- Identify the reactants (\(A, B\)) and products (\(C, D\)) along with their stoichiometric coefficients (\(a, b, c, d\)).
- Apply the Law of Mass Action, which defines the equilibrium constant as the ratio of product concentrations to reactant concentrations, each raised to their respective stoichiometric coefficients.
- The resulting expression is \(\frac{[C]^c [D]^d}{[A]^a [B]^b}\), which corresponds to option (c).
- The correct choice is \(\boxed{c}\).
Explanation
- Problem Analysis and Data Identification
The system represents a general reversible chemical reaction given by the equation \[a A + b B \longleftrightarrow c C + d D.\] In this equation, \(A\) and \(B\) represent the reactants, while \(C\) and \(D\) represent the products. The lowercase letters \(a, b, c,\) and \(d\) are the respective stoichiometric coefficients of these chemical species. We need to determine the mathematical expression for the equilibrium constant \(K_{eq}\).
- Applying the Law of Mass Action
According to the Law of Mass Action, for any reversible chemical reaction at equilibrium at a constant temperature, the equilibrium constant \(K_{eq}\) is defined as the product of the concentrations of the products raised to the power of their stoichiometric coefficients, divided by the product of the concentrations of the reactants raised to the power of their stoichiometric coefficients. Mathematically, for the reaction \(aA + bB \longleftrightarrow cC + dD\), the concentration of species \(X\) is denoted by \([X]\). Therefore, the products are \([C]^c\) and \([D]^d\), and the reactants are \([A]^a\) and \([B]^b\). The ratio is constructed as \[K_{eq} = \frac{[C]^c [D]^d}{[A]^a [B]^b}.\] Comparing this expression with the provided options, it corresponds exactly to option (c).
- Conclusion
Based on the derivation, the correct equilibrium constant expression is \(K_{eq} = \frac{[C]^c [D]^d}{[A]^a [B]^b}\). This matches option (c). Thus, the correct choice is \(\boxed{c}\).
Examples
In industrial chemical production, such as the synthesis of ammonia (the Haber-Bosch process: \(N_2 + 3H_2 \longleftrightarrow 2NH_3\)), understanding the equilibrium constant expression allows chemical engineers to predict the final concentrations of reactants and products. By manipulating conditions like pressure or temperature to alter \(K_{eq}\), they can maximize the yield of the desired product (ammonia) while minimizing waste, thereby increasing the efficiency and profitability of the manufacturing process.