The optimal conditions for the Haber process involve high pressure (approximately $200$ atmospheres), moderate temperature (around $450^\circ \text{C}$), and the use of an iron catalyst.

The Haber process, often referred to as the Haber-Bosch process, is a widely used method for the large-scale synthesis of ammonia. This process entails the direct combination of nitrogen and hydrogen gases to produce ammonia. Notably, the reaction is both reversible and exothermic, indicating that it releases heat during the process.

The ideal conditions for the Haber process include a high pressure of about $200$ atmospheres, a moderate temperature near $450^\circ \text{C}$, and the incorporation of a catalyst—typically iron. These conditions are strategically selected to maximize ammonia yield while also taking into account economic and safety considerations.

High pressure favors the forward reaction that generates ammonia since there are fewer gas molecules on the product side of the reaction equation (with $4$ molecules on the left and $2$ on the right). According to Le Chatelier’s principle, increasing the pressure shifts the equilibrium towards the side with fewer gas molecules, thereby enhancing the ammonia yield. However, it is important to note that extremely high pressures can be both costly and hazardous, leading to the compromise value of approximately $200$ atmospheres.

The exothermic nature of the reaction implies that lower temperatures would favor the formation of ammonia, in accordance with Le Chatelier’s principle. Yet, operating at low temperatures would result in a reaction rate that is too slow to be economically feasible. Thus, a moderate temperature of around $450^\circ \text{C}$ is selected as a balance between maximizing yield and maintaining a reasonable reaction rate.

Lastly, a finely divided iron catalyst is employed to accelerate the reaction rate without altering the position of the equilibrium. The catalyst offers a surface for the reactant molecules to adsorb onto, increasing the likelihood of effective collisions and thereby expediting the reaction process.