Question

Challenge The reaction \(\mathrm{COCl}_{2}(\mathrm{g}) \rightleftharpoons \mathrm{CO}(\mathrm{g})+\mathrm{Cl}_{2}(\mathrm{g})\) reaches equilibrium at 900 \(\mathrm{K}\) . \(K_{\mathrm{eq}}\) is \(8.2 \times 10^{-2} .\) If the equilibrium concentrations of \(\mathrm{CO}\) and \(\mathrm{Cl}_{2}\) are \(0.150 \mathrm{M},\) what is the equilibrium concentration of \(\mathrm{COCl}_{2}\) ?

Short answer

The equilibrium concentration of COCl₂ is approximately 0.274 M.

Step-by-step solution

Write the balanced chemical equation and the equilibrium expression

The balanced chemical equation is given as: \[COCl_{2}(g) \rightleftharpoons CO(g) + Cl_{2}(g)\] The equilibrium constant (Kₑq) expression for the reaction is: \[K_{eq} = \frac{[CO][Cl_2]}{[COCl_2]}\]

Substitute the known values in the equilibrium expression

We know the equilibrium constant (Kₑq) is equal to 8.2 × 10⁻², and the equilibrium concentrations of CO and Cl₂ are both 0.150 M. Let x be the equilibrium concentration of COCl₂. We can write the equilibrium expression as: \[8.2 \times 10^{-2} = \frac{(0.150)(0.150)}{x}\]

Solve for the equilibrium concentration of COCl2

Now, we will solve the equation for x (the equilibrium concentration of COCl₂): \(8.2 \times 10^{-2} = \frac{0.0225}{x}\) By cross multiplying to rearrange the equation for x, we get: \(x = \frac{0.0225}{8.2 \times 10^{-2}}\) Now, we can calculate the value of x: \(x \approx 0.274\) So the equilibrium concentration of COCl₂ is approximately 0.274 M.

Key concepts

Chemical Equilibrium

Chemical equilibrium is a fundamental concept in chemistry that occurs when a reversible chemical reaction reaches a state where the rates of the forward and reverse reactions are equal, resulting in no net change in the concentrations of reactants and products over time. This dynamic balance means that while the reactants and products are being formed and consumed continuously, their amounts remain constant.
For example, in the reaction given in the exercise \(COCl_2(g) \rightleftharpoons CO(g) + Cl_2(g)\), chemical equilibrium is achieved when the rate at which \(COCl_2\) decomposes into \(CO\) and \(Cl_2\) is equal to the rate at which \(CO\) and \(Cl_2\) combine to form \(COCl_2\). It is important to note that achieving equilibrium does not mean that the reactants and products are in equal amounts but rather that their ratios are stable.
Understanding chemical equilibrium is essential when predicting the outcome of chemical reactions and in various applications such as chemical manufacturing, pharmaceuticals, and environmental science.

Equilibrium Constant

The equilibrium constant (\(K_{eq}\)) is a dimensionless value that expresses the ratio of the concentrations of products to the concentrations of reactants for a reaction at equilibrium, each raised to the power of their respective coefficients in the balanced chemical equation.

Understanding \(K_{eq}\) and Its Significance

The value of \(K_{eq}\) provides insight into the position of equilibrium. A high \(K_{eq}\), much greater than 1, suggests a reaction that favors the formation of products, while a low \(K_{eq}\), less than 1, indicates that the reactants are more favored. In the provided exercise, the equilibrium constant is \(8.2 \times 10^{-2}\), suggesting that at 900 K, the reactants are somewhat favored over the products.

Calculating \(K_{eq}\)

When calculating the equilibrium constant, it's critical to use the equilibrium concentrations of the reactants and products. Keep in mind that pure solids and liquids do not appear in the \(K_{eq}\) expression because their concentrations do not change. Gases and aqueous solutions are included, as their concentrations can vary.
By understanding the value of the equilibrium constant, you can predict how changes in conditions such as temperature or concentration will shift the chemical equilibrium, affecting the yield of product in industrial processes.

Equilibrium Expressions

Equilibrium expressions, also known as the law of mass action, define the relationship between the concentrations of reactants and products at chemical equilibrium. The general form of the equilibrium expression for a reaction \(aA + bB \rightleftharpoons cC + dD\) is given by the equation:
\[K_{eq} = \frac{[C]^c[D]^d}{[A]^a[B]^b}\]
where \([A]\), \([B]\), \([C]\), and \([D]\) represent the molar concentrations of the chemical species and \(a\), \(b\), \(c\), and \(d\) are the stoichiometric coefficients from the balanced equation.

Applying Equilibrium Expressions

In the context of the exercise, the balanced chemical equation is \(COCl_2(g) \rightleftharpoons CO(g) + Cl_2(g)\), with the equilibrium expression being \(K_{eq} = \frac{[CO][Cl_2]}{[COCl_2]}\). By substituting known values into this expression, we can solve for the unknown concentration. This application is crucial for not just academic exercises but also in industrial processes where achieving desired concentrations of substances is essential for productivity and cost-effectiveness.
Understanding how to derive and manipulate these expressions allows you to calculate the concentrations of various components in a chemical system at equilibrium, making it a powerful tool in both theoretical and practical chemistry.