Question

Determine \(K_{c}\) for each of the following equilibria from the value of \(K\) : (a) \(2 \mathrm{SO}_{2}(\mathrm{~g})+\mathrm{O}_{2}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{SO}_{3}(\mathrm{~g}), K=3.4\) at \(1000 . \mathrm{K}\) (b) \(\mathrm{NH}_{4} \mathrm{HS}(\mathrm{s}) \rightleftharpoons \mathrm{NH}_{3}(\mathrm{~g})+\mathrm{H}_{2} \mathrm{~S}(\mathrm{~g}), K=9.4 \times 10^{-2}\) at \(24^{\circ} \mathrm{C}\)

Short answer

For both (a) and (b), the equilibrium constant Kc is equal to the given K values. Hence, Kc for (a) is 3.4, and for (b) is 9.4 x 10^{-2}.

Step-by-step solution

Understanding the Equilibrium Constant

The equilibrium constant for a reaction, given as K, can be expressed in terms of the concentrations of the reactants and products. For a gaseous reaction, the equilibrium constant is expressed as Kp when in terms of partial pressures, or Kc when based on concentrations. Since we are given K and asked for Kc, we assume the values of K provided are already in terms of concentration, hence for gaseous reactions, K is Kc.

Handling Solids in the Equilibrium Expression

In the equilibrium expression for Kc, the concentration of solids is not included because it remains constant. This means that for part (b), the equilibrium constant Kc will be the same as the given K because the solid \(NH_{4}HS\) does not appear in the expression for Kc.

Interpreting K for Each Equilibrium

For part (a), the equilibrium involves only gases, so K is equivalent to Kc. Therefore, Kc for reaction (a) will be the same as the given K value of 3.4 at 1000 K. For part (b), since solids are not included in the equilibrium expression and the given K is already based on the concentrations of the gaseous products, the value of Kc will be the same as the given K, which is 9.4 x 10^{-2} at 24°C.

Key concepts

Chemical Equilibrium

Chemical equilibrium is a fundamental concept in chemistry that describes the state of a chemical reaction in which the rates of the forward and reverse reactions are equal, resulting in no net change in the concentration of reactants and products over time. It is important to understand that this does not mean the reactants and products are equal in concentration, but rather that their concentrations have stabilized to constant values.

When a system reaches equilibrium, it is said to be 'closed' as the conditions remain constant provided external factors such as temperature or pressure are not altered. The position of the equilibrium, favoring either reactants or products, is determined by various factors including the nature of the reactants and products, temperature, and pressure. Understanding the principles of chemical equilibrium is essential for predicting the outcome of chemical reactions and for the synthesis of materials in industries.

Reaction Concentration

The concentration of a chemical species in a given volume significantly impacts reaction rates and the state of chemical equilibrium. In the context of equilibrium, reaction concentration refers to the molarity (mol/L) of reactants and products at equilibrium. These concentrations are used to express the equilibrium constant (Kc) which quantifies the ratio of product concentrations to reactant concentrations.

It's worth noting that the concentration of pure solids and liquids are not included in the calculation of Kc because their concentrations do not change during the reaction. Thus, they are incorporated into the equilibrium constant. Factors like temperature, which can shift the equilibrium position, will change the concentrations of reactants and products at equilibrium but do not affect the value of the equilibrium constant at a given temperature.

Kc Expression

The Kc expression is a mathematical representation of the equilibrium constant for reactions where the reactants and products are in the same phase, typically gaseous or aqueous. The expression is derived by taking the product of the concentrations of the products, each raised to the power of their stoichiometric coefficients, and dividing by the product of the concentrations of the reactants, each also raised to the power of their stoichiometric coefficients.

For example, in the reaction \(2SO_2(g) + O_2(g) \rightleftharpoons 2SO_3(g)\), the Kc expression would be \[Kc = \frac{[SO_3]^2}{[SO_2]^2[O_2]}\] where \[ [ ] \] represent the concentration of the species at equilibrium. The exponents reflect the coefficients in the balanced chemical equation. A larger Kc value indicates a greater extent of reaction, favoring product formation, while a smaller Kc indicates a less favorable formation of products. It's crucial to understand the meaning and use of the Kc expression to solve problems related to chemical equilibrium in gaseous and aqueous systems.