Question

Consider the reaction: $$\mathrm{A}(g) \rightleftharpoons \mathrm{B}(g)+\mathrm{C}(g)$$ Find the equilibrium concentrations of \(\mathrm{A}, \mathrm{B},\) and \(\mathrm{C}\) for each value of \(K_{\mathrm{cr}}\) Assume that the initial concentration of \(\mathrm{A}\) in each case is 1.0 \(\mathrm{M}\) and that the reaction mixture initially contains no products. Make any appropriate simplifying assumptions. \(\begin{equation}\begin{array}{l}{\text { a. } K_{\mathrm{c}}=1.0} \\ {\text { b. } K_{\mathrm{c}}=0.0010} \\ {\text { c. } K_{\mathrm{c}}=1.0 \times 10^{-5}}\end{array}\end{equation}\)

Short answer

For each case, the equilibrium concentrations are obtained by solving a quadratic equation or implementing an approximation if x is significantly smaller than the initial concentration. Once the value of x is determined, [A] = 1.0 - x, [B] = x, and [C] = x.

Step-by-step solution

Understanding the Problem

We have a reversible chemical reaction where A converts into B and C with no initial concentration of products B and C. We need to find the equilibrium concentrations of A, B, and C for different values of the equilibrium constant, Kc. We start with 1.0 M of A.

Writing the Equilibrium Expression

For the reaction A(g) <=> B(g) + C(g), the equilibrium constant expression is given by Kc = [B][C]/[A], where square brackets denote concentrations of substances at equilibrium.

Setting Up Initial and Equilibrium Concentrations

Let x be the change in concentration of A that reacts to form B and C at equilibrium. Thus, at equilibrium, [A] = 1.0 - x, [B] = x, and [C] = x.

Case a: Kc = 1.0

For Kc = 1.0, we substitute the equilibrium concentrations into the equilibrium expression: 1.0 = [B][C]/[A] = x*x / (1.0 - x), which simplifies to x^2 = 1.0 - x. We solve this quadratic equation to find the equilibrium concentrations.

Simplifying and Solving the Quadratic (Case a)

To solve x^2 = 1.0 - x, rearrange the terms to get x^2 + x - 1.0 = 0. Find x using the quadratic formula: x = (-b ± sqrt(b^2 - 4ac)) / 2a, where a = 1, b = 1, and c = -1.

Case b: Kc = 0.0010

Substitute the equilibrium concentrations into the equilibrium expression for Kc = 0.0010 to get 0.0010 = x*x / (1.0 - x) and solve for x.

Case c: Kc = 1.0 x 10^-5

Substitute into the equilibrium expression for Kc = 1.0 x 10^-5 to obtain 1.0 x 10^-5 = x*x / (1.0 - x) and solve for x.

Applying the Quadratic Formula (Cases b and c)

For each case, apply the quadratic formula or an approximation if x is small compared to 1, which implies that 1-x approximately equals 1. Solve for x to find the concentrations.

Interpreting the Solutions

Use the calculated value of x to find the equilibrium concentrations [A] = 1.0 - x, [B] = x, and [C] = x for each case.

Key concepts

Equilibrium Constant Expression

Understanding the equilibrium constant expression is crucial in the study of chemical equilibria. It quantifies the relative concentrations of reactants and products in a reversible reaction at equilibrium. For a general reaction where \( A(g) \rightleftharpoons B(g) + C(g) \) the equilibrium constant expression \( K_c \) is defined as \( K_c = [B][C]/[A] \), with \( [A] \) representing the concentration of reactant A and \( [B] \) and \( [C] \) as the concentrations of products B and C, respectively, all measured at equilibrium. The square brackets denote molarity (M), which is moles per liter.

In equilibrium constant expressions, the concentrations of products appear in the numerator, and the concentrations of reactants appear in the denominator. If the value of \( K_c \) is large, it suggests that the equilibrium heavily favors the formation of products. Conversely, a small \( K_c \) indicates that the reactants are favored. Understanding this relationship helps us predict the extent of a reaction and the concentrations of all species involved at equilibrium.

Reversible Reaction

A reversible reaction is one where the conversion of reactants to products and the reverse process, the formation of reactants from products, occur simultaneously. These reactions reach a state of balance, or chemical equilibrium, where the rate of the forward reaction equals the rate of the reverse reaction. This dynamic equilibrium does not mean the chemical reactions have stopped; rather, both reactions continue to occur at the same rate, resulting in constant concentrations of reactants and products.

An understanding of reversible reactions is fundamental in calculating equilibrium concentrations, as changes in conditions can shift the point of equilibrium, demonstrating Le Chatelier's Principle. The concept of the reaction quotient (Q) is also used to predict the direction in which a reversible reaction will proceed to achieve equilibrium given certain initial concentrations.

Equilibrium Concentration Calculations

To calculate equilibrium concentrations, one often employs a methodical approach known as the ICE (Initial, Change, Equilibrium) table. This process begins by recording the initial concentrations of reactants and products, the change in concentration as the system moves towards equilibrium, and finally, the equilibrium concentrations. In the exercise at hand, we started with an initial concentration of 1.0 M for reactant A and zero for products B and C. Upon reaching equilibrium, the concentration of A decreased by \( x \) while the concentrations of B and C each increased by \( x \).

The change in concentration is represented by the variable \( x \) and is determined through the relationship prescribed by the balanced chemical equation. Once \( x \) is found, usually through solving a quadratic equation derived from the equilibrium constant expression, the equilibrium concentrations can be precisely calculated, leading to a deeper understanding of the system's dynamics at equilibrium.

Quadratic Formula in Chemistry

The quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) is a valuable tool in chemistry for solving equilibrium problems where the mathematical relationship between reactants and products results in a quadratic equation. When equilibrium constant expressions simplify into a quadratic form, such as \( ax^2 + bx + c = 0 \), the quadratic formula is used to find values of \( x \) that satisfy the equilibrium condition.

In our case, after simplifying the equilibrium constant expression, we obtained a quadratic equation representing the changes in species concentrations. Calculating the discriminant, \( b^2 - 4ac \), helps determine the nature of the solutions. A positive discriminant indicates two real and distinct solutions, a zero discriminant a single real solution, and a negative discriminant means that no real solutions exist. The solutions for \( x \) represent the shifts in concentration needed to achieve equilibrium, which are then used to calculate the final equilibrium concentrations of A, B, and C.