Question

The Haber process is the principal industrial route for converting nitrogen into ammonia: $$ \mathrm{N}_{2}(g)+3 \mathrm{H}_{2}(g) \longrightarrow 2 \mathrm{NH}_{3}(g) $$ (a) What is being oxidized, and what is being reduced? (b) Using the thermodynamic data in Appendix \(\mathrm{C}\), calculate the equilibrium constant for the process at room temperature. (c) Calculate the standard emf of the Haber process at room temperature.

Short answer

(a) Nitrogen (N) is being reduced, and hydrogen (H) is being oxidized. (b) The equilibrium constant for the Haber process at room temperature is approximately 6.31 × 10^3. (c) The standard emf of the Haber process at room temperature is approximately 0.09 V.

Step-by-step solution

1. Identify the Oxidized and Reduced elements

First, let's assign oxidation numbers to the elements in the reactants and products. Nitrogen in \(N_2\) has an oxidation number of 0, as do the hydrogens in \(H_2\). In ammonia (\(NH_3\)), nitrogen has an oxidation number of -3, and hydrogen has an oxidation number of +1. Since the nitrogen's oxidation number changes from 0 to -3, it gains electrons and thus is being reduced. The hydrogen's oxidation number changes from 0 to +1, meaning that it loses electrons and is being oxidized. (a) Nitrogen (N) is being reduced, and hydrogen (H) is being oxidized.

2. Calculate the Equilibrium Constant

We will use the Gibbs Free Energy (\(ΔG\)) to calculate the equilibrium constant (K) for this process at room temperature (298 K). First, we should know the values of Gibbs Free Energy for nitrogen (\(N_2\)), hydrogen (\(H_2\)), and ammonia (\(NH_3\)). According to Appendix C: \(ΔG^o_{N_2} = 0\;kJ/mol\) \(ΔG^o_{H_2} = 0\;kJ/mol\) \(ΔG^o_{NH_3} = -16.45\;kJ/mol\) Next, calculate the total change in Gibbs Free Energy (\(ΔG_{total}\)) for the reaction: \(ΔG_{total} = ΔG_{products} - ΔG_{reactants}\) \(ΔG_{total} = 2 × ΔG^o_{NH_3} - (ΔG^o_{N_2} + 3 × ΔG^o_{H_2})\) \(ΔG_{total} = 2 × −16.45 - (0 + 3 × 0)\) \(ΔG_{total} = -32.90\;kJ/mol\) Now, we can use the equation K = \(e^{(-ΔG_{total}/RT)}\) to calculate the equilibrium constant, where R is the ideal gas constant (8.314 J/mol·K) and T is the temperature in Kelvin (298K). K = \(e^{[-(-32.9 × 10^3)/(8.314 × 298)]}\) K ≈ 6.31 × 10^3 (b) The equilibrium constant for the Haber process at room temperature is approximately 6.31 × 10^3.

3. Calculate the Standard emf

The standard emf (Eº) of a reaction is related to the change in Gibbs Free Energy by the equation \(ΔG_{total} = -nFE^o\), where n is the number of moles of electrons transferred and F is the Faraday's constant (96,485 C/mol). In the Haber process, 3 moles of hydrogen, each with an oxidation number of 0, is converted to 6 moles of hydrogen in ammonia, each with an oxidation number of +1. Therefore, the total number of moles of electrons transferred in the Haber process is 6. Now, we can calculate the standard emf (Eº) using the Gibbs Free Energy (\(ΔG_{total}\)) calculated earlier: \(E^o = \frac{-ΔG_{total}}{nF}\) \(E^o = \frac{(-(-32.9 × 10^3))}{(6 × 96,485)}\) Eº ≈ 0.09 V (c) The standard emf of the Haber process at room temperature is approximately 0.09 V.

Key concepts

Oxidation-Reduction in the Haber Process

The Haber Process is a classic example of an oxidation-reduction reaction. In such reactions, one substance is oxidized (loses electrons), and another is reduced (gains electrons). Let's break it down:

• Nitrogen (\( \text{N}_2 \)) is reduced during the reaction. Initially, nitrogen molecules each have an oxidation number of 0. After the reaction, in ammonia (\( \text{NH}_3 \)), nitrogen has an oxidation number of -3. This change indicates that nitrogen gains electrons, undergoing reduction.
• Hydrogen (\( \text{H}_2 \)) is oxidized. Initially having an oxidation number of 0, hydrogen in ammonia possesses an oxidation number of +1. This change marks the loss of electrons by hydrogen, identifying it as oxidized.

The sum total of oxidation and reduction must balance, meaning the number of electrons lost equals the number gained. This balance is essential in assessing and understanding redox reactions.

Understanding the Equilibrium Constant

The equilibrium constant (\( K \)) is a crucial aspect of the Haber Process. It quantifies the relationship between the concentrations of products and reactants at equilibrium, giving us insight into the reaction's favorability and extent. Here's a simple explanation:

• We calculate \( K \) using Gibbs Free Energy (\( \Delta G \)). The formula is\[ K = e^{(-\Delta G_{total}/RT)} \]where \( R \) is the gas constant (8.314 J/mol·K) and \( T \) is temperature (298 K, here at room temperature).
• The Gibbs Free Energy (\( \Delta G_{total} \)) is derived from the Gibbs energies of the products minus the reactants. For ammonia formation, it specifically depends on the energy changes from nitrogen and hydrogen's conversion to ammonia.
• A large value of \( K \) implies that the equilibrium favors products, meaning the reaction proceeds significantly toward ammonia production.

By understanding the equilibrium constant, students grasp the dynamics and tendencies of chemical reactions.

Standard emf of the Haber Process

The standard electromotive force (emf, \( E^o \)) relates to the energy changes and electron flow in redox reactions like the Haber Process. This measure reveals the reaction's capacity to do electrical work by transferring electrons:

• In hydrogen's oxidation within the Haber Process, electrons are transferred. For each molecule of hydrogen oxidized, electrons are transferred, calculated using the number of moles of electrons involved (6 in this process).
• The formula to calculate the standard emf is:\[ E^o = \frac{-\Delta G_{total}}{nF} \]where n is the moles of electrons transferred, and F (Faraday's constant) is 96,485 C/mol.
• The standard emf, around 0.09 V here, helps understand the energy efficiency of producing ammonia. A positive \( E^o \) indicates a spontaneous reaction, suitable for energy transformations.

This concept ties thermodynamic principles with electrochemical data, aiding in a comprehensive understanding of the process dynamics.