Jamison Koehler put up a post on the fraction of your breath that a breathalyzer uses to estimate your blood alcohol ratio. He’s had expert training in DUI defense and therefore DUI-related technology, whereas I’ve just had some basic math and science education. But it’s an interesting subject, and I thought I’d try to explain what I think is happening from a science-ish point of view. DUI experts are invited to explain where I go wrong.

Jamison’s post begins it with a charming anecdote about his daughter (which makes me imagine the whole post as a rehearsal for a closing argument in a DUI trial), and then he gets into some of the math behind breathalyzer operation:

To be more precise: The amount of ethanol present in a DUI breath sample is measured in terms of grams per 210 liters.

…

The Intoximeter EC/IR II – the breath test machine used in both DC and most jurisdictions in Virginia — requires a breath sample of at least 1500 cubic centimeters (or 1.5 liters) before it can provide a result. Of this amount, McGarry says, the machine measures only 2 cubic centimeters (or 0.002 liters).

You need to multiply 0.002 by 105,000 to equal 210. This means that any error in the measurement of ethanol in the 2 cubic centimeter breath sample will be magnified 105,000 times.

That’s true if we’re talking about the absolute amount of alcohol in the sample. For example, if the 2 cubic centimeter sample chamber contains 1 milligram (1/1000 of a gram) of alcohol (an unrealistically large amount that makes the math simple), then the full 210 liters would contain 105 grams of alcohol. Now if the breathalyzer measurement was off by 10% on the high side, then it would measure 1.1 milligrams of alcohol which would appear to mean the 210 liters of breath would have 115.5 grams of alcohol. So a measurement error of 0.1 milligrams in the sample results in a reported error of 10.5 grams.

I’m puzzled by that line of reasoning, however, because the 210 liters is not the size of a breath sample, and it’s not the volume of a pair of lungs. I’m pretty sure it’s not anything real. It’s just an artifact of the way the calculations are set up.

The legal limit for Blood Alcohol Content (BAC) while driving is 0.08 grams of alcohol per 100 milliliters of blood. Since alcohol is measured by weight and blood by volume, you can’t technically express this as a percentage because the units aren’t the same. Nevertheless, we implicitly assume that blood has the same density as water (it’s close), so a milliliter weighs one gram. That works out to o.o8 grams of alcohol/100 grams of blood, and we can cancel out the units to get our familiar 0.08% that we’re always hearing about.

Breathalyzers, as the name suggests, do not measure alcohol in the blood directly. Instead, they measure alcohol in the breath, and use that to produce an estimate of alcohol in the blood. For reasons too complicated for me to figure out breathalyzers perform their calculations on the assumption (ripe for attack by DUI defense lawyers) that the ratio of alcohol to blood volume is 2100 times the ratio of alcohol to breath volume. So, if we take the ratio of 0.08 grams of alcohol/100 milliliters of blood and divide by 2100 to convert to breath concentration, we get 0.0000380952 grams of alcohol/100 milliliters of breath.

That’s a really annoying number to work with, so instead of dividing the 0.08 grams of alcohol by 2100, let’s multiply the 100 milliliters of breath by 2100 to get 0.08 grams of alcohol/210000 milliliters of breath. Now divide the bottom number by 1000 to convert to liters, and we get 0.08 grams/210 liters. That’s a handy way to express it because there’s our familiar 0.08 legal limit again. And that’s where the 210 liters shows up in the calculations.

But that’s just because of the convention we’ve chosen to express the ratio. It would have been just as accurate express it as 0.000380952 grams of alcohol/liter of breath. Or we could go in the opposite direction and measure the alcohol in kilograms, which works out to 0.08 kilograms of alcohol/210000 liters of breath. If the absolute measurement error grows by 105,000 when converting to 210 liters, then converting to 210000 liters implies a multiplier of 105 *million*, which sounds even worse.

But it’s not, because however you choose to describe it, you’re still just measuring a ratio between the amounts of two substances, and therefore an error of 10% is an error of 10% regardless of the units you choose. Multiplying it out to 210 liters doesn’t really mean anything. In other words, I think Jamison is engaging in the time-honored defense strategy known as *muddying the waters*.

McGarry also uses the analogy of a railroad car full of grain. Can you take a small sample of that grain, test it and then be confident you know the contents of that railroad car?

Now this is a much more interesting question. Let me see if I can figure out an answer…

Let’s assume for the sake of simplicity that the grain car contains 1 billion grains. Jamison says the breathalyzer uses a sample that is 1/105,000 of the theoretical 210 liters. Rounding a bit, we can calculate that 1/100000 of a billion grains is a sample of 10,000 grains. If we’re trying to estimate, say, how much of the grain is too small (or rotting, deformed, whatever), is testing a sample of 10000 grains good enough to estimate how much of the grain in the car is defective?

My answer is the favorite answer of lawyers everywhere: It depends.

In particular, it depends on how you select the sample. The ideal would be to choose the sample completely at random: Start by numbering each of the grains from 1 to 1,000,000,000. Then use a random number generator to generate a set of 10,000 random number between 1 and 1,000,000,000 inclusive. Now for each random number, find the grain corresponding to that number and inspect it. (A simpler way to do that would be to let the grain out of the car through a tiny hole and count the grains as they fall out. Whenever your count is equal to one of the 10,000 random numbers on your list, inspect that grain.) Tally which ones are acceptable and which ones are defective.

When you’re done, calculate the fraction that’s defective in the sample, and use it as the estimate of the defective fraction in the entire grain car. With a random sample of 10,000 grains, you can have very high confidence in your result. The chance of a significant difference between the sample defective fraction and the defective fraction for the entire grain car is vanishingly small. The tiny sample will tell you a lot.

That kind of random sampling would be a lot of work, so you may be tempted to try something simpler. Perhaps you could take a scoop that holds 10,000 grains and dip it into the grain in the car to take a sample. That’s a lot easier, but it’s no longer a truly random sample. It’s a *mechanical* sample, which might not be as random as we’d like: Perhaps all the small grains have settled to the bottom, so a scoop off the top will be biased in favor of large grains, and it will therefore cause you to underestimate the number of under-sized grains; or maybe the grain car was filled by dumping in bushels of grain, and one of the bushels was filled with small grains. That batch of small grain will probably still be clumped together. If your sample misses it, you’ll underestimate the fraction of defective grain in the car, but if you happen to dig your scoop right into it, you’ll overestimate the fraction of defective grain in the car.

You can alleviate the problem somewhat by scooping smaller samples from several different places in the car and combining them into one 10,000-grain sample, but it’s still a mechanical sample rather than a truly random one, which means there’s still a pretty good chance of significant error.

So is there is any way you can use a scoop and still get the benefits of random sampling? Maybe. You could try stirring the grain thoroughly before taking a scoop. You’d have to be very careful and thorough about this, so that every grain from anywhere in the car before stirring has exactly the same chance as any other grain of ending up in the scoop. The scoop still takes a mechanical sample, but it’s a sample of grains that have themselves have been thoroughly randomized, which is just as good.

But randomizing an entire car full of grain is a difficult task. How do you make sure that grains that start in the bottom have an equal chance at ending up in the sample as grains that start on top? How do you make sure grains don’t get stuck in the corners? It requires a carefully designed stirring mechanism. Laboratories often buy expensive stirring and shaking equipment to get a sufficient randomization, and even those are usually intended for liquids. (Hmm…maybe we could fill the grain car with water, stir the floating mass of grain thoroughly, and then drain the water out. That would make for pretty good randomization, but wetting the grain brings its own problems.)

The breath analysis situation is a little simpler, however, because nature lends a hand in the form of turbulence and Brownian motion (the jiggling motion of gas molecules as they bounce around against each other and the sides of their container), which are about as random as any natural processes can get. Add a little alcohol vapor to a container of air, and in just a little while it will be diffused evenly throughout the entire container — faster if it is stirred or agitated to cause turbulence — so that even a tiny sample will contain enough randomized molecules to be an accurate representation of the alcohol concentration of the entire container.

However, the diffusion of the alcohol vapor through the air isn’t instantaneous. It takes a bit of time, and happens quickest near the highest concentration. In addition, if fresh air is circulating, it continues to dilute the alcohol vapor. This is why when you open a bottle of liquor, you can easily smell it at the bottle opening — the air inside is already saturated with vapors — but the farther away you get, the weaker the smell, as the alcohol is diffusing outward and mixing with fresh air and being carried away.

(In a perfectly sealed room with nowhere for the alcohol to go, the entire room would eventually become saturated to the same extent, and the smell wouldn’t depend on how close you were to the bottle. Also, I am neglecting the effects of gravity: If you’re mixing gases that have significantly different densities and there isn’t much turbulence, the heavier gases will eventually sink to the bottom of the container or room instead of mixing evenly.)

Now let’s think about what happens in your body, in your lungs, when there’s alcohol in your blood. As the blood flows through your lungs, alcohol diffuses across thin membranes in the alveolar sacs deep within the branching airways of your lungs, and it is therefore in these sacs that the alcohol concentration is the highest.

When you take a breath of fresh air, your lungs fill with alcohol-free air. This begins to mix with the alcohol-laden alveolar air due to Brownian motion and turbulence within your lungs. The longer you hold the air in your lungs, the more thorough the mixing. As you exhale, you first push out the air in your mouth and throat. This air was the farthest from the alveolar sacs, and is therefore has the lowest concentration. As you continue to exhale, you are forcing out air that was closer and closer to the alveolar sacs, and therefor higher in alcohol content. This is why police administering a breathalyzer test want you to give a long, slow exhale. It gives the alcohol more time to saturate your breath, and it brings up air from deep inside your lungs where the alcohol concentration is highest.

I’m almost done, but I should probably mention one of the possible sources of error in a breathalyzer test, which is that alcohol on your breath can come from other sources besides your blood.

Remember that in a person near the legal limit for DUI, the fraction of alcohol in the blood is 0.08%, and it is the diffusion of that tiny amount of alcohol into the lungs that is ultimately measured by the breathalyzer. An alcoholic beverage like beer, on the other hand, typically has about 5% alcohol. That’s not very much as alcoholic beverages go, but it’s more than 60 times the alcohol concentration in the blood at the legal limit.

So if there’s any beer in your mouth — because you just took a drink (or just threw up) — the alcohol from the beer will diffuse into your breath the same way the alcohol from blood diffused into the air in your lungs. There’s a lot more air in your lungs than in your mouth, and the convoluted surface area of your lungs is hundreds of square feet, meaning that diffusion happens much faster than with beer evaporating off the interior of your mouth. Nevertheless, the equilibrium point for air directly exposed to beer is still 60 times the legal limit, so any beer in your mouth will push the alcohol content of your breath in that direction. And even a little of that could put you over the limit.

Then there’s the beer in your stomach. Compared to your mouth, that’s a sealed system, with little fresh air getting in, which means your stomach gases should get pretty close to their saturation level of 60-times the legal limit. One burp before or during the breath test, and you’re in a lot of trouble.

It’s even worse if you drink hard liquor, because that could have an alcohol concentration hundreds of times the legal blood limit, which means an even greater chance of corrupting the DUI results.

I suppose you could always try arguing that the measured BAC is so high that it couldn’t possibly be right. But it would have to be pretty damned high.