Chemistry - Why is the Haber process carried out at such high temperatures?
Solution 1:
As others have pointed out, it is purely kinetics, but you may still wonder, why.
For a reaction to actually occur (in both directions) and thus for an equilibrium to be reached, you need to overcome the activation energy. In the case of the Haber-Bosch process, this involves breaking the highly stable $\ce{N#N}$ triple bond. Even with the catalysts used, the energy required to break apart $\ce{N2}$ is still enormous. Therefore, a lower temperature may give a better yield of ammonia theoretically (i.e. based on equilibrium and Le Châtelier considerations) but the reaction speed would be a lot slower.
Even if you consider a batch-wise process of generating ammonia (which, as orthocresol points out, isn’t the case), it is more efficient to run two batches in half the time for two sets of $15~\%$ than to run a single batch for twice the time to get an overall yield of maybe (note: This yield is a ballpark estimate) $25~\%$ — with a batch size where $1~\%$ yield is equivalent to $100~\mathrm{kg}~\ce{NH3}$, two runs at higher temperature give $3~\mathrm{t}\ \ce{NH3}$ and one run at lower temperature $2.5~\mathrm{t}\ \ce{NH3}$ in the same time frame.
Solution 2:
The answer mainly has to do with kinetic considerations, as aml points out. I want to point out another thing.
In a typical industrial setting, you don't just mix the $\ce{N2}$ with the $\ce{H2}$ at a certain $T$ and $p$, collect the ammonia, and throw the unused reactants away. That would be horribly inefficient.
Greenwood & Earnshaw, Chemistry of the Elements (2nd ed) writes:
The gas leaving the catalyst beds contains about $15\%~\ce{NH3}$; this is condensed by refrigeration and the remaining gas mixed with more incoming synthesis gas and recycled.
The "synthesis gas" that they speak of is a mixture of $\ce{H2}$ and $\ce{N2}$ that is freshly produced via a series of reactions.
So, at first glance, the "yield" is absolutely rubbish. But reusing the reactants allows you to get high overall yields (albeit with more steps).
Solution 3:
While it is true that the Haber Process would be much more efficient at a lower temperature it's carried out at a higher one because it happens much faster at the higher temperature and while the industry wants an efficient process they also need it to happen fast enough to be commercially viable.
So you are correct but it's an example of practical realities coming before theoretical ideals. The use of the word optimum is misleading in this context