What is the ammonia, nitrite, nitrate and pH in numbers?
How often and how do you clean the filter?
How often do you do water changes and how much do you change?
Do you gravel clean the substrate when you do a water change?
Do you dechlorinate the new water before adding it to the aquarium?
Do you have chlorine or chloramine in the tap water?
The Corydoras in the first video appears to have a creamy patch on its head. This is excess mucous caused by something in the water irritating the fish. The other Corydoras does not appear to have it so it's either an injury or external protozoan infection like Costia.
The second video shows a lot of damage to the fish's tail and heavy breathing. I would treat the tank with salt and see how it goes. But post water quality results and filter and tank cleaning info first.
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Before you add salt, do the following.
Wipe the inside of the glass down with a clean fish sponge. This removes the biofilm on the glass and the biofilm will contain lots of harmful bacteria, fungus, protozoans and various other microscopic life forms.
Do a 75% water change and gravel clean the substrate. The water change and gravel cleaning will reduce the number of disease organisms in the water and provide a cleaner environment for the fish to recover in. It also removes a lot of the gunk and this means any medication can work on treating the fish instead of being wasted killing the pathogens in the gunk.
Make sure any new water is free of chlorine/ chloramine before it is added to the tank.
Clean the filter if it hasn't been done in the last 2 weeks. However, if the filter is less than 6 weeks old, do not clean it. Wash the filter materials/ media in a bucket of tank water and re-use the media. Tip the bucket of dirty water on the garden/ lawn. Cleaning the filter means less gunk and cleaner water with fewer pathogens so any medication (if needed) will work more effectively on the fish.
Increase surface turbulence/ aeration to maximise the dissolved oxygen in the water.
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SALT
You can add rock salt (often sold as aquarium salt), swimming pool salt, or any non iodised salt (sodium chloride) to the aquarium at the dose rate of 1 heaped tablespoon per 20 litres (5 gallons) of water. If there is no improvement after 48 hours you can double that dose rate so there is 2 heaped tablespoons of salt per 20 litres.
Keep the salt level like this for at least 2 weeks but no longer than 4 weeks otherwise kidney damage can occur. Kidney damage is more likely to occur in fish from soft water (tetras, Corydoras, angelfish, Bettas & gouramis, loaches) that are exposed to high levels of salt for an extended period of time, and is not an issue with livebearers, rainbowfish or other salt tolerant species.
The salt will not affect the beneficial filter bacteria, fish, plants, shrimp or snails.
After you use salt and the fish have recovered, you do a 10% water change each day for a week using only fresh water that has been dechlorinated. Then do a 20% water change each day for a week. Then you can do bigger water changes after that. This dilutes the salt out of the tank slowly so it doesn't harm the fish.
If you do water changes while using salt, you need to treat the new water with salt before adding it to the tank. This will keep the salt level stable in the tank and minimise stress on the fish.
When you first add salt, add the salt to a small bucket (2 litres) of tank water and dissolve the salt. Then slowly pour the salt water into the tank near the filter outlet. Add the salt over a couple of minutes.
