# Let's learn about coffee and alcohol!

## Linear vs Exponential decay

I came across a write up on metabolism of alcohol. So today’s post isn’t about something you should buy to make your life better, instead it is about something that you should know[1]. Learning things takes time and time is money so maybe it is really the same. Regardless, the point today is that when you are throwing things like coffee and alcohol into your face, you should understand how long they will affect you.

First, read the block quote below and feel smarter for knowing a new thing about yourself:

Why Alcohol Has a Steady State Metabolism Rather Than a Half Life

When a drug like valium is broken down by the human body the resultant metabolites are harmless. It is for this reason that drugs like valium are broken down as quickly as the body can process them–and hence they have a half life. The half life of valium is 35 hours on the average. This means that if you take a 10 mg dose of valium, then 35 hours later half of it will have been metabolized and only 5 mg will remain. In another 35 hours half of this will be metabolized and only 2.5 mg will remain and so on. When we plot the metabolism of valium on a graph we get an exponential curve–in other words–drugs which have a half life have an exponential rate of decay. Chemists refer to this as a First Order Reaction.

Alcohol, on the other hand, shows a steady state metabolism not an exponential metabolism. The body of the average human metabolizes around 13 ml of alcohol per hour regardless. When we plot the metabolism of alcohol on a graph we get a straight line–in other words the rate of decay of alcohol is linear. Chemists refer to this as a Zero Order Reaction. The reason why alcohol has a steady state metabolism rather than a half-life metabolism is because the primary decay product of alcohol metabolism–acetaldehyde–is poisonous. The body must eliminate the acetaldehyde produced by the breakdown of alcohol before any more alcohol can be processed in order to avoid acetaldehyde poisoning. This slows down the rate of alcohol metabolism to a Zero Order Reaction rather than a First Order Reaction. –Source[2]

Wow, cool, what the hell does all that mean. Well, luckily, if you don’t feel like thinking too hard about it [3] I’ll explain it in terms of caffeine and alcohol[4]. Let’s assume it is 8pm and you are trying to decide between taking 3 shots of whiskey or 3 shots of espresso [5]. You have to wake up for work tomorrow and want to know how much of one or the other will still be in your body when you go to sleep at 11pm [6].

Now that we know that alcohol is a Zero Order Reaction it is possible to approximate the amount of alcohol in your body after drinking: $$A_t = -kt + A_0$$ I know. It is an equation. Either you see a lot of equations and think “thanks idiot, that’s obvious” or you don’t and you are thinking “thanks idiot, I’m done reading this site forever”. But stay with me for a look at each of the terms[7]:

$A_0$ is the initial amount of alcohol. A shot is 44 ml (1.5 ounces) and at 40% alcohol that is 52.8 ml of alcohol[8]

$-kt$ is your body working to remove alcohol at a constant rate. The rate quoted, 13ml per hour, varies but sounds pretty good here. The time will be 3 hours since we are looking at 8 to 11pm

$A_t$ is the answer we are looking for. Nice.

A quick look at the graph above says that at t = 3 you will have 13.8 ml of alcohol still in you at 11pm. So that is like being sober and taking a shot just before bed [9]. Tasty.

## Alcohol (ml) vs time (h)

What about the espresso? This time it is a First Order Reaction so the equation is: $$A_t = A_0 e{-kt}$$

$A_0$ is the initial amount of caffeine. An average espresso shot has 64 mg of caffeine [10], so that means three shots would give 192 mg

$-kt$ is again a that your body removes caffeine but now it depends on the amount of caffeine still in your system. We can use a quoted median half-life of 5.7 hours [11] and some basic math that you don’t care about to find k

$A_t$ is still the answer. Still nice.

Back to the graph at t = 3 and your body is only down to 133 mg of caffeine aka an espresso double shot just as you crawl into bed. Maybe you should think about some of the life decisions you are making.

## Caffeine (mg) vs time (h)

A few caveats. I ignored absorption of the alcohol/caffeine. It would probably take 30–60 minutes for everything to hit your blood stream and THEN your body would start removing it. Adjust everything above depending on how much you care. It is also important to note all the half lives and absorption rates vary greatly between individuals. Caffeine half life had a mean of 5.7 hours in the study I referenced but it varied from 1.5 to 9.5 hours. It sure would be cool if someone made an app so that you could test how you specifically respond to things like that. I wonder who would take the time to do something like that… oh… me… that is what I’m working on. Cool.

1. Does that make me unpatriotic? Sorry capitalism, can we still be buddies?  ↩

3. Remember, we had a deal, I get to do all the thinking, you get to… … … tolerate all the yelling?  ↩

4. If you need to know more about valium then maybe talk to your doctor, dealer, or probably some shady site somewhere on the internet with those weird people who ask really specific medical questions in the comments  ↩

5. Woah, your life sounds a lot more exciting than mine  ↩

6. Nevermind, false alarm, you are as boring as the rest of us  ↩

7. Notice that I don’t actually give any reason to stick with me despite the equation… but seriously be a gem and keep reading, would it help if I say please?  ↩

8. Which is a lot for you. You are drunk now. On a work night. What would your mom think?  ↩

9. Fine from a hangover perspective but pretty weird from a social perspective  ↩

10. Why is alcohol always discussed as a volume while caffeine is discussed by mass? No clue. If you want to convert then you can assume a density for alcohol of about 0.8 g/ml  ↩

11. The source for that value is actually pretty great. But you won’t read it, will you :(  ↩