Question

A compound, 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?

Short answer

The reaction will proceed to the left, towards the formation of compound X.

Step-by-step solution

Write down the given information.

We have the equilibrium reaction: $$ 2 \mathrm{X}(g) \rightleftharpoons \mathrm{A}(g)+3 \mathrm{C}(g) \quad K=1.1 \times 10^{-3} $$ We are given the initial partial pressures of each gas: $$ P_{\mathrm{X}}=0.250 \mathrm{~atm}, \quad P_{\mathrm{A}}=0.250 \mathrm{~atm}, \quad P_{\mathrm{C}}=0.250 \mathrm{~atm} $$ And the equilibrium constant K: $$ K = 1.1 \times 10^{-3} $$

Calculate the reaction quotient Q.

To calculate the reaction quotient Q, we need to substitute the values of the given partial pressures into the following expression: $$ Q = \frac{P_{\mathrm{A}} \cdot P_{\mathrm{C}}^3}{P_{\mathrm{X}}^2} $$ Plug in the values for the partial pressures: $$ Q = \frac{(0.250) \times (0.250)^3}{(0.250)^2} = \frac{0.250^4}{0.250^2} $$ Now simplify the expression and calculate the value of Q: $$ Q = 0.250^2 = 0.0625 $$

Compare Q and K to determine the direction of the reaction.

Now, we compare the reaction quotient Q with the equilibrium constant K to determine the direction of the reaction: - If Q > K, the reaction will proceed to the left (toward the reactants). - If Q < K, the reaction will proceed to the right (toward the products). - If Q = K, the reaction is already at equilibrium. Since we have calculated Q (0.0625) and K (1.1 × 10⁻³), we can compare their values: $$ 0.0625 > 1.1 \times 10^{-3} $$ Since Q > K, the reaction will proceed to the left, towards the reactants. Hence, in this case, the reaction will proceed in the direction of the formation of the compound X.

Key concepts

Equilibrium Constant

The equilibrium constant, denoted as K, is a fundamental concept in chemistry that quantifies the ratio of concentrations of products to reactants at equilibrium for a reversible reaction.

In the context of the provided exercise, the equilibrium constant is given for the decomposition of compound X into products A and C, with the formula \[ K = 1.1 \times 10^{-3} \]. This numerically small value of K suggests that at equilibrium, the concentration (or partial pressure in case of gases) of the reactants is higher compared to the products.

Understanding the magnitude of K helps predict whether reaction favors the formation of products or reactants. A low value, such as in our example, indicates that the reactants are favored at equilibrium. Hence, only a minute amount of reactants will be converted into products at equilibrium.

Reaction Quotient

The reaction quotient, Q, represents the ratio of the current concentrations or partial pressures of reaction components. Unlike the equilibrium constant, Q can be calculated at any point of the reaction, not just at equilibrium.

By calculating Q for our given scenario: \[ Q = \frac{{P_{\mathrm{A}} \cdot P_{\mathrm{C}}^3}}{{P_{\mathrm{X}}^2}} \], and substituting the pressures provided, we found out that \[ Q = 0.0625 \].

By comparing this value to the K, we can evaluate the direction in which the reaction will proceed. For Q > K, the reaction will move towards reactants, indicating that it has exceeded the equilibrium concentrations of products, and for Q < K, the reaction will progress towards products, showing that there's still room for more product formation before equilibrium is reached.

Le Chatelier's Principle

Le Chatelier's principle provides a qualitative insight into how a system at equilibrium reacts to external changes. It states that if a dynamic equilibrium is disturbed by changing the conditions, the position of equilibrium will shift to counteract the change.

In the exercise, we determined that Q > K, indicating that there are more products than required at equilibrium. According to Le Chatelier's principle, the system will respond by favoring the reverse reaction to produce more reactants and thus decrease the concentrations of the products to re-establish equilibrium.

This principle is crucial when predicting the effects of changes in concentration, pressure, or temperature on a system. For example, increasing the pressure of a gaseous system will shift the equilibrium towards the side with fewer moles of gas. Understanding Le Chatelier's principle helps in controlling chemical reactions and optimizing yields in industrial processes.