Problem 31 - Chemistry: Principles and Reactions

Question

A compound, \(\mathrm{X}\), decomposes at \(131^{\circ} \mathrm{C}\) according to the following equation: $$2 \mathrm{X}(g) \rightleftharpoons \mathrm{A}(g)+3 \mathrm{C}(g) \quad K=1.1 \times 10^{-3}$$ If a flask initially contains \(\mathrm{X}, \mathrm{A}\), and \(\mathrm{C}\), all at partial pressures of \(0.250 \mathrm{~atm}\), in which direction will the reaction proceed?

Step-by-Step Solution

Answer

Answer: The reaction will proceed in the reverse direction to reach equilibrium.

Step 1: Understand the reaction and write the equilibrium constant expression.

Given the reaction: $$2 \mathrm{X}(g) \rightleftharpoons \mathrm{A}(g)+3 \mathrm{C}(g) \quad K=1.1 \times 10^{-3}$$ We can write the equilibrium constant expression as: $$K = \frac{[\mathrm{A}][\mathrm{C}]^3}{[\mathrm{X}]^2}$$

Step 2: Calculate the initial reaction quotient (Q).

Using the initial partial pressures of X, A, and C, we can calculate the initial reaction quotient: $$Q = \frac{[\mathrm{A}][\mathrm{C}]^3}{[\mathrm{X}]^2} = \frac{(0.250)(0.250)^3}{(0.250)^2}$$

Step 3: Simplify the expression for the reaction quotient (Q).

After simplification, we have: $$Q=\frac{(0.250)^4}{(0.250)^2} = (0.250)^2$$

Step 4: Evaluate the reaction quotient (Q) and compare to the equilibrium constant (K).

We have: $$Q = (0.250)^2 = 0.0625$$ Now compare Q to K: $$Q > K \Rightarrow (0.0625 > 1.1 \times 10^{-3})$$

Step 5: Determine the direction of the reaction.

Since Q is greater than K, the reaction will proceed in the reverse direction to reach the equilibrium. The decomposition of X will decrease, and the formation of A and C will increase until Q equals to K, reaching the equilibrium.

Key Concepts

Reaction Quotient

The reaction quotient, often represented as \( Q \), provides insight into the current state of a chemical reaction compared to its position at equilibrium. It is calculated similarly to the equilibrium constant \( K \), but at any given moment in time, not necessarily at equilibrium. For a general reaction:

• \( ext{aA} + ext{bB} ightleftharpoons ext{cC} + ext{dD} \)
• The reaction quotient \( Q \) is \( Q = \frac{[ ext{C}]^c [ ext{D}]^d}{[ ext{A}]^a [ ext{B}]^b} \)

For the reaction \( 2 ext{X}(g) ightleftharpoons ext{A}(g) + 3 ext{C}(g) \), the initial quotient \( Q \) is:
\[ Q = \frac{[ ext{A}][ ext{C}]^3}{[ ext{X}]^2} \]
By substituting the initial conditions, we find \( Q = 0.0625 \). The comparison of \( Q \) with \( K \) determines how the reaction will shift to achieve equilibrium.

Partial Pressures

Partial pressure is the pressure each gas in a mixture would exert if it were alone in the container. In a reaction involving gases, the partial pressure of each component contributes to the reaction's behavior. Using the pressure values lets you calculate both the equilibrium constant expression and the reaction quotient:

• Initial pressures for \( ext{X}, ext{A}, \) and \( ext{C} \) are all 0.250 atm.
• The equilibrium constant expression and \( Q \) both utilize these pressures.

Partial pressures can fluctuate as reactions proceed to reach equilibrium, affecting the value of \( Q \) and the reaction dynamics.

Chemical Equilibrium

Chemical equilibrium occurs when the rate of the forward reaction equals the rate of the reverse reaction, resulting in constant concentrations of both reactants and products. This state is described by the equilibrium constant \( K \). For our reaction, \( K = 1.1 \times 10^{-3} \) at \( 131^\circ C \).
When you initially measure the imbalance using \( Q \), any deviation informs you about the shift needed to achieve equilibrium:

• If \( Q < K \), the forward reaction is favored.
• If \( Q > K \), the reverse reaction is favored.
• If \( Q = K \), the system is at equilibrium.

Understanding \( Q \) in relation to \( K \) helps predict whether a system will shift to produce more products or reactants.

Reaction Direction

The direction in which a reaction proceeds is determined by comparing \( Q \) and \( K \). If the initial \( Q \) is larger than \( K \), like in this exercise, the reaction moves in the reverse direction to decrease the concentration of products and increase reactants until equilibrium is reached.
To predict the reaction direction:

• Calculate \( Q \) using initial conditions as shown: \( Q = 0.0625 \).
• Compare \( Q \) to \( K \): if \( Q > K \), the reaction shifts left (reverse reaction is favored).
• Given \( Q = 0.0625 > K = 1.1 \times 10^{-3} \),
  • The reaction favors the decomposition of \( ext{A} \) and \( ext{C} \) back into \( ext{X} \).

Such understanding aids in controlling reaction conditions effectively in various chemical processes.