Question

Is the Haber process for the industrial synthesis of ammonia spontaneous or nonspontaneous under standard conditions at \(25^{\circ} \mathrm{C} ?\) At what temperature \(\left({ }^{\circ} \mathrm{C}\right)\) does the changeover occur? \(\mathrm{N}_{2}(g)+3 \mathrm{H}_{2}(g) \longrightarrow 2 \mathrm{NH}_{3}(g) \quad \Delta H^{\circ}=-92.2 \mathrm{~kJ} ; \Delta S^{\circ}=-199 \mathrm{~J} / \mathrm{K}\)

Short answer

The Haber process is spontaneous at 25°C, and the changeover occurs at about 190°C.

Step-by-step solution

Understand the Gibbs Free Energy Formula

To determine if the reaction is spontaneous, use the Gibbs free energy formula: \( \Delta G^{\circ} = \Delta H^{\circ} - T \Delta S^{\circ} \), where \( \Delta G^{\circ} \) is the change in free energy, \( \Delta H^{\circ} \) is the change in enthalpy, \( T \) is the temperature in Kelvin, and \( \Delta S^{\circ} \) is the change in entropy.

Convert Units

Convert \( \Delta S^{\circ} \) from \( \mathrm{J/K} \) to \( \mathrm{kJ/K} \) to match the units of \( \Delta H^{\circ} \).\[ \Delta S^{\circ} = -199 \mathrm{~J/K} = -0.199 \mathrm{~kJ/K} \]

Calculate \( \Delta G^{\circ} \) at 25°C

Substitute the values into the Gibbs free energy equation at \( 25^{\circ} \mathrm{C} \) which is \( 298 \mathrm{~K} \).\[ \Delta G^{\circ} = -92.2 \mathrm{~kJ} - (298 \mathrm{~K}) \times (-0.199 \mathrm{~kJ/K}) \]\[ \Delta G^{\circ} = -92.2 \mathrm{~kJ} + 59.302 \mathrm{~kJ} \]\[ \Delta G^{\circ} = -32.898 \mathrm{~kJ} \]

Interpret \( \Delta G^{\circ} \) at 25°C

Since \( \Delta G^{\circ} < 0 \), the process is spontaneous at \( 25^{\circ} \mathrm{C} \).

Determine the Temperature for Changeover

Find the temperature where \( \Delta G^{\circ} = 0 \). Set up the equation:\[ 0 = \Delta H^{\circ} - T \Delta S^{\circ} \]\[ T = \frac{\Delta H^{\circ}}{\Delta S^{\circ}} \]\[ T = \frac{-92.2 \mathrm{~kJ}}{-0.199 \mathrm{~kJ/K}} \approx 463.32 \mathrm{~K} \]

Convert Changeover Temperature to Celsius

Convert the temperature from Kelvin to Celsius.\[ T_{\mathrm{changeover}} (^{\circ}C) = 463.32 \mathrm{~K} - 273.15 \approx 190.17^{\circ}C \]

Key concepts

Gibbs Free Energy

The concept of Gibbs Free Energy (\( \Delta G^{\circ} \)) is crucial for understanding the spontaneity of chemical reactions, like the Haber process. Gibbs Free Energy links several thermodynamic quantities and measures the maximum reversible work that may be performed by a thermodynamic system at constant temperature and pressure. To predict reaction spontaneity, we use:
\[ \Delta G^{\circ} = \Delta H^{\circ} - T \Delta S^{\circ} \]
Here,

• \( \Delta G^{\circ} \): Change in Gibbs free energy
• \( \Delta H^{\circ} \): Change in enthalpy
• \( T \): Absolute temperature in Kelvin
• \( \Delta S^{\circ} \): Change in entropy

When \( \Delta G^{\circ} < 0 \), the process is spontaneous under the given conditions. If \( \Delta G^{\circ} > 0 \), it is nonspontaneous. For the Haber process, \( \Delta G^{\circ} \) is negative at 25°C, indicating spontaneous ammonia synthesis under these conditions.

Enthalpy Change

Enthalpy change (\( \Delta H^{\circ} \)) is the total heat absorbed or released in a reaction at constant pressure. It plays a major role in determining the spontaneity of reactions. For the Haber process:
\[ \Delta H^{\circ} = -92.2 \text{ kJ/mol} \]
This indicates that the reaction releases heat, making it exothermic. Exothermic reactions often favor spontaneity since they lower the energy of the system. The negative \( \Delta H^{\circ} \) is one reason why the Haber process can proceed spontaneously at certain temperatures. This release of heat helps drive the formation of ammonia.

Entropy Change

Entropy change (\( \Delta S^{\circ} \)) refers to the disorder or randomness within a system. It is crucial for understanding how the spontaneity of a reaction is influenced by temperature. In the Haber process:
\[ \Delta S^{\circ} = -199 \text{ J/K}\]
This negative value indicates a decrease in disorder as nitrogen and hydrogen gases form solid ammonia. Generally, a decrease in entropy acts against spontaneity, especially at higher temperatures where the \( T \Delta S^{\circ} \) term becomes significant.
Thus, while a negative \( \Delta S^{\circ} \) doesn't favor spontaneity, it is balanced by the exothermic nature (negative \( \Delta H^{\circ} \)), allowing spontaneity under standard conditions.

Spontaneity

Spontaneity is a vital concept for predicting whether a reaction will proceed on its own under set conditions. For the Haber process, determining spontaneity involves evaluating the \( \Delta G^{\circ} \).
At standard conditions (25°C), the negative Gibbs free energy (\( \Delta G^{\circ} = -32.898 \text{ kJ/mol} \)) implies that the Haber process is spontaneous.
However, as temperature shifts, so does spontaneity potential due to entropy's temperature dependence. The changeover temperature, calculated to be approximately 190.17°C, marks the point where the balance of \( \Delta H^{\circ} \) and \( T \Delta S^{\circ} \) shifts and \( \Delta G^{\circ} \) reaches zero, i.e., the process is no longer spontaneous. Understanding such balance aids in optimizing industrial conditions for ammonia synthesis.