Problem 69 - Chemistry: The Central Science

Question

Nitric oxide (NO) reacts readily with chlorine gas as follows: $$ 2 \mathrm{NO}(g)+\mathrm{Cl}_{2}(g) \rightleftharpoons 2 \mathrm{NOCl}(g) $$ At \(700 \mathrm{~K}\) the equilibrium constant \(K_{p}\) for this reaction is \(0.26\). Predict the behavior of each of the following mixtures at this temperature: (a) \(P_{\mathrm{NO}}=0.15 \mathrm{~atm}, P_{\mathrm{Cl}_{2}}=0.31 \mathrm{~atm}\), and \(P_{\mathrm{NOCl}}=0.11 \mathrm{~atm} ;\) (b) \(\mathrm{P}_{\mathrm{NO}}=0.12 \mathrm{~atm}, P_{\mathrm{Cl}_{2}}=0.10 \mathrm{~atm}\) and \(\quad P_{\mathrm{NOCl}}=0.050 \mathrm{~atm} ; \quad\) (c) \(\quad P_{\mathrm{NO}}=0.15 \mathrm{~atm}\), \(P_{\mathrm{C}_{2}}=0.20 \mathrm{~atm}\), and \(P_{\mathrm{NOCl}}=5.10 \times 10^{-3} \mathrm{~atm}\)

Step-by-Step Solution

Answer

For all three cases (a, b, and c), the reaction will proceed to the right, forming more NOCl and consuming the reactants (NO and Cl₂), as the reaction quotient Qp is less than the equilibrium constant Kp for each case.

Step 1: Write the expression for the equilibrium constant Kp and reaction quotient Qp.

The equilibrium constant Kp is given by: \[K_p = \frac{(P_{NOCl})^2}{(P_{NO})^2 \cdot P_{Cl_2}}\] The expression for the reaction quotient Qp is the same as the equilibrium constant Kp, except we use the initial partial pressures instead of the equilibrium partial pressures: \[Q_p = \frac{(P'_{NOCl})^2}{(P'_{NO})^2 \cdot P'_{Cl_2}}\]

Step 2: Calculate the reaction quotient Qp for case (a).

For case (a), the initial partial pressures are given as \(P_{NO} = 0.15 \mathrm{~atm}\), \(P_{Cl_2} = 0.31 \mathrm{~atm}\), and \(P_{NOCl} = 0.11 \mathrm{~atm}\). We will plug these values into the Qp formula: \[Q_p = \frac{(0.11)^2}{(0.15)^2 \cdot 0.31} = 0.1811\]

Step 3: Compare Qp and Kp for case (a).

For case (a), Qp = 0.1811 and Kp = 0.26. Since Qp < Kp, the reaction will proceed to the right, forming more NOCl and consuming the reactants (NO and Cl2).

Step 4: Calculate the reaction quotient Qp for case (b).

For case (b), the initial partial pressures are given as \(P_{NO} = 0.12 \mathrm{~atm}\), \(P_{Cl_2} = 0.10 \mathrm{~atm}\), and \(P_{NOCl} = 0.050 \mathrm{~atm}\). We will plug these values into the Qp formula: \[Q_p = \frac{(0.050)^2}{(0.12)^2 \cdot 0.10} = 0.1736\]

Step 5: Compare Qp and Kp for case (b).

For case (b), Qp = 0.1736 and Kp = 0.26. Since Qp < Kp, the reaction will proceed to the right, forming more NOCl and consuming the reactants (NO and Cl2).

Step 6: Calculate the reaction quotient Qp for case (c).

For case (c), the initial partial pressures are given as \(P_{NO} = 0.15 \mathrm{~atm}\), \(P_{Cl_2} = 0.20 \mathrm{~atm}\), and \(P_{NOCl} = 5.10 \times 10^{-3} \mathrm{~atm}\). We will plug these values into the Qp formula: \[Q_p = \frac{(5.10 \times 10^{-3})^2}{(0.15)^2 \cdot 0.20} = 0.00011378\]

Step 7: Compare Qp and Kp for case (c).

For case (c), Qp = 0.00011378 and Kp = 0.26. Since Qp < Kp, the reaction will proceed to the right, forming more NOCl and consuming the reactants (NO and Cl2). Based on the comparisons of Qp and Kp for each case, the reaction will proceed to the right in all three cases, and the system will evolve toward the equilibrium position.

Key Concepts

Reaction Quotient

The reaction quotient, represented by \(Q_p\) for gas-phase reactions, helps us understand if a system is at equilibrium or how it will shift to reach equilibrium. To calculate \(Q_p\), we use the same expression as the equilibrium constant, but with current or initial pressures:

• If \(Q_p < K_p\), the system will shift to form more products.
• If \(Q_p > K_p\), the system will shift to form more reactants.
• If \(Q_p = K_p\), the system is at equilibrium.

In practical terms, determine the current pressures of reactants and products, then plug them into the \(Q_p\) formula for comparison with \(K_p\). This comparison reveals the direction the reaction will naturally proceed to achieve equilibrium.

Equilibrium Constant

The equilibrium constant \(K_p\) expresses the ratio of the concentrations (or partial pressures for gases) of products to reactants at equilibrium. It provides insight into the composition of a reaction mixture at equilibrium, with different values indicating:

• A large \(K_p\) (much greater than 1) means products predominate.
• A small \(K_p\) (much less than 1) means reactants predominate.
• When \(K_p\) is around 1, it suggests significant amounts of both reactants and products.

The formula used in this particular reaction is:\[K_p = \frac{(P_{NOCl})^2}{(P_{NO})^2 \cdot P_{Cl_2}}\]This guides us in comparing with \(Q_p\) to predict shifts as the reaction moves toward equilibrium.

Partial Pressure

Partial pressure is a crucial concept in gas reactions, where each gas in a mixture exerts a pressure as if it were alone in the container. The total pressure of the gas mixture is the sum of the individual partial pressures. For the reaction involving nitric oxide and chlorine:

• Partial pressure is denoted by \(P_{NO}\), \(P_{Cl_2}\), and \(P_{NOCl}\).
• These values are used to calculate both the reaction quotient \(Q_p\) and the equilibrium constant \(K_p\).
Understanding partial pressures helps us determine the initial conditions and calculate how the system responds to changes, aiding in prediction and analysis of the shift in equilibrium.

Le Chatelier's Principle

Le Chatelier's Principle helps predict how a system at equilibrium responds to changes in concentration, pressure, or temperature. It's like a balance scale for chemical reactions. When the balance is disturbed, the system shifts to counteract the change.
For example:

• If the concentration of reactants increases, the reaction will shift towards products to restore balance.
• If pressure increases, the reaction may shift toward the side with fewer gas molecules.

In our reaction, knowing \(Q_p < K_p\) and considering Le Chatelier’s Principle, it helps predict that increasing the concentration of \(NOCl\) or decreasing that of \(NO\) and \(Cl_2\) helps shift the reaction toward equilibrium. The dynamic nature of equilibrium points is crucial for optimizing reactions and understanding chemical behavior.