It’s ~6 hours. A person takes in 200mg at 6:00 each morning. They have 12.5mg in their system at 6:00 the next morning. The cycle continues. Each morning, they take in 200mg of caffeine and have more caffeine in their system than the day before until they have thousands of mgs of caffeine in their system. Yes?
In: Biology
using that math, never.
After 12 days of this you wake up with 13 and 1/3rd mg of caffeine in your system, add 200mg brings you up to 213 + 1/3mg of caffeine (640/3), divide that by 16 (4 half-lives) and you get 40/3, or 13+1/3, the amount you started the day with. after that, the total amount will not increase
It’s a general rule, but that’s not quite how it works. Your body is always trying to keep a certain balance of things like water, salt, sugar, acidity etc within itself. When you take in caffeine your body starts to immediately get rid of that from the moment it’s entered your bloodstream, mostly via the kidneys. They’re constantly filtering your blood and taking out stuff that it doesn’t want or has too much of. That just takes time. But due to the way that your kidneys work it’s easier to filter out stuff that there’s lots of, so if you ingest a lot of caffeine your kidneys can filter out more caffeine at once.
So yeah, maybe there is still some leftover caffeine in your blood the next morning, but consequently your body will excrete more caffeine so that it doesn’t slowly build up in your blood over time. Of course that only works if you keep drinking water, so stay hydrated folks!
Putting aside for the moment that this is a rule of thumb.
You are ignoring the fact that the following morning the person has slightly more caffeine in their system, so the rate at which it leaves their system is higher. This continues each day until the amount removed over a 24 hour period is equal to 200mg, at which point you have a steady state.
The extra 12.5mg of caffeine also has the same halflife. The next day, it will have reduced to 0.78mg.
Plus the 12.5mg, and another 200 mg, adds up to 213.28mg. Another day, and the new 12.5mg will have reduced to 0.78mg, and the 0.78mg from the first day will have reduced to 0.05mg.
Your amount of caffeine will never increase towards infinity. Mathematically, it will increase towards (but never quite reaching) some certain value. That value depends on what the halflife time is and how much you are adding each time.
You can visualize it this way: What would happen if you started with 800mg of caffeine in your system, and add 200mg each day?
First day: 1000mg
Second day: The 1000mg has reduced to 62.5mg, + 200mg = 262.5mg
Third day: The 262.5mg has reduced to 16.4mg, +200mg = 216.4mg
As you can see, we are not ending up with more and more caffeine in the system.
(editing in progress. I used the wrong half life)
You’re talking about what’s known as an infinite series in math (assuming you drink coffee forever. IN this case it’s:
200 * (1 + 1/8 + 1/64 + 1/512 + ….)
Infinite series come in two flavors: convergent and nonconvergent. A convergent infinite series (which this one is) will not grow infinitely, but instead converge on a finite number.
In this case you’ll finish your coffee each morning with an amount of caffeine in your system that will get closer and closer to 228.571mg. So while caffeine accumulates, it doesn’t do so infinitely. In fact, you end up with not that much more than the first day.
Of course, this is an idealized case ignores the real world problems that the halflife probably depends on a lot of things (possibly even including how much caffeine is in your system), and the amount of caffeine in a cup is going to vary from day to day, etc.
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