Problem 49 - Chemistry: An Atoms-Focused Approach

Question

Calculate the value of the equilibrium constant \(K_{c}\) for the reaction, $$ 2 \mathrm{D} \rightleftharpoons \mathrm{A}+2 \mathrm{B} $$ given the following information: $$ \begin{array}{rlrl} A+2 B & F C & K_{c} & =3.3 \\ C & \rightleftharpoons 2 D & K_{c} & =0.041 \end{array} $$

Step-by-Step Solution

Answer

Answer: The equilibrium constant (\(K_c\)) for the reaction \(2D \rightleftharpoons A + 2B\) is approximately 80.49.

Step 1: Write down the given reactions and their equilibrium constants

First, let's write down the given reactions and their corresponding equilibrium constants: Reaction 1: \(A + 2B \rightleftharpoons C\) with \(K_{c1} = 3.3\) Reaction 2: \(C \rightleftharpoons 2D\) with \(K_{c2} = 0.041\) We want to find \(K_c\) for the reaction: \(2D \rightleftharpoons A + 2B\)

Step 2: Combine the given reactions to obtain the desired reaction

Now, we need to manipulate the given reactions in such a way that we obtain the desired reaction. We can accomplish this by reversing Reaction 2 and then adding it to Reaction 1. This will give us the following reaction: $$2D \stackrel{K_{c2}^{-1}}{\rightleftharpoons} C \stackrel{K_{c1}}{\rightleftharpoons} A + 2B$$

Step 3: Calculate the equilibrium constant for the desired reaction

To find the equilibrium constant for the overall reaction, we need to multiply the equilibrium constants of the individual reactions. Since we reversed Reaction 2, we will use its inverse value for the calculation: $$K_c = K_{c1} \times K_{c2}^{-1}$$ $$K_c = 3.3 \times \frac{1}{0.041}$$ Now, we can calculate the value of \(K_c\): $$K_c \approx 3.3 \times 24.39 \approx 80.49$$

Step 4: Write down the answer

The equilibrium constant (\(K_c\)) for the reaction \(2D \rightleftharpoons A + 2B\) is approximately 80.49.

Key Concepts

Chemical Equilibrium

In any chemical reaction, the components involved, namely the reactants and products, may achieve a state of balance known as chemical equilibrium. At equilibrium, the rates at which reactants are converted into products and products are converted back into reactants are equal. This means no net change in the concentrations of the reactants and products over time, even though the reactions are still occurring.
One key feature of chemical equilibrium is that it can be described by the equilibrium constant, denoted as \(K_c\) for reactions in solution. This constant directly reflects the ratio of concentrations of products to reactants, raised to the power of their respective coefficients in the chemical equation. While \(K_c\) indicates the position of equilibrium, it does not give information about how quickly equilibrium is reached.
**Factors Influencing Chemical Equilibrium**

• Concentration changes
• Pressure and volume changes, especially for gases
• Temperature changes, as it can affect the energy profile of the reaction

Reversible Reactions

Reversible reactions are those where the reactants form products and simultaneously, the products gradually re-form into the reactants. Both forward and backward reactions occur until equilibrium is established. Consider the reaction involved in your exercise: \(2D \rightleftharpoons A + 2B\).
Here, the reaction can proceed both in the forward direction, forming A and 2B from 2D, and in the reverse direction, reconstituting 2D from A and 2B. The double arrow symbolizes this reversibility. A reversible reaction doesn't necessarily mean equal amounts of reactants and products at equilibrium. Instead, the proportions depend on their initial concentrations and the equilibrium constant.
**Key Characteristics of Reversible Reactions**

• Dynamic equilibrium: Both forward and reverse reactions continue occurring at equilibrium
• Influenced by external conditions like temperature and pressure
• Possibility of product accumulation, which can shift the reaction towards the formation of reactants

Le Chatelier's Principle

Le Chatelier's Principle is a fundamental concept used to predict the effect of changes in conditions on a chemical reaction at equilibrium. It states that if a dynamic equilibrium is disturbed by changing the conditions, the system adjusts itself to counteract the change and a new equilibrium is established. This principle is vital for understanding how reactions shift when external parameters like concentration, pressure, or temperature are modified.
**Applications of Le Chatelier's Principle**

• **Concentration:** Adding more reactants or products will shift the equilibrium to the opposite side to reduce the added component.
• **Pressure:** Increasing pressure will shift the equilibrium towards the side with fewer moles of gas. Conversely, decreasing pressure favors the side with more moles of gas.
• **Temperature:** For exothermic reactions, an increase in temperature shifts the equilibrium towards the reactants, while for endothermic reactions, it favors the products.

It's essential to understand that while these shifts can affect the concentrations of reactants and products, the value of \(K_c\) remains unchanged unless there's a change in temperature. Le Chatelier's Principle provides a qualitative viewpoint to anticipate reaction behavior under varying conditions.