Question

For the equilibrium $$ \mathrm{Br}_{2}(g)+\mathrm{Cl}_{2}(g) \rightleftharpoons 2 \mathrm{BrCl}(g) $$ at \(400 \mathrm{~K}, K_{c}=7.0\). If \(0.25 \mathrm{~mol}\) of \(\mathrm{Br}_{2}\) and \(0.55 \mathrm{~mol}\) of \(\mathrm{Cl}_{2}\) are introduced into a \(3.0-\mathrm{L}\) container at \(400 \mathrm{~K}\), what will be the equilibrium concentrations of \(\mathrm{Br}_{2}, \mathrm{Cl}_{2}\), and \(\mathrm{BrCl}\) ?

Short answer

The equilibrium concentrations of \(\mathrm{Br}_{2}\), \(\mathrm{Cl}_{2}\), and \(\mathrm{BrCl}\) are approximately 0.074 M, 0.149 M, and 0.064 M, respectively.

Step-by-step solution

Write the equation for the equilibrium constant

Based on the given equilibrium, the first step is to write the equilibrium constant expression. For the given equilibrium, we have: \[K_c = \frac{[\mathrm{BrCl}]^2}{[\mathrm{Br}_{2}] [\mathrm{Cl}_{2}]}\] where \([\mathrm{BrCl}]\), \([\mathrm{Br}_{2}]\), and \([\mathrm{Cl}_{2}]\) represent the equilibrium concentrations of each species.

Set up the ICE table

Next, set up an ICE (Initial, Change, Equilibrium) table to track the changes in the concentrations of all species as the reaction proceeds towards equilibrium. \[\begin{array}{c|c|c|c} \text{Species} & \text{Initial (M)} & \text{Change (M)} & \text{Equilibrium (M)} \\ \hline \mathrm{Br}_{2} & \frac{0.25}{3.0} & -x & \frac{0.25}{3.0} - x \\ \mathrm{Cl}_{2} & \frac{0.55}{3.0} & -x & \frac{0.55}{3.0} - x \\ \mathrm{BrCl} & 0 & +2x & 2x \\ \end{array}\] In this table, x represents the change in concentration for the reactants and products and is related to the stoichiometric coefficients.

Substitute the values in the equilibrium constant expression

Now, substitute the equilibrium concentrations from the ICE table into the equilibrium constant expression: \[K_c = \frac{(2x)^2}{\left(\frac{0.25}{3.0} - x\right) \left(\frac{0.55}{3.0} - x\right)}\]

Solve for x

Now, plug in the given value of \(K_c = 7.0\) and solve for x: \[7.0 = \frac{(2x)^2}{\left(\frac{0.25}{3.0} - x\right) \left(\frac{0.55}{3.0} - x\right)}\] Through some algebraic manipulation or using numerical methods, we find that: \[x \approx 0.032\]

Calculate the equilibrium concentrations

Finally, plug the value of x back into the equilibrium concentrations in the ICE table to find the equilibrium concentrations of all species: \[\left[\mathrm{Br}_{2}\right]_{eq} = \frac{0.25}{3.0} - x \approx 0.074\] \[\left[\mathrm{Cl}_{2}\right]_{eq} = \frac{0.55}{3.0} - x \approx 0.149\] \[\left[\mathrm{BrCl}\right]_{eq} = 2x \approx 0.064\] Therefore, the equilibrium concentrations of \(\mathrm{Br}_{2}\), \(\mathrm{Cl}_{2}\), and \(\mathrm{BrCl}\) are approximately 0.074 M, 0.149 M, and 0.064 M, respectively.

Key concepts

Equilibrium Constant

The equilibrium constant, denoted as K_c when concentrations are used, is a numerical value that expresses the ratio of the concentrations of products to reactants at chemical equilibrium, each raised to the power of their stoichiometric coefficients. It provides a way to understand the extent of a reaction; a larger K_c suggests that the products are favored at equilibrium, while a smaller value indicates that the reactants are favored.

In the given example, the equilibrium constant expression for the reaction Br₂(g) + Cl₂(g) ⇌ 2 BrCl(g) is represented as \[K_c = \frac{[\mathrm{BrCl}]^2}{[\mathrm{Br}_{2}] [\mathrm{Cl}_{2}]}\]. It communicates that at 400 K, the system will reach equilibrium when the concentration of bromine chloride squared is seven times the product of the concentrations of bromine and chlorine. Understanding this concept is crucial as it sets the groundwork for predicting how the system behaves under different initial conditions and how the concentration of each species settles at equilibrium.

ICE Table

An ICE table—which stands for Initial, Change, and Equilibrium—is a systematic method to track the concentrations of reactants and products over the course of a reaction until it reaches equilibrium. It's an invaluable tool in solving equilibrium problems.

For the given reaction, an ICE table helps you visualize the initial molarity of the reactants, calculate the shift in molarities as the reaction proceeds toward equilibrium (represented by the variable x in the example), and derive the final equilibrium molarities. The table allows us to set up a clear path for numerical substitution into the equilibrium constant expression. Thus, it simplifies the problem-solving process and can be particularly helpful in complex systems where multiple reactions or phases are occurring.

Equilibrium Concentrations

Equilibrium concentrations refer to the molar amounts of reactants and products present when a chemical reaction has reached a state of balance—when the rate of the forward reaction equals the rate of the reverse reaction.

In the example provided, by solving the equation derived from the equilibrium constant and the ICE table, we determine the 'shift' in concentration, x, and ultimately the equilibrium concentrations of all chemical species involved. It's critical to ensure that the calculated values are physically plausible; for instance, concentrations cannot be negative. If seemingly unrealistic values emerge, this may indicate an error in assumptions or calculations, or that the system may not actually reach equilibrium under the given conditions. By understanding equilibrium concentrations, we gain a complete picture of the reaction's behavior and can predict product yields, necessary reactant quantities, and how the system responds to changes in conditions.