Question

For the reaction $$ 2 \mathrm{CO}_{2}(g) \rightleftharpoons 2 \mathrm{CO}(g)+\mathrm{O}_{2}(g) $$ an analysis of an equilibrium mixture is performed. At a particular temperature, it is found that \([\mathrm{CO}]=0.11 M,\left[\mathrm{O}_{2}\right]=0.055 M,\) and \(\left[\mathrm{CO}_{2}\right]=1.4 M .\) Calculate \(K\) for the reaction.

Short answer

The equilibrium constant (K) for the reaction \(2\mathrm{CO}_{2}(g) \rightleftharpoons 2\mathrm{CO}(g)+\mathrm{O}_{2}(g)\) is approximately \(3.40\times10^{-4}\) when the equilibrium concentrations are \([\mathrm{CO}]=0.11\,M\), \([\mathrm{O}_2]=0.055\,M\), and \([\mathrm{CO}_2]=1.4\,M\).

Step-by-step solution

Recall the formula for the equilibrium constant (K)

When a chemical reaction is at equilibrium, the concentrations of the species are related by the equilibrium constant (K). In general, for a reaction of the form: $$ a\mathrm{A} + b\mathrm{B} \rightleftharpoons c\mathrm{C} + d\mathrm{D} $$ The equilibrium constant K is given by: $$ K = \frac{[\mathrm{C}]^c [\mathrm{D}]^d}{[\mathrm{A}]^a [\mathrm{B}]^b} $$ In our case, the reaction is: $$ 2\mathrm{CO}_2(g) \rightleftharpoons 2\mathrm{CO}(g) + \mathrm{O}_2(g) $$ So, we'll have: $$ K = \frac{[\mathrm{CO}]^2 [\mathrm{O}_2]}{[\mathrm{CO}_2]^2} $$

Plug in the given equilibrium concentrations

We are given the equilibrium concentrations as \([\mathrm{CO}] = 0.11\,M\), \([\mathrm{O}_2] = 0.055\,M\), and \([\mathrm{CO}_2] = 1.4\,M\). Plug these values into the expression for K: $$ K = \frac{(0.11)^2\times 0.055}{(1.4)^2} $$

Calculate the equilibrium constant (K)

Now, perform the calculations to find K: $$ K = \frac{0.0121\times 0.055}{1.96} $$ $$ K = \frac{0.0006655}{1.96} \sim 3.40\times10^{-4} $$ Thus, the equilibrium constant, K, for the given reaction is approximately \(3.40\times10^{-4}\).

Key concepts

Equilibrium Constant

When a chemical reaction reaches a state where the rate of the forward reaction equals the rate of the reverse reaction, it is said to be at dynamic equilibrium. At this point, the concentration of the reactants and products remains constant over time, although they are not necessarily equal. The equilibrium constant, denoted as K, is a numerical value that expresses the ratio of product concentrations to reactant concentrations at equilibrium, each raised to the power of their coefficients in the balanced chemical equation.

The equilibrium constant provides insight into the position of equilibrium and the extent of the reaction. A high value of K (significantly larger than 1) indicates that, at equilibrium, the reaction mixture is predominantly products. Conversely, a low value of K (significantly smaller than 1) suggests that the reactants are favored. The precise value of K is determined by the specific reaction conditions, such as temperature. It's important to note that K only changes with temperature; it is not affected by changes in concentration or pressure.

Equilibrium Concentrations

Understanding the concept of equilibrium concentrations is vital for grasping the nature of chemical equilibrium. These are the concentrations of the reactants and products in a reaction mixture when the reaction has reached equilibrium. To calculate the equilibrium constant, you need the concentrations of each species at equilibrium.

The given equilibrium concentrations enable us to apply the equilibrium constant expression. Substituting the equilibrium concentrations into the equilibrium constant expression allows us to solve for the numerical value of K. In practical terms, knowing the equilibrium concentrations can inform us about the efficiency of a reaction and predict how the system will respond to external disturbances, according to Le Châtelier's Principle.

Chemical Reaction Equilibrium

Grasping the principle of chemical reaction equilibrium is essential for understanding how reactions proceed in nature and industry. It is the state in a chemical reaction where the rate of the forward reaction equals the rate of the backward reaction, causing no net change in the concentration of reactants and products. However, this does not imply that the reaction has stopped; rather, the reactants and products are being created and consumed at an equal rate.

Chemical reaction equilibrium is dynamic and can be shifted by changes in concentration, pressure, or temperature, as stated by Le Châtelier's Principle. This principle helps to predict how the equilibrium position will shift if external conditions are altered. For instance, increasing the concentration of products typically pushes the reaction towards the reactants to re-establish equilibrium. Depending on external conditions and the nature of the reaction, equilibrium can favor either the reactants or the products to varying degrees.