Consider the Haber synthesis of gaseous NH₃ (∆H°_f = -46.1 kJ/mol; ∆G°_f = -16.5 kJ/mol: (d) What are the equilibrium constants K_p and K_c for the reaction at 350 K? Assume that ∆H° and ∆S° are independent of temperature.

Verified step by step guidance

1. First, write down the balanced chemical equation for the Haber process: N₂(g) + 3 H₂(g) ⇌ 2 NH₃(g).
2. Use the given standard Gibbs free energy of formation (∆G°_f) to calculate the standard Gibbs free energy change for the reaction (∆G°_rxn) using the formula: ∆G°_rxn = ∑∆G°_f(products) - ∑∆G°_f(reactants).
3. Calculate the equilibrium constant K at 350 K using the relationship between Gibbs free energy and the equilibrium constant, K = e^{(-∆G°_rxn / RT)}, where R is the gas constant (8.314 J/mol·K) and T is the temperature in Kelvin.
4. To find K_p, use the equation K_p = K_c(RT)^Δn, where Δn is the change in moles of gas (moles of gaseous products minus moles of gaseous reactants) for the reaction.
5. To find K_c, use the relationship K_c = K_p / (RT)^Δn. Calculate K_c using the value of K_p obtained in the previous step and the same values for R and T.

Key Concepts

Haber Synthesis

The Haber synthesis is a chemical reaction that combines nitrogen gas (N₂) and hydrogen gas (H₂) to produce ammonia (NH₃). This process is crucial for producing fertilizers and is represented by the equation N₂(g) + 3H₂(g) ⇌ 2NH₃(g). Understanding this reaction involves recognizing its enthalpy change (∆H°) and Gibbs free energy change (∆G°), which indicate the energy dynamics and spontaneity of the reaction.

Equilibrium Constants (Kp and Kc)

Equilibrium constants, Kp and Kc, quantify the ratio of products to reactants at equilibrium for a given reaction. Kp is used for gas-phase reactions and is expressed in terms of partial pressures, while Kc is used for reactions in solution and is expressed in terms of molar concentrations. The relationship between Kp and Kc can be derived from the ideal gas law and the reaction's stoichiometry, and they are temperature-dependent.

Gibbs Free Energy and Temperature

Gibbs free energy (G) is a thermodynamic potential that helps predict the direction of chemical reactions. The change in Gibbs free energy (∆G) at a given temperature indicates whether a reaction is spontaneous (∆G < 0) or non-spontaneous (∆G > 0). The relationship between Gibbs free energy and equilibrium constants is given by the equation ∆G° = -RT ln(K), where R is the gas constant and T is the temperature in Kelvin, allowing for the calculation of Kp and Kc at specific temperatures.