Question

When heated, a gaseous compound A dissociates as follows: $$ \mathrm{A}(g) \rightleftarrows \mathrm{B}(g)+\mathrm{C}(g) $$ In an experiment, A was heated at a certain temperature until its equilibrium pressure reached \(0.14 P\), where \(P\) is the total pressure. Calculate the equilibrium constant \(K_{P}\) of this reaction.

Short answer

The equilibrium constant \(K_{P}\) is 5.283.

Step-by-step solution

Define Initial and Equilibrium Conditions

Initially, we have only A, and its partial pressure is written as \(P_{A} = P\). At equilibrium, the pressure of A is reduced to \(0.14P\), so \(P_{A_{eq}} = 0.14P\). Let the change in pressure be \(x\), then the pressure of B and C each becomes \(x\). Since the dissociation of A to B and C is a 1:1:1 reaction, we have:\(x = P - 0.14P = 0.86P\). Thus, \(P_{B_{eq}} = P_{C_{eq}} = 0.86P\).

Express Total Pressure at Equilibrium

The total pressure at equilibrium remains \(P\). It's the sum of the partial pressures of A, B, and C, which is:\[P_{eq} = P_{A_{eq}} + P_{B_{eq}} + P_{C_{eq}} = 0.14P + 0.86P + 0.86P = P\]. This confirms our expressions for individual partial pressures are consistent with the total pressure remaining constant at P.

Write Expression for Equilibrium Constant \(K_{P}\)

The equilibrium constant \(K_P\) for the reaction \(\mathrm{A}(g) \rightleftharpoons \mathrm{B}(g)+\mathrm{C}(g)\) is defined as:\[K_P = \frac{P_{B_{eq}} \cdot P_{C_{eq}}}{P_{A_{eq}}}\]. Substituting the equilibrium pressures, we have \[K_P = \frac{(0.86P) \cdot (0.86P)}{0.14P}\].

Simplify the Expression for \(K_{P}\)

Simplify the expression obtained in Step 3:\[K_P = \frac{0.86^2 P^2}{0.14P} = \frac{0.7396P^2}{0.14P}\].By cancelling out the common \(P\) term, we get:\[K_P = \frac{0.7396P}{0.14} = 5.283\].

Key concepts

Dissociation Reaction

A dissociation reaction involves the breaking down of a compound into its constituent parts. In the given problem, the gaseous compound A dissociates into two different gases B and C. This can be represented by the equation: \( \text{A}(g) \rightleftharpoons \text{B}(g) + \text{C}(g) \). The equation with the double arrow indicates that the reaction is reversible. This means it can proceed in both forward and backward directions. When heat is applied, compound A breaks down into B and C, establishing a dynamic equilibrium. This process is crucial in understanding and calculating equilibrium situations for gases in closed systems. Recognizing dissociation reactions helps us to comprehend how concentration changes affect equilibrium positions.

Partial Pressure

In a mixture of gases, the partial pressure of a component is the pressure it would exert if it alone occupied the entire volume. For chemical reactions involving gases, like the dissociation of A into B and C, partial pressures play a crucial role. Consider the initial partial pressure of A to be \( P \). The dissociation reduces the partial pressure of A to \( 0.14P \). As it dissociates, the gases B and C are formed, each having a partial pressure of \( 0.86P \). Understanding partial pressures helps us in determining the equilibrium constant \( K_P \). It also provides insights into the distribution of gas molecules at equilibrium.

Le Chatelier's Principle

Le Chatelier's Principle gives us insight into how a system at equilibrium reacts to external changes. According to this principle, if any change is imposed on a system at equilibrium, the system shifts in a direction that counteracts the change and re-establishes equilibrium. In the context of the dissociation reaction \( \text{A}(g) \rightleftharpoons \text{B}(g) + \text{C}(g) \), increasing the temperature shifts the equilibrium towards the products, B and C. This is because dissociation is endothermic, requiring heat to proceed in the forward direction. Le Chatelier's principle helps predict how changes like pressure, temperature, or concentration affect the equilibrium state and thus guides us in controlling reactions in practical scenarios.

Equilibrium Expression

The equilibrium expression is a mathematical equation that describes the relationship between the concentrations (or partial pressures for gases) of reactants and products at equilibrium. For the dissociation reaction \( \text{A}(g) \rightleftharpoons \text{B}(g) + \text{C}(g) \), the equilibrium constant \( K_P \) is expressed as: \[ K_P = \frac{P_{B_{eq}} \cdot P_{C_{eq}}}{P_{A_{eq}}} \] Where \( P_{B_{eq}} \), \( P_{C_{eq}} \), and \( P_{A_{eq}} \) are the partial pressures of B, C, and A at equilibrium, respectively. These expressions allow chemists to calculate the composition of gas mixtures at equilibrium and predict how the system will respond to changes. By using the equilibrium expression, we can solve for unknowns in chemical reactions and thus gain deeper insights into reaction dynamics.