|How long does Carbon last in the system?
Entirely dependent on what the carbon you HAVE is supposed to absorb, AND what quality the carbon is.
Not usually longer than eight weeks - and if there is a lot of organic or other chemical debris in the system, the carbon is shorter-lived. On the other hand, If the water is clean and clear and there are no additions of dyes or other chemicals for the carbon to bind and there's a relatively low stocking density, it could conceivably last three months.
One of the variables in this is simply: How much carbon, relative to the mass of water, are you using? For example, a gallon of carbon on a ten gallon fish tank is going to function very well, and for a very very long time. A gallon of carbon on a six thousand gallon fish pond will likely have no affect at all.
Here's an empirical rule of thumb - If you used a gallon jug of carbon for two or three hundred gallons of water, you would get both good function and reasonable life out of it. To know if carbon is "spent" take a handful of it and toss it in a drinking glass. Add pond water. Add one drop of methylene blue or malachite green to the tumbler. Stir. If the color is removed from the water, your carbon has life. Never rely on the carbon after 3 months. Many would argue a lower number along the lines of eight weeks.