Dinitrogen tetroxide decomposes to nitrogen dioxide: N₂O₄(g) → 2 NO₂(g) ΔH°_rxn = 55.3 kJ At 298 K, a reaction vessel initially contains 0.100 atm of N₂O₄. When equilibrium is reached, 58% of the N₂O₄ has decomposed to NO₂. What percentage of N₂O₄ decomposes at 388 K? Assume that the initial pressure of N₂O₄ is the same (0.100 atm).

Verified step by step guidance

1. Step 1: Write down the balanced chemical equation for the reaction. In this case, it is N2O4(g) ⇌ 2NO2(g).
2. Step 2: Use the given information to calculate the equilibrium constant (K) at 298 K. The equilibrium constant is given by K = [NO2]^2 / [N2O4]. Since the reaction is in terms of pressure, we can use the partial pressures of the gases. The initial pressure of N2O4 is 0.100 atm, and 58% of it decomposes, so the equilibrium pressure of N2O4 is 0.100 atm * (1 - 0.58) = 0.042 atm. The pressure of NO2 at equilibrium is twice the amount of N2O4 that decomposed, so it is 2 * 0.100 atm * 0.58 = 0.116 atm.
3. Step 3: Substitute these values into the equation for K to find its value at 298 K.
4. Step 4: Use the Van't Hoff equation to find the equilibrium constant at 388 K. The Van't Hoff equation is ln(K2/K1) = -ΔHrxn/R * (1/T2 - 1/T1), where K1 and K2 are the equilibrium constants at temperatures T1 and T2 respectively, ΔHrxn is the enthalpy change of the reaction, and R is the ideal gas constant. In this case, K1 is the equilibrium constant at 298 K, T1 is 298 K, ΔHrxn is 55.3 kJ, R is 8.314 J/(mol*K), and T2 is 388 K.
5. Step 5: Solve the Van't Hoff equation for K2, which is the equilibrium constant at 388 K. Then, use this value to calculate the percentage of N2O4 that decomposes at 388 K. The percentage of N2O4 that decomposes is given by the ratio of the equilibrium pressure of NO2 to the initial pressure of N2O4, multiplied by 100%.

Key Concepts

Le Chatelier's Principle

Le Chatelier's Principle states that if a dynamic equilibrium is disturbed by changing the conditions, the position of equilibrium shifts to counteract the change. In the context of temperature changes, increasing the temperature of an exothermic reaction will favor the endothermic direction, while decreasing the temperature will favor the exothermic direction. This principle helps predict how the equilibrium will shift when the temperature is altered.

Equilibrium Constant (K)

The equilibrium constant (K) is a numerical value that expresses the ratio of the concentrations of products to reactants at equilibrium for a given reaction at a specific temperature. For the reaction N2O4 ⇌ 2 NO2, K can be calculated using the partial pressures of the gases involved. Changes in temperature will affect the value of K, which in turn influences the extent of the reaction and the concentrations of the reactants and products at equilibrium.

Reaction Enthalpy (ΔHrxn)

Reaction enthalpy (ΔHrxn) indicates the heat change associated with a chemical reaction at constant pressure. A positive ΔHrxn, such as 55.3 kJ for the decomposition of N2O4, signifies that the reaction is endothermic, absorbing heat from the surroundings. This information is crucial for understanding how temperature changes can affect the equilibrium position and the extent of decomposition of N2O4 at different temperatures.