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When the biological media in a filter is sized properly for the fish the tank, it will serve to remove toxic ammonia and nitrite, and help to covert those toxic substances to, or store “less toxic”
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From your statement above, how does the biological media "remove" the toxic ammonia and nitrite? Don't the bacteria only "convert" from one form to the next by breaking the bonds between oxygen, hydrogen, and nitrogen?
This is a good thread, thanks for starting it.

The thing to remember when we talk about the nitrogen cycle is, nitrate doesn't go away unless it is consumed by either additional bacteria (anaerobic)or plants.
Anaerobic bacteria will consume nitrate and produce nitrogen. Plants will produce oxygen and carbon dioxide.
The process is cellular and molecular and totally invisible. Thats why, when a filter manufacturer makes claims without disclosing the dangers of nitrate accumulation in favor of high flow rates and massive mechanical filtration, they are doing a disfavor to the average aquarist who may not know about the clear and present danger.....and I'm not talking about a Tom Clancy novel-made-movie starring Han Solo......a person can have absolutely crystal clear water, 0 ammonia and still have chemicals in the water that are harmful.
Unless the aquarist purges the media of this stuff, it is still in the system, and still in the tank, unless removed by water changes, or some other chemical reactor, or substance.
In my filtration plant work, biological filters needed to be backwashed every 5 days, otherwise non-beneficial bacteria would proliferate, and particles would break through, which lead to a lessening of filter efficiency.
I agree, that ammonia and nitrite in reality, are only converted to nitrate.
But I used the word removed, as a simple way of putting that those substances are mostly removed as toxic danger to the fish.
I believe filtration is most often misunderstood, especially by noobs, where it is seen as a kind of panacea, where a filter is often turned on, and until it backs up, or fish die, is forgotten about.
I am also trying to break the process down into a form where its complexity is described in enough detail to be payed attention to, but not so that the new aquarists eyes (and attention span) are glassed over with a myriad of technical terms, and at the same time,without being overly dumbed down.
Another point.
I often see that many new aquarists think the beneficial bacteria are free floating in water, and they will transfer old water to a new tank, thinking that is a way of cycling it.
This is not the case.
99.9% of beneficial are sessile, meaning they live attached to something.
The glass sides, the substrate, the biological media, plants, but not as plankton in the tank water. Using old water could be a shock buffer, if tap water is drastically different from tank water, due to a lack of water changes allowing the tank to acidify, or temperature regulation, but is really of "no" use in cycling a tank.