Question

At \(450^{\circ} \mathrm{C}\), the equilibrium constant \(K_{\mathrm{c}}\) for the HaberBosch synthesis of ammonia is 0.16 for the reaction written as $$ 3 \mathrm{H}_{2}(\mathrm{~g})+\mathrm{N}_{2}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{NH}_{3}(\mathrm{~g}) $$ Calculate the value of \(K_{\mathrm{c}}\) for the same reaction written as $$ \frac{3}{2} \mathrm{H}_{2}(\mathrm{~g})+\frac{1}{2} \mathrm{~N}_{2}(\mathrm{~g}) \rightleftharpoons \mathrm{NH}_{3}(\mathrm{~g}) $$

Short answer

The equilibrium constant for the second reaction is 0.4.

Step-by-step solution

Understanding the Problem

The given reaction for the synthesis of ammonia is \[3 \text{H}_2(g) + \text{N}_2(g) \rightleftharpoons 2 \text{NH}_3(g)\]and its equilibrium constant is \(K_c = 0.16\) at \(450^{\circ} \text{C}\). We are asked to find the equilibrium constant \(K'_{c}\) for the same reaction but written differently:\[\frac{3}{2} \text{H}_2(g) + \frac{1}{2} \text{N}_2(g) \rightleftharpoons \text{NH}_3(g)\]

Relationship Between Equilibrium Constants

Notice that the second reaction is just the stoichiometry of the first reaction divided by 2. For a reaction scaled by a factor \(n\), the new equilibrium constant \(K'_{c}\) is related to the original \(K_c\) by \[K'_{c} = K_{c}^{1/n}\]where \(n\) is the factor by which the stoichiometry is divided.

Calculate the New Equilibrium Constant

In this case, \(n = 2\) as the stoichiometry of all reactants and products is halved. Thus, \[K'_{c} = (K_{c})^{1/2} = (0.16)^{1/2}\]Calculate the square root of \(0.16\) to find \(K'_{c}\).

Perform the Calculation

Calculate \[(0.16)^{1/2} = 0.4\]So, the new equilibrium constant \(K'_{c}\) is 0.4.

Key concepts

Haber-Bosch process

The Haber-Bosch process is an essential method for synthesizing ammonia. It involves the direct combination of nitrogen from the air with hydrogen gas, typically derived from natural gas. This reaction occurs under high pressure and temperature, facilitated by a catalyst.
The primary reaction is:

• 3H₂(g) + N₂(g) ↔ 2NH₃(g)

This process is vital for producing fertilizers, supporting global agriculture. Its industrial relevance stems from the balance between reaction conditions and efficiency. The Haber-Bosch process not only revolutionized agriculture but also deeply impacted chemical engineering and industrial practices.
Understanding this process helps grasp other key concepts in chemistry, such as equilibrium and reaction kinetics.

Chemical Equilibrium

In chemistry, equilibrium refers to the state where the rates of the forward and reverse reactions equalize. This means the concentrations of reactants and products remain steady over time.
In the context of the Haber-Bosch process:

• The equilibrium is represented by the reversible reaction 3H₂(g) + N₂(g) ↔ 2NH₃(g).

At equilibrium, the concentration of ammonia (NH₃) does not change unless external conditions such as pressure or temperature are altered.
Understanding chemical equilibrium involves recognizing how the equilibrium constant (K_c) indicates the position of equilibrium. A higher K_c value suggests more products relative to reactants at equilibrium.

Stoichiometry

Stoichiometry involves the calculation of reactants and products in chemical reactions. It uses the coefficients in a balanced chemical equation to determine how much reactant is needed or how much product will form.
In our ammonia synthesis problem, stoichiometry helps adjust the reaction:

• Original: 3H₂ + N₂ ↔ 2NH₃
• Modified: 1.5H₂ + 0.5N₂ ↔ NH₃

This adjustment means changing the equilibrium constant. If a reaction’s stoichiometry is divided by a factor, the new equilibrium constant relates to the original by a power of this division factor.
Using stoichiometry correctly ensures accurate chemical reactions and calculations.

Ammonia Synthesis

Ammonia synthesis is the process of creating ammonia from hydrogen and nitrogen molecules. It's a key part of the Haber-Bosch process and fundamental in producing fertilizers.
The synthesis process:

• Involves combining hydrogen (H₂) and nitrogen (N₂).
• Occurs under high pressure and temperature with a catalyst to speed up the reaction.

Ammonia is crucial for multiple applications, primarily in fertilizers, which aid in food production globally.
The synthesis is a perfect example of applied chemical principles like equilibrium and kinetics, showing how theoretical chemistry translates to practical applications.

Reaction Quotient

The reaction quotient (Q) is a measure used to determine the direction a reaction will proceed to reach equilibrium. It’s calculated like the equilibrium constant (K_c) but at non-equilibrium conditions.
For the reaction:

• 3H₂ + N₂ ↔ 2NH₃

The reaction quotient is given by:
\[ Q = \frac{[NH_3]^2}{[H_2]^3[N_2]} \]If Q = K_c, the system is at equilibrium. If Q < K_c, the reaction will proceed forward, creating more products. If Q > K_c, the reaction will shift to produce more reactants.
Understanding Q helps predict reaction shifts and manage chemical processes effectively.