As a non-drinker, I have never had much interest in alcohol. That said, alcohol has been a big part of my life. My mother earned her living working in bars, starting as a waitress and eventually managing them. Even in retirement, she works part-time selling pull tabs in bars for local sports groups. I have also been the designated driver for literally hundreds of social occasions. At work, I have had numerous co-workers who have derived much pleasure brewing beer and wine. It is always interesting hearing them discuss their various brews.
But alcohol must be used in moderation. To help guide people in their consumption, there are numerous charts and tables that tell people how much they can drink and remain under the legal limits. While prowling around the Wolfram Alpha web site, I noticed that they have a Blood Alcohol Content (BAC) calculator. I started looking at the output and I saw that there was some interesting math going on there. Let’s dig in …
The Wikipedia provides a very complete definition of BAC and I will start there with a slight paraphrasing of their definition.
BAC is the concentration of alcohol in a person’s blood. It is most commonly used as a metric of intoxication for legal or medical purposes. It is usually expressed as a fractional percentage in terms of volume of alcohol per liter of blood in the body.
So BAC is unitless quantity because it is the ratio of two volumes. This definition of BAC is not universal. For example, California actually uses grams (gm) of alcohol per deciliter (100 mL = dL) of blood. For this post, we will use the more common ratio of volumes, which is expressed mathematically in Equation 1.
Key Analysis Assumptions
The analysis assumptions in the creation of the BAC tables vary because a person’s response to alcohol varies by person. Here are the assumptions that I could identify.
- The alcohol evenly spreads through all the water of the body, not just the blood.
- The amount of water in the body is proportional to body weight
This assumption is highly variable between individuals. Quoting from the Wikipedia, “In a newborn infant, this may be as high as 75 percent of the body weight, but it progressively decreases from birth to old age, most of the decrease occurring during the first 10 years of life. Gender also affects the percentage of water for an individual. This is because women, on average, have a higher body fat percentage. Higher body fat percentage correlates with lower body water percentages. As such, obesity decreases the percentage of water in the body, sometimes to as low as 45 percent.”
- Every drink has the same amount of alcohol.
In fact, the amount of alcohol varies by drink. To simplify the discussion, most tables assume a drink contains about 0.5 oz of alcohol by weight. Wolfram Alpha assumes 0.533 gm per drink. Since ethyl alcohol (ethanol) has a density of 0.789 g/cm3, 0.533 oz of alcohol is equivalent to 0.648 fluid ounces. This amount of alcohol corresponds roughly to the following common drinks:
- 12 fluid ounces of beer
- 1.25 fluid ounces of 100 proof liquor
- 4 fluid ounces of 25 proof table wine
- The body eliminates alcohol at a fixed rate.
Quoting the Wikipedia, “The rate of elimination in the average person is commonly estimated at .015 to .020 gm/dL per hour, although again this can vary from person to person and in a given person from one moment to another.” To convert the units from gm/(dL hr) to the BAC’s unitless over volume/volume, we need to apply the density of ethanol as follows: .
Everything I have been able to find on the web uses the model shown in Equation 2 with different parameters.
- w is the weight in pounds
- n is the number of drinks
- t is the time since consumption (hours)
- α is the percentage of water in a body (weight of water/body weight = ~75%)
- γ is volume of ethanol in a drink (0.648 fluid ounces)
- β is the rate of elimination (~0.021% per hour)
- ρH20 is the density of water (1 gm/cm3 = 0.065198 lb/fluid ounce)
If we substitute the values shown into Equation 2 we get Equation 3.
I have looked at a number of web sites on BAC levels and they all make different assumptions about their definition of a drink, body water percentages, and elimination rate. The variation in the assumptions all reflect that fact that these characteristics vary by individual. Other important factors, like how long you were drinking and whether you were consuming food while drinking, are completely ignored. Looks to me like it is difficult to know how many drinks you can have and still drive. Sounds like not drinking and driving is still the best way to go.