The Chemical Kinetics of Sobering Up
Throughout societies, cultures and the age’s alcohol has been consumed and enjoyed. My first taste of Yorkshire bitter was a sip from my father’s glass at the age of four, and as I screwed up my face, he said to me,
Believe me son, one day you’ll love it!
As good as beer, wine and spirits are, alcohol is often abused and people get intoxicated. A quick check on the Internet shows a plethora of pills and concoctions that claim to quickly sober you up. Only time will work.
Of course the main reason people want to sober up faster is so they can drive home. In many countries, the allowed Blood Alcohol Content (BAC) is low (half a beer) and the fines high so that a designated driver is chosen who does not imbibe.
How long does it take to sober up? It is important that you know. Let us suppose that by “sober” I mean a BAC of half a beer. Moreover, let us assume that if you drink one beer, then the time it takes to reduce your BAC to half a beer is 30 minutes. That is the half life to metabolize one beer to half a beer is 30 minutes.
The metabolism of alcohol in the liver is a REDOX reaction. The alcohol is oxidized through the reduction of nicotinamide adenine dinucleotide (NAD+ to NADH). This is catalyzed by the enzyme, alcohol dehydrogenase, and the basic simplified formula is
CH3CH2OH + NAD+ —> CH3CH=O + NADH + H+
The bye-products of the oxidation, the aldehydes, ketones and acids, are what give a hangover. Hangovers are best treated by drinking water rather than taking aspirin, but only time can eliminate the alcohol from the blood which accumulates the more one drinks. This is because the liver can only oxidize about an ounce (28 grams) of ethanol per hour.
The study of the rates of reactions is called Chemical Kinetics and a big part of Physical Chemistry. The metabolism of alcohol, expressed in the above equation, follows Zeroth Order Chemical Kinetic. The most common kinetics is First Order. In that case, like for carbon-14 dating and many other exponential processes, the half life is independent of how much you start with. For example, in radioactive decay if you have a half life of, say, 10 hours, then if you start with 1 gram or 10,000 grams, after 10 hours there remain respectively 0.5 grams and 5,000 grams.
This does not work for Zeroth Order rates. In that case, the half life is given by,
Here [Alcohol]o is the initial concentration and the unit I will use is “number of beers consumed”. The quantity k is the rate constant, and it can easily be found because we know the half life is 30 minutes for one beer, hence
Now we can answer the question. If it takes 30 minutes to reduce one beer to half a beer, then for two beers the half life is double,
Ok, so what is the problem? The problem is, if you start with 2 beers then after 60 minutes you are down to 1 beer level, and you would fail the breathalyzer test. We need to get down to half a beer, and that takes another 30 minutes.
So the conclusion here is that it takes 30 minutes for the BAC from one beer to drop to the legal limit of half a beer, but it takes three times longer, 90 minutes, to reduce two beers to the legal limit. (Of course I am tacitly assuming that the alcohol is consumed in a short time, and this is often the case with shooters and shots.)
The above plot is from my tutorials on General Chemistry. The initial concentration here is an arbitrary amount of 40, and you can easily see that the constant metabolism gives half lives that get shorter and shorter as the initial concentration drops.
So what about four beers (or four shots)? The time to reduce 4 beers to 2, using the above formula, is two hours. The time to go from 2 beers to 1 is one hour, and the time to go from 1 beer to the legal limit is 30 minutes. It therefore takes a whooping 3.5 hours to reach the legal limit: 4 beers take seven times longer to metabolize than 1 beer.
A quick rule of thumb is to use the fact that the liver metabolizes about 1 ounce per hour. So count the number of drinks and that gives you the number of hours to eliminate the alcohol completely. On the other hand, if you have only one drink an hour, you should be ok to drive 30 minutes after your last drink.