Question

The equilibrium constant for the reaction $$2 \mathrm{NOCl}(g) \rightleftharpoons 2 \mathrm{NO}(g)+\mathrm{Cl}_{2}(g)$$ has the value \(9.2 \times 10^{-6}\) at a particular temperature. The system is analyzed at equilibrium, and it is found that the concentrations of \(\mathrm{NOCl}(g)\) and \(\mathrm{NO}(g)\) are \(0.44 \mathrm{M}\) and \(1.5 \times 10^{-3} \mathrm{M},\) respectively. What is the concentration of \(\mathrm{Cl}_{2}(g)\) in the equilibrium system under these conditions?

Short answer

The concentration of Cl₂(g) in the equilibrium system under these conditions is \(0.792\: M\).

Step-by-step solution

Write down the equation for the equilibrium constant

The equilibrium constant K relates the equilibrium concentrations of the products to the reactants in a chemical reaction. In this problem, the reaction can be written as: \(2 \mathrm{NOCl}(g) \rightleftharpoons 2 \mathrm{NO}(g)+\mathrm{Cl}_{2}(g)\) The equilibrium constant (K) for this reaction is given by the expression: \[K = \frac{[\mathrm{NO}]^2 [\mathrm{Cl}_2]}{[\mathrm{NOCl}]^2}\]

Substitute the given values into the equation

We are given the equilibrium concentrations of NOCl and NO as 0.44 M and 1.5 x 10^{-3} M, respectively. The equilibrium constant is also provided as \(9.2 \times 10^{-6}\). We can substitute these values into the equilibrium constant expression to solve for the concentration of Cl₂: \[(9.2 \times 10^{-6}) = \frac{(1.5 \times 10^{-3})^2 [\mathrm{Cl}_2]}{(0.44)^2}\]

Solve for the concentration of Cl₂

To find the concentration of Cl₂, we will rearrange the equation and simplify: \[[\mathrm{Cl}_2] = \frac{9.2 \times 10^{-6} \times (0.44)^2}{(1.5 \times 10^{-3})^2}\] Now, calculate the concentration: \[[\mathrm{Cl}_2] = \frac{9.2 \times 10^{-6} \times 0.1936}{2.25 \times 10^{-6}}\] \[[\mathrm{Cl}_2] = \frac{1.78032 \times 10^{-6}}{2.25 \times 10^{-6}}\] \[[\mathrm{Cl}_2] = 0.792\] Thus, the concentration of Cl₂(g) in the equilibrium system under these conditions is 0.792 M.

Key concepts

Chemical Equilibrium

Chemical equilibrium occurs in a chemical reaction when the rates of the forward and reverse reactions are equal, leading to no net change in the concentrations of reactants and products over time. This does not mean the reactions have stopped; rather, they continue to occur simultaneously at a constant rate.

At equilibrium, the system's properties remain constant, provided external conditions like temperature and pressure do not change. However, reaching equilibrium does not imply that reactants and products are present in equal amounts. It simply denotes a balance in reaction rates, resulting in a stable ratio of concentration for the reactants and products, which is described by the equilibrium constant, K.

Equilibrium Concentrations

Equilibrium concentrations refer to the amounts of reactants and products present in a reaction mixture when the reaction has reached equilibrium. These concentrations do not change as long as the system's conditions remain constant. Determining the equilibrium concentrations is vital for calculating the equilibrium constant, and for understanding the extent to which a reaction has proceeded.

One can calculate unknown equilibrium concentrations using the equilibrium constant and the known concentrations of other species in the reaction. This typically involves setting up an equation based on the stoichiometry of the reaction and solving for the unknown.

K Expression

The K expression, or equilibrium constant expression, is a mathematical representation of the equilibrium constant (K) for a given chemical reaction. It is a ratio of the concentrations of products, each raised to the power of their stoichiometric coefficients, to the concentrations of reactants, similarly raised to the power of their stoichiometric coefficients.

In terms of a general reaction, \(aA + bB \rightleftharpoons cC + dD\), the K expression is given by: \[ K = \frac{[C]^c[D]^d}{[A]^a[B]^b} \]
where \( [A] \) and \( [B] \) are the molar concentrations of the reactants, and \( [C] \) and \( [D] \) are the molar concentrations of the products at equilibrium. The brackets represent concentration in molarity (mol/L). It's important to note that pure solids and liquids do not appear in the K expression because their concentrations do not change.

Reactant and Product Concentrations

Reactant and product concentrations are the amounts of substances present in a solution, typically measured in molarity (M), which is moles per liter. These concentrations can vary during a reaction, but at equilibrium, they reach constant values. It's important for chemists to be able to calculate these concentrations at any point in a reaction to understand the reaction's progress and yield.

For the reaction explained in the step by step solution, using the known equilibrium constant and the given concentrations of some reactants or products, equilibrium concentrations of the other species can be mathematically deduced. The stoichiometry of the balanced equation is crucial as it indicates the proportion in which reactants combine and products form, a key factor in determining these concentrations at equilibrium.