Question

Consider the reaction: $$\begin{array}{c}{2 \mathrm{H}_{2} \mathrm{S}(g) \Longrightarrow 2 \mathrm{H}_{2}(g)+\mathrm{S}_{2}(g)} \\ {K_{\mathrm{p}}=2.4 \times 10^{-4} \mathrm{at} 1073 \mathrm{K}}\end{array}$$ A reaction mixture contains 0.112 atm of \(\mathrm{H}_{2}, 0.055\) atm of \(\mathrm{S}_{2},\) and 0.445 atm of \(\mathrm{H}_{2} \mathrm{S} .\) Is the reaction mixture at equilibrium? If not, in what direction will the reaction proceed?

Short answer

The reaction mixture is not at equilibrium because the reaction quotient (Qp = 3.48 x 10^-3) is greater than the equilibrium constant (Kp = 2.4 x 10^-4). The reaction will proceed to the left.

Step-by-step solution

Write The Expression For The Equilibrium Constant \textbf{Kp}

First, write the expression for the equilibrium constant (Kp) for the given reaction: $$K_{\text{p}} = \frac{(P_{{H}_2})^2 \cdot (P_{{S}_2})}{(P_{{H}_2S})^2}$$where \(P_{substance}\) denotes the partial pressure of each substance.

Substitute The Given Partial Pressures Into the Kp Expression

Substitute the given partial pressures into the Kp expression to find the reaction quotient Qp: $$Q_{\text{p}} = \frac{(0.112)^2 \cdot 0.055}{(0.445)^2}$$

Calculate the Reaction Quotient Qp

Calculate Qp using the provided partial pressures: $$Q_{\text{p}} = \frac{(0.112)^2 \times 0.055}{(0.445)^2} = \frac{(0.012544) \times 0.055}{0.198025} \approx 3.48 \times 10^{-3}$$

Compare Qp to Kp

Compare the calculated Qp with the given Kp to determine the direction of the reaction: - If Qp > Kp, the reaction will proceed to the left. - If Qp < Kp, the reaction will proceed to the right. - If Qp = Kp, the system is at equilibrium.

Determine the Direction of the Reaction

Since \(Q_{\text{p}} (3.48 \times 10^{-3})\) is greater than \(K_{\text{p}} (2.4 \times 10^{-4})\), the reaction will proceed to the left in order to reach equilibrium.

Key concepts

Equilibrium Constant (Kp)

The equilibrium constant (Kp) is a fundamental concept in chemical equilibrium that expresses the ratio of the product of partial pressures of the products to the product of partial pressures of the reactants, each raised to the power of their stoichiometric coefficients in a balanced chemical equation. This is typically applied to gas-phase reactions. For our example reaction, the expression for Kp is

\[\begin{equation}K_{\text{p}} = \frac{(P_{{H}_2})^2 \cdot (P_{{S}_2})}{(P_{{H}_2S})^2}\end{equation}\]
It's important to grasp that the equilibrium constant is only affected by the temperature, and a specific temperature value should always be mentioned, as seen in the given problem statement where it specifies a temperature of 1073 K.

Reaction Quotient (Qp)

When dealing with chemical equilibria, it's often necessary to determine how far a system is from reaching equilibrium. This is where the reaction quotient (Qp) comes in. Qp serves as a 'snapshot' of a system's condition at a given moment before it has reached equilibrium. The expression for Qp mirrors that of Kp, involving the partial pressures of the gases involved:\[\begin{equation}Q_{\text{p}} = \frac{(P_{{H}_2})^2 \cdot (P_{{S}_2})}{(P_{{H}_2S})^2}\end{equation}\]
By comparing Qp to Kp, you determine the direction in which the reaction must proceed to achieve equilibrium. If Qp is greater than Kp, the system will shift towards the reactants (to the left). If Qp is less than Kp, the system will shift towards the products (to the right).

Partial Pressure

In gas-phase equilibria, partial pressure—which is the pressure that each gas in a mixture contributes to the total pressure—is key. The partial pressure is denoted by P and depends on several factors, including the volume of the system, the temperature, and the amount of gas in moles. The partial pressures of each gas are utilized in calculating both Qp and Kp. In our example problem, we use the given partial pressures, 0.112 atm for \[\begin{equation}H_{2}\end{equation}\],0.055 atm for \[\begin{equation}S_{2}\end{equation}\],and 0.445 atm for \[\begin{equation}H_{2}S\end{equation}\],to determine Qp and assess the system's state with respect to equilibrium.

Le Chatelier's Principle

Le Chatelier's principle is an insightful guideline that predicts the response of a system in equilibrium to external changes. This could be alterations in concentration, pressure, volume, or temperature. The principle states that if a dynamic equilibrium is disturbed by changing the conditions, the position of equilibrium will shift in a direction that tends to cancel out the change. It's an essential factor in predicting the behavior of reactions in response to environmental shifts. For example, if a reactant is added to a system, the equilibrium will shift towards the formation of products in order to minimize the disturbance. This principle helps us understand that in the provided reaction, due to the reaction quotient being greater than the equilibrium constant, the reaction will shift to the left to lower the pressures of the products and raise the pressure of the reactants, thereby moving towards a new equilibrium state.