Yes, the carbon is now worthless as carbon. You might get some more life out of it if you crush it to open up new surfaces. The biowheel is supposed to hold enough of the biology to safely change cartridges without re-cycling the tank. Do the rest of us change cartridges every month? Nope. I rinse in used tank water if the cartridge isn't bad, sink water if it is, even rub it with a brush. I throw out the cartridge when it starts to fall apart. Now bettas eat high protein food and that leaves a scum on the water, fresh carbon should tank that out, and its great for taking out meds or odd colors from the water. But I don't know why fresh carbon would be essential.
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