Question

Consider the reaction: $$ \mathrm{N}_{2}(g)+\mathrm{O}_{2}(g) \rightleftarrows 2 \mathrm{NO}(g) $$ If the equilibrium partial pressures of \(\mathrm{N}_{2}, \mathrm{O}_{2},\) and \(\mathrm{NO}\) are \(0.15,0.33,\) and 0.050 atm, respectively, at \(2200^{\circ} \mathrm{C}\) what is \(K_{P}\) ?

Short answer

\(K_{P} \approx 0.0505\).

Step-by-step solution

Write the Equilibrium Expression

For the given reaction \( \mathrm{N}_{2}(g)+\mathrm{O}_{2}(g) \rightleftarrows 2 \mathrm{NO}(g) \), the equilibrium constant \( K_{P} \) in terms of partial pressures is expressed as: \[K_{P} = \frac{{(P_{\mathrm{NO}})^2}}{{P_{\mathrm{N}_2} \cdot P_{\mathrm{O}_2}}}\]

Insert Given Equilibrium Partial Pressures

Substitute the given partial pressures into the equilibrium expression. \[P_{\mathrm{N}_2} = 0.15 \text{ atm}, \quad P_{\mathrm{O}_2} = 0.33 \text{ atm}, \quad P_{\mathrm{NO}} = 0.050 \text{ atm}\] Insert these values into the equation:\[K_{P} = \frac{{(0.050)^2}}{{0.15 \cdot 0.33}}\]

Calculate \(K_{P}\)

Now, perform the calculation:1. Compute \((0.050)^2 = 0.0025.\)2. Compute the product \(0.15 \times 0.33 = 0.0495.\)3. Compute the division: \[K_{P} = \frac{{0.0025}}{{0.0495}} \approx 0.0505\]

Final Result

The calculated equilibrium constant \(K_{P}\) for the reaction at \(2200^{\circ} \mathrm{C}\) is approximately 0.0505.

Key concepts

Partial Pressure

Partial pressure refers to the pressure exerted by a specific gas in a mixture of gases. Each gas in a mixture has its partial pressure, representing its contribution to the total pressure. In chemical equilibrium, understanding partial pressure is crucial because it helps calculate equilibrium constants like \( K_P \), which are based on partial pressures.
To find the partial pressure, use Dalton's Law of Partial Pressures. According to this law, the total pressure of a gas mixture is the sum of the partial pressures of each individual gas. Therefore, the partial pressure of a gas can be expressed as:

• \( P_{gas} = X_{gas} \times P_{total} \)

Where \( X_{gas} \) is the mole fraction of the gas and \( P_{total} \) is the total pressure of the gas mixture.
In the exercise, the equilibrium constants like \( K_P \) depend on the partial pressures of reacting gases like \( \mathrm{N}_2 \), \( \mathrm{O}_2 \), and \( \mathrm{NO} \). By understanding and substituting these into the equilibrium expression, we can find meaningful values for reaction predictions.

Reaction Quotient

The reaction quotient, \( Q \), is a dimensionless number that helps us understand the state of a reaction relative to equilibrium. While the equilibrium constant, \( K \), describes a system at equilibrium, \( Q \) can describe a system at any point during the reaction.
Calculating \( Q \) involves the same expression used for the equilibrium constant but can be determined by current concentrations or partial pressures of the reactants and products. It looks like:

• \( Q = \frac{{(P_{\mathrm{NO}})^2}}{{P_{\mathrm{N}_2} \cdot P_{\mathrm{O}_2}}} \)

By comparing \( Q \) with \( K \):

• If \( Q = K \), the system is at equilibrium.
• If \( Q < K \), the reaction will proceed forward to form more products.
• If \( Q > K \), the reaction will shift backward to form more reactants.

Understanding \( Q \) helps to predict how a reaction will proceed under varying conditions, which is essential for optimizing industrial chemical processes.

Gas Equilibrium

Gas equilibrium refers to a state where the rate of the forward reaction equals the rate of the reverse reaction for a gas-phase system. At this point, the concentrations or partial pressures of reactants and products remain constant over time.
In the given reaction \( \mathrm{N}_2(g)+\mathrm{O}_2(g) \rightleftarrows 2\mathrm{NO}(g) \), the gases reach equilibrium quickly due to rapid molecular interactions, especially at high temperatures. Gauging gas equilibrium involves using partial pressures in equilibrium expressions, as seen with \( K_P \) calculations.
Key characteristics of gas equilibrium include:

• Reversibility: Reactions can proceed in both forward and reverse directions.
• Constant State: Partial pressures remain constant over time at equilibrium.
• Dependence: Equilibrium is affected by changes in pressure, temperature, and concentration.

Knowing about gas equilibrium helps in predicting the impact of changing conditions on a chemical system.

Chemical Equilibrium

Chemical equilibrium is the state of balance in a reversible reaction when the rate of the forward reaction equals the rate of the backward reaction. At this point, the concentrations of reactants and products remain unchanged, but they are not necessarily equal.
This state is defined by the equilibrium constant (\( K \)), which can be expressed in terms of concentrations as \( K_c \), or partial pressures as \( K_P \), for gas-phase reactions. These constants provide specific insights into the reaction's extent.

• \( K_P \) is used when dealing with gases, such as in the given reaction scenario.
• The magnitude of \( K \) indicates the extent of the reaction. A large \( K \) (much greater than 1) means products are favored, while a small \( K \) (much less than 1) indicates reactants are favored.

Understanding chemical equilibrium is crucial for manipulating reactions to optimize yields and processes effectively, aligning with the goals of synthetic chemistry and industrial production.