Problem 72 - Chemistry The Central Science In Si Units

Question

If \(K_{c}=1\) for the equilibrium \(3 \mathrm{~A}(g) \rightleftharpoons 2 \mathrm{~B}(g),\) what is the relationship between [A] and [B] at equilibrium?

Step-by-Step Solution

Answer

At equilibrium, \([A]^3 = [B]^2\).

Step 1: Write the expression for the equilibrium constant

The equilibrium constant expression for the reaction \(3 \mathrm{~A}(g) \rightleftharpoons 2 \mathrm{~B}(g)\) is derived from the equation: \[K_c = \frac{[B]^2}{[A]^3}.\] We know that at equilibrium, the concentration of the reactants and products raise to the power of their respective coefficients in the balanced equation.

Step 2: Substitute the equilibrium constant value

Given that \(K_c = 1\), substitute it into the equilibrium expression. This yields: \[1 = \frac{[B]^2}{[A]^3}.\]

Step 3: Rearrange the expression to find the relationship between [A] and [B]

Rearrange \(1 = \frac{[B]^2}{[A]^3}\) to express one concentration in terms of the other. Multiply both sides by \([A]^3\) to get: \[[A]^3 = [B]^2.\] This shows that at equilibrium, the concentration of \(A\) cubed is equal to the concentration of \(B\) squared.

Key Concepts

Chemical Equilibrium

In chemical reactions, equilibrium is the state where the rates of the forward and reverse reactions are equal. At this point, the concentrations of the reactants and products remain constant over time. It doesn't mean that the reactions stop; rather, they occur at the same rate in both directions.

In the given reaction, \(3 \mathrm{~A}(g) \rightleftharpoons 2 \mathrm{~B}(g)\), chemical equilibrium is achieved when the conversion between \(A\) and \(B\) happens at a steady ratio. The equilibrium constant \(K_c\) is a crucial part in understanding this balance, as it provides a snapshot of this concentration relationship at a certain temperature.

The concept of equilibrium is fundamental in predicting the behavior of chemical systems in various conditions. It is a dynamic equilibrium, meaning while the microscopic changes continue, the macroscopic observations remain constant. Understanding how to manipulate or predict the state of equilibrium can aid in the efficient use of materials and energy in chemical processes.

Concentration Relationship

The concentration relationship in chemical equilibrium is a way of understanding how the quantities of reactants and products compare as the system stabilizes. In the given expression, \(K_c = \frac{[B]^2}{[A]^3}\), the equilibrium constant \(K_c\) gives us valuable information on how these concentrations relate to one another at equilibrium.

The equation dictates that for every increase in the concentration of \([B]\), the concentration of \([A]\) cubed must increase proportionally to maintain a constant \(K_c\). Given \(K_c = 1\), it's evident that the concentrations are directly linked by the equilibrium expression.

Understanding these relationships helps in designing chemical processes, allowing chemists to predict how changing concentrations will affect the position of equilibrium. This is essential in industrial and laboratory settings, where one may need to optimize the output of a desired product.

Reaction Stoichiometry

Reaction stoichiometry refers to the quantitative relationship between reactants and products in a balanced chemical equation. This concept is key when working with chemical equilibrium because it determines the precedence and proportion of substances involved.

In the equation \(3 \mathrm{~A}(g) \rightleftharpoons 2 \mathrm{~B}(g)\), stoichiometry tells us three moles of \(A\) react to form two moles of \(B\). Each molecule's concentration is raised to the power of its stoichiometric coefficient within the equilibrium expression. This relationship is crucial for converting between reactants and products within the equilibrium state.

The stoichiometry ensures that, regardless of starting amounts, once equilibrium is reached, the moles of \(A\) and \(B\) will always adhere to the ratio provided by the balanced equation. Thus, stoichiometry is instrumental in calculating and predicting the concentrations necessary to achieve equilibrium under various conditions.