Question

The reaction \(\mathrm{N}_{2} \mathrm{O}_{4}(g) \rightleftharpoons 2 \mathrm{NO}_{2}(g)\) has \(K_{\mathrm{P}}=0.140\) at \(25^{\circ} \mathrm{C}\). In a reaction vessel containing these gases in equilibrium at this temperature, the partial pressure of \(\mathrm{N}_{2} \mathrm{O}_{4}\) was 0.250 atm. (a) What was the partial pressure of the \(\mathrm{NO}_{2}\) in the reaction mixture? (b) What was the total pressure of the mixture of gases?

Short answer

The partial pressure of NO2 in the reaction mixture was approximately 0.210 atm, and the total pressure of the mixture of gases was approximately 0.670 atm.

Step-by-step solution

Write the Expression for the Equilibrium Constant Kp

The equilibrium constant, Kp, is expressed in terms of partial pressures of the gases involved in the equilibrium reaction. For the given reaction, it can be written as: \( K_{p} = \frac{(P_{NO_{2}})^2}{P_{N_{2}O_{4}}} \)

Substitute the Known Values into the Equation

Substitute the given values for Kp and the partial pressure of N2O4 into the equation to solve for the partial pressure of NO2. \( 0.140 = \frac{(P_{NO_{2}})^2}{0.250} \)

Solve for Partial Pressure of NO2

Isolate the term \((P_{NO_{2}})^2\) by multiplying both sides of the equation by the partial pressure of N2O4, and then take the square root of both sides to find PNO2. \( (P_{NO_{2}})^2 = 0.140 \times 0.250 \Rightarrow P_{NO_{2}} = \sqrt{0.140 \times 0.250} \)

Calculate the Total Pressure of the Gas Mixture

Since the reaction is at equilibrium, the total pressure of the gas mixture is the sum of the partial pressures of the individual gases. The total pressure P_total is given by: \( P_{total} = P_{N_{2}O_{4}} + 2 \times P_{NO_{2}} \)

Substitute the Known Values to Find Total Pressure

Substitute the known partial pressure of N2O4 and the calculated partial pressure of NO2 into the equation for P_total. \( P_{total} = 0.250 + 2 \times P_{NO_{2}} \)

Key concepts

Equilibrium Constant (Kp)

The equilibrium constant for a reaction involving gases, denoted as Kp, is a measure of the extent of the reaction when it reaches a state of balance, or equilibrium. At this point, the rate of the forward reaction equals the rate of the reverse reaction, and the concentrations of the reactants and products remain constant over time.

For the reaction N₂O₄(g) ↔ 2NO₂(g), we write the Kp expression by raising the partial pressures of the products to the power of their stoichiometric coefficients and dividing by the corresponding expression for the reactants:
\[\begin{equation}K_{p} = \frac{\left(P_{NO_{2}}\right)^2}{P_{N_{2}O_{4}}}\end{equation}\]
The value of Kp provided, 0.140, indicates the relative amounts of products and reactants at equilibrium at a given temperature, which in this case is 25^oC.

Partial Pressure

Partial pressure represents the individual pressure exerted by a particular gas in a mixture of gases. It is directly proportional to its molar fraction in the mixture. The total pressure exerted by the gas mixture is the sum of the partial pressures of all gases present.

In our example, the partial pressure of N₂O₄ is given as 0.250 atm, and the partial pressure of NO₂ can be calculated using the formula derived from the equilibrium constant expression. By determining the partial pressures of each component, one can also compute the total pressure of the gas mixture in the system.

Reaction Quotient

The reaction quotient, Q, is a measure that indicates the direction in which a reaction will proceed to reach equilibrium. It is calculated using the same formula as the equilibrium constant, Kp, but with the initial partial pressures instead of the equilibrium partial pressures.

If Q > Kp, the reaction will proceed toward the reactants to reach equilibrium. Conversely, if Q < Kp, the reaction will proceed toward the products. When Q = Kp, the reaction is already at equilibrium. Calculating Q can predict shifts in the equilibrium position due to changes in concentration, pressure, or volume.

Le Chatelier's Principle

Le Chatelier's principle states that if a dynamic equilibrium is disturbed by changing the conditions, the system responds by shifting its equilibrium position in a way that counteracts the change. This can include changes in concentration, pressure, volume, or temperature.

For example, increasing the pressure of a gas reaction by decreasing volume generally shifts the equilibrium toward the side with fewer moles of gas. If temperature changes, the direction of the shift will favor the exothermic or endothermic reaction, depending on whether the temperature is decreased or increased respectively. Understanding Le Chatelier's principle helps predict how a reaction will respond to external changes and is crucial for controlling industrial chemical processes.