Question

An equilibrium mixture of the following reaction has \(\left[\mathrm{I}_{2}\right]=0.0112 \mathrm{M}\) and \(\left[\mathrm{Cl}_{2}\right]=0.0155 \mathrm{M}\) at \(25^{\circ} \mathrm{C}\). What is the concentration of ICl? $$ \begin{gathered} \mathrm{I}_{2}(g)+\mathrm{Cl}_{2}(g) \rightleftharpoons 2 \mathrm{ICl}(g) \\ K_{\mathrm{eq}}=81.9 \text { at } 25{ }^{\circ} \mathrm{C} \end{gathered} $$

Short answer

The concentration of ICl is approximately 0.042 M.

Step-by-step solution

Write the expression for the equilibrium constant

For the reaction \(\mathrm{I}_{2}(g) + \mathrm{Cl}_{2}(g) \rightleftharpoons 2 \mathrm{ICl}(g)\), the equilibrium constant expression is \[K_{\mathrm{eq}} = \frac{\left[\mathrm{ICl}\right]^2}{\left[\mathrm{I}_{2}\right] \cdot \left[\mathrm{Cl}_{2}\right]}\]

Determine unknown concentration using initial concentrations and the equilibrium constant

Using the equilibrium concentrations of \(\left[\mathrm{I}_{2}\right] = 0.0112 \mathrm{M}\) and \(\left[\mathrm{Cl}_{2}\right] = 0.0155 \mathrm{M}\), and the equilibrium constant \(\mathrm{K_{eq}} = 81.9\), we have \[81.9 = \frac{\left[\mathrm{ICl}\right]^2}{0.0112 \cdot 0.0155}\]Solve for \(\left[\mathrm{ICl}\right]\).

Calculate the concentration of ICl

Rearrange the equation to isolate \(\left[\mathrm{ICl}\right]\):\[\left[\mathrm{ICl}\right]^2 = 81.9 \cdot 0.0112 \cdot 0.0155\]Take the square root of both sides: \[\left[\mathrm{ICl}\right] = \sqrt{81.9 \cdot 0.0112 \cdot 0.0155}\]Calculate the value to find the concentration of ICl.

Key concepts

Equilibrium Constant Expression

The equilibrium constant expression is fundamental to understanding chemical equilibrium. It provides the relationship between the concentrations of the reactants and products at equilibrium in a reversible chemical reaction. For the reaction given, \(\mathrm{I}_{2}(g) + \mathrm{Cl}_{2}(g) \rightleftharpoons 2 \mathrm{ICl}(g)\), the expression is written using the law of mass action:
\[K_{\mathrm{eq}} = \frac{[\mathrm{ICl}]^2}{[\mathrm{I}_{2}] \cdot [\mathrm{Cl}_{2}]}\]
Here, \( K_{\mathrm{eq}} \) represents the equilibrium constant at a given temperature, and the square brackets denote the molar concentrations of the gases. Reactants are in the denominator and products in the numerator. The stoichiometry of the reaction is reflected in the exponents, with the coefficient of ICl being 2 and hence its concentration is squared. Understanding this relationship is critical for predicting how changes in concentration, pressure, or temperature can affect the system's equilibrium position.

Equilibrium Concentrations Calculation

Calculating equilibrium concentrations requires the use of the equilibrium constant expression and the known equilibrium constant (\( K_{eq} \)). Based on the reaction at hand, the known concentrations of \( \mathrm{I}_{2} \) and \( \mathrm{Cl}_{2} \) can be substituted into the expression to solve for the unknown concentration of ICl. Here's how the calculation is performed:
With \( [\mathrm{I}_{2}] = 0.0112 \mathrm{M} \) and \( [\mathrm{Cl}_{2}] = 0.0155 \mathrm{M} \), and \( K_{eq} = 81.9 \), the expression becomes
\[81.9 = \frac{[\mathrm{ICl}]^2}{0.0112 \cdot 0.0155}\]
From this, we must isolate \( [\mathrm{ICl}] \) and solve the equation to find the equilibrium concentration of ICl. The calculation involves algebraic manipulation, such as taking square roots, to obtain the final concentration.

Equilibrium Constant (Keq)

The equilibrium constant, denoted as \( K_{eq} \), is a dimensionless number that describes the ratio of product concentrations to reactant concentrations at equilibrium. Each reaction at a specific temperature has its unique equilibrium constant. A high \( K_{eq} \) value indicates a greater concentration of products, suggesting that the reaction favors the formation of products under given conditions. Conversely, a lower \( K_{eq} \) implies a reaction that favors the reactants. In this problem, \( K_{eq} = 81.9 \) tells us that at 25 degrees Celsius, the reaction produces a significant amount of ICl compared to \( \mathrm{I}_{2} \) and \( \mathrm{Cl}_{2} \). This quantitative measure is critical for chemists to predict how much product can be formed and to control reaction conditions effectively.