Question

At \(25^{\circ} \mathrm{C}, K_{\mathrm{c}}=0.145\) for the following reaction in the solvent \(\mathrm{CCl}_{4}\) $$ 2 \mathrm{BrCl} \rightleftharpoons \mathrm{Br}_{2}+\mathrm{Cl}_{2} $$ If the initial concentration of \(\mathrm{BrCl}\) in the solution is \(0.050 M,\) what will the equilibrium concentrations of \(\mathrm{Br}_{2}\) and \(\mathrm{Cl}_{2}\) be?

Short answer

At equilibrium, the concentrations of \(\mathrm{Br}_{2}\) and \(\mathrm{Cl}_{2}\) will be equal to the value of \(x\) calculated from the equilibrium expression.

Step-by-step solution

Define the Equilibrium Expression

Write the equilibrium constant expression for the reaction according to the law of mass action as follows: \[ K_{\mathrm{c}} = \frac{[\mathrm{Br}_{2}][\mathrm{Cl}_{2}]}{[\mathrm{BrCl}]^2} \]

Set Up an ICE Table

Create an ICE (Initial, Change, Equilibrium) table to keep track of the molar concentrations of the reactants and products throughout the reaction.

Express Equilibrium Concentrations in Terms of Change

Express the equilibrium concentrations in terms of the change that occurs as the system reaches equilibrium. Let the change in concentration of \(\mathrm{BrCl}\) be \(-2x\), then the change for \(\mathrm{Br}_{2}\) and \(\mathrm{Cl}_{2}\) will be \(+x\) each, because of the stoichiometry of the reaction.

Write Out the Equilibrium Concentrations

The equilibrium concentrations can be written as: \[[\mathrm{Br}_{2}] = [\mathrm{Cl}_{2}] = x\] and \[[\mathrm{BrCl}] = 0.050 - 2x\]

Substitute the Equilibrium Concentrations into the Equilibrium Expression

Substitute the expressions for the equilibrium concentrations into the equilibrium constant expression and solve for \(x\).

Solve for x

Solve the equation \(0.145 = \frac{x^2}{(0.050 - 2x)^2}\) for \(x\).

Calculate the Equilibrium Concentrations

Once \(x\) has been found, substitute it back into the expressions for the equilibrium concentrations to find the concentrations of \(\mathrm{Br}_{2}\) and \(\mathrm{Cl}_{2}\).

Key concepts

Equilibrium Constant Expression

The equilibrium constant expression quantifies the ratio of product concentrations to reactant concentrations at chemical equilibrium, each raised to the power of their coefficients in the balanced chemical equation. For the reaction
\(2 \mathrm{BrCl} \rightleftharpoons \mathrm{Br}_2 + \mathrm{Cl}_2\), the expression is
\[ K_{\mathrm{c}} = \frac{[\mathrm{Br}_2][\mathrm{Cl}_2]}{[\mathrm{BrCl}]^2} \]
This equation reflects the law of mass action, stating that the rate of a reaction is proportional to the product of the concentrations of the reactants. In essence, the equilibrium constant, \( K_{\mathrm{c}} \), is pivotal in predicting the extent of the reaction and determining the concentrations of substances present when the chemical system is at equilibrium.

ICE Table

An ICE (Initial, Change, Equilibrium) table is an invaluable tool for solving equilibrium problems, providing a structured method for organizing and calculating changes in concentration. It starts with the 'Initial' concentrations of reactants and products, 'Change' represents the shift in concentrations as the reaction moves towards equilibrium, and 'Equilibrium' gives the final concentrations once the system is stable.
When dealing with the given equation,
\(2 \mathrm{BrCl} \rightleftharpoons \mathrm{Br}_2 + \mathrm{Cl}_2\),
we can set up an ICE table that helps visualize how the initial concentration of \(\mathrm{BrCl}\) decreases by \(-2x\) while the \(\mathrm{Br}_2\) and \(\mathrm{Cl}_2\) each increase by \(+x\), tracking the stoichiometric ratios.

Stoichiometry

Stoichiometry is the section of chemistry concerned with the relative quantities of reactants and products in chemical reactions. In the context of equilibrium, stoichiometry dictates the ratios in which compounds react and form products. For the given reaction, for every 2 moles of \(\mathrm{BrCl}\) that react, 1 mole of \(\mathrm{Br}_2\) and 1 mole of \(\mathrm{Cl}_2\) are produced. This is a 2:1:1 ratio and is crucial for establishing the relationship between the concentration changes of all species involved when setting up an ICE table and solving for equilibrium concentrations.

Equilibrium Concentrations

Equilibrium concentrations refer to the concentrations of each species when the forward and reverse rates of a chemical reaction are equal, indicating that the system has reached a state of balance and concentrations remain constant over time. By using the equilibrium constant expression and the stoichiometric relationships found in the balanced chemical equation, we can solve for these concentrations mathematically. For instance, after setting up the expressions for \(\mathrm{Br}_2\) and \(\mathrm{Cl}_2\) as \(x\) and for \(\mathrm{BrCl}\) as \(0.050 - 2x\), it's possible to find the specific value of \(x\) that satisfies the equilibrium condition. This process allows for the determination of the amounts of each substance present in a mixture at equilibrium.

Law of Mass Action

The law of mass action states that for a reversible reaction at equilibrium and at a constant temperature, a certain ratio of the concentration of reactants to the concentration of products always exists. This value is known as the equilibrium constant. It is a fundamental concept when discussing chemical equilibrium, as it relates the concentrations of each chemical species to one another. It implies that if the system is disturbed, the concentrations will shift in such a way as to re-establish the original equilibrium constant value, a concept known as Le Chatelier's principle. Applying the law of mass action to the given reaction, and understanding that the reaction quotient (Q) will adjust until it equals the equilibrium constant (K), can help students grasp the dynamic yet stable nature of chemical equilibrium.