations of reactants and products remain constant over time.
At this point, the forward and reverse reactions occur at the same rate. This does not mean that the reactants and products are equal in concentration, but that their ratios remain steady. For example, in the equation \( A + B \rightleftharpoons 2C \), the reaction reaches equilibrium when the formation of \( C \) from \( A \) and \( B \) equals the reformation of \( A \) and \( B \) from \( C \).A key characteristic of chemical equilibrium is its dynamic nature, meaning that both the forward and reverse reactions continue to occur. This is equilibrium in a closed system, where no additional reactants or products enter or leave the system.

• Reversible reactions can reach equilibrium.
• Equilibrium is a balance between forward and reverse reaction rates.
• It is not limited to equal concentrations of reactants and products.

Reaction Stoichiometry

Reaction stoichiometry determines how substances react and form products in specific ratios. It is based on the balanced equation of a reaction where each element is accounted for and conserved.
Stoichiometry involves understanding the mole ratios of reactants and products, which directly influences how equilibrium constants are expressed. For the reaction \( A + B \rightleftharpoons 2C \), the mole ratio of A to C is 1:2, indicating that one mole of A reacts to form two moles of C.When reaction coefficients are multiplied, as in the change from \( A + B \rightleftharpoons 2C \) to \( 2A + 2B \rightleftharpoons 4C \), this affects the equilibrium expression. Reaction stoichiometry must be meticulously followed to accurately calculate changes in equilibrium constants.

• Stoichiometry is based on the conservation of mass.
• It allows predictions of how much product forms from a given amount of reactant.

Equilibrium Constant Relationships

Equilibrium constant relationships help us understand how changes in the chemical equation affect the equilibrium constant.
The equilibrium constant \( K \) is a measure of the extent of a reaction at equilibrium and depends on the reaction stoichiometry. When the multiplication of a reaction by a factor occurs, the new equilibrium constant is the original constant raised to the power of that factor.For instance, if reaction \( A + B \rightleftharpoons 2C \) with \( K_1 \) is multiplied by 2 to become \( 2A + 2B \rightleftharpoons 4C \), the equilibrium constant changes to \( K_1^2 \). This is because each component of the reaction's balanced equation, including its stoichiometric coefficients, becomes exponentiated in the equilibrium expression.
Understanding these relationships allows for prediction and manipulation of chemical equilibria under different conditions, crucial for both laboratory and industrial processes.

• Equilibrium constants change with stoichiometry adjustments.
• Exponentiation of \( K \) is consistent with the multiplication factor.
• It highlights how sensitive equilibria can be to reaction conditions.