Question

Write the general equilibrium constant expression for each of the following: (a) \(2 \mathrm{~A} \rightleftarrows \mathrm{C}\) (b) \(A+2 B \rightleftarrows 3 C\) (c) \(2 \mathrm{~A}+3 \mathrm{~B} \rightleftarrows 4 \mathrm{C}+\mathrm{D}\)

Short answer

(a) \(K_a = \frac{[C]}{[A]^2}\); (b) \(K_b = \frac{[C]^3}{[A][B]^2}\); (c) \(K_c = \frac{[C]^4[D]}{[A]^2[B]^3}\)."

Step-by-step solution

Understanding the Equilibrium Constant Expression

The equilibrium constant expression for a chemical reaction is derived from the concentrations of the reactants and products at equilibrium. For a general reaction of the form \(aA + bB \rightleftarrows cC + dD\), the equilibrium constant \(K\) is expressed as \(K = \frac{[C]^c[D]^d}{[A]^a[B]^b}\). Here, \([X]\) denotes the concentration of substance \(X\) at equilibrium.

Expression for Reaction (a)

For the reaction \(2 \mathrm{~A} \rightleftarrows \mathrm{C}\), the equilibrium expression is based on the coefficients of the balanced equation. The equilibrium constant \(K_a\) is given by \(K_a = \frac{[C]}{[A]^2}\). This reflects 1 mole of \(C\) produced for every 2 moles of \(A\).

Expression for Reaction (b)

In the reaction \(A+2 B \rightleftarrows 3 C\), we write the equilibrium constant \(K_b\) considering one mole of \(A\), two moles of \(B\), and three moles of \(C\). The expression is \(K_b = \frac{[C]^3}{[A][B]^2}\).

Expression for Reaction (c)

For the complex reaction \(2 \mathrm{~A}+3 \mathrm{~B} \rightleftarrows 4 \mathrm{C}+\mathrm{D}\), we identify the stoichiometry: two moles of \(A\), three moles of \(B\), four moles of \(C\), and one mole of \(D\). The equilibrium constant \(K_c\) is \(K_c = \frac{[C]^4[D]}{[A]^2[B]^3}\).

Key concepts

Chemical Reactions

Chemical reactions are processes where substances, known as reactants, are transformed into different substances, called products. These transformations involve breaking chemical bonds in the reactants and forming new bonds in the products. In an equation, reactions are represented using chemical formulas. For example, in the reaction \(2 \mathrm{~A} \rightleftarrows \mathrm{C}\), the reactant "A" is transformed into the product "C". The double arrow \(\rightleftarrows\) signifies that the reaction can proceed in both forward and reverse directions, indicating a state of dynamic equilibrium. This means that at some point, the rate of formation of products equals the rate of formation of reactants, resulting in no net change in concentrations over time. This concept of dynamic equilibrium is crucial for understanding how chemical reactions behave under different conditions and is fundamental when calculating the equilibrium constant.

Concentration

Concentration in chemistry refers to the amount of a substance in a given volume. It's a significant factor in determining the rate and direction of chemical reactions. In equilibrium expressions, concentrations of reactants and products are represented by square brackets, such as \([A]\) for the concentration of compound A.

Concentrations are crucial when working with equilibrium constants because they help determine the relative proportions of reactants and products at equilibrium. For instance, in a reaction like \(A + 2 B \rightleftarrows 3 C\), the equilibrium constant expression \(K_b = \frac{[C]^3}{[A][B]^2}\) uses these concentrations to quantify the balance between reactants and products. Understanding this relationship allows chemists to predict how changes in concentration, such as adding more reactant or product or removing a substance, will shift the equilibrium position.

Stoichiometry

Stoichiometry is the part of chemistry that deals with the relative quantities of reactants and products in a chemical reaction. It relies on the balanced chemical equation to determine how much of each substance is involved. A key aspect of stoichiometry is using the coefficients from balanced equations to relate moles of each reactant and product.

For instance, in the equation \(2 \mathrm{~A} + 3 \mathrm{~B} \rightleftarrows 4 \mathrm{C} + \mathrm{D}\), stoichiometry tells us that 2 moles of A react with 3 moles of B to produce 4 moles of C and 1 mole of D. This ratio is crucial for writing the equilibrium constant expression \(K_c = \frac{[C]^4[D]}{[A]^2[B]^3}\).

Understanding stoichiometry ensures the correct setup of equilibrium expressions and calculations, helping chemists accurately describe and predict the behavior of chemical reactions.