The substrate in a goldfish tank may be sand, pea-sized or pebble-sized gravel, or you may decide to forgo gravel altogether and just have a bare-bottomed tank. Many gold fish spend their days digging through the substrate searching for food, so the gravel must not be dangerous for them to ingest or disturb. Goldfish produce a lot of waste, so the substrate must be easy to clean. The additional filtration provided by living plants benefits a messy fish like the goldfish, so the substrate should also support plant life.
If swallowed, sand is irritating to the goldfish's gills over the long term. Plants can be planted directly in the sand without pots, but the sand may become so densely packed that it suffocates the plants' root systems. Sand is difficult to clean using a gravel vacuum or siphon, because the lightweight, small particles are sucked into the siphon along with the waste. Undisturbed sections of sand can become densely packed, and air pockets in those sections become isolated from the rest of the water and air in the tank. Over time, those trapped air pockets can turn into toxic gas, which can kill the goldfish if accidentally released, such as through the goldfish's habitual burrowing.
Pea-sized gravel is large enough that although the goldfish can get pieces into its mouth, they will be too large for the goldfish to swallow, which means they won't irritate its gills when they're expelled. Plants can be planted in this gravel directly - the plants' root systems grow between the gravel pieces and won't suffocate. The space between the gravel pieces is large enough that the gravel gets adequate aeration and toxic gas pockets are unlikely to form. Pea-sized gravel is heavy and large enough to be cleaned easily with a gravel vacuum, but packs together densely enough that the waste tends to sit on top of the gravel rather than sinking between the pieces and on the tank floor.
Pebbles are so large that the goldfish can't fit the pieces into their mouths, which lowers the risk of ingestion even further. The pebbles are too large and the spaces between the pebbles too great for plants. They wouldn't have anything to anchor on, so any living plants in the tank must be kept in pots. The large spaces between the large pebbles make toxic gas build up unlikely, but waste tends to sink between those spaces onto the bottom of the tank, making the tank difficult to siphon clean. Though a goldfish is unlikely to be so small that it's in danger of getting crushed beneath a pebble or in a crevice between pebbles, in a community tank that could be a danger to other fish.
Bare-bottomed tanks deny the goldfish the ability to engage in its natural digging behavior. Any plants in a bare-bottomed tank must be kept in pots. Additionally, the tank's natural filtration may be impaired because without substrate, there is less surface area in the tank for "good" bacteria (which metabolizes ammonia and fish waste into less harmful chemicals) to grow on. However, the tank is easy to clean; waste is easy to siphon out of the tank, and the bottom of the tank can be scrubbed with an algae brush.
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