The Widmark formula explained

The formula

Erik Widmark worked out how alcohol distributes in the body and how fast it leaves. His result, still used in forensic medicine, fits on one line:

BAC = A / (r × W) − β × t

• A — grams of pure alcohol drunk.
• W — body mass in kilograms.
• r — the Widmark factor: the share of the body that alcohol spreads into as if it were water.
• β — the elimination rate, how many per mille the liver clears per hour.
• t — hours since the first drink.

The result is in per mille (grams of alcohol per litre of blood, which is numerically the same as g/l). Divide by ten for US % BAC.

Step 1: grams of alcohol

The label tells you the volume and the strength. Multiply them, then multiply by the density of ethanol, 0.789 g/ml at 20 °C. A 0.5 l beer at 5 % holds 500 × 0.05 × 0.789 = 19.7 g. A 125 ml glass of 12 % wine holds 11.8 g. A 4 cl shot of 40 % spirit holds 12.6 g. The standard drink sizes page lists the common glasses in grams.

Using grams avoids the confusion of "drinks" and "units", which mean 14 g in the US, 8 g in the UK, 10 g in much of Europe and 20 g in Japan.

Step 2: the distribution factor

Alcohol dissolves in water, not in fat, so it spreads through the watery part of the body. If the whole body were water the factor would be 1. It is less, because of fat, bone and other tissue: on average 0.68 for men and 0.55 for women. Those are the figures this site uses unless you ask for more precision.

Multiply the factor by body weight and you have, in effect, the litres of "water" the alcohol dilutes into, scaled to blood. An 80 kg man: 0.68 × 80 = 54.4. A 60 kg woman: 0.55 × 60 = 33.0.

Step 3: the peak

Divide grams by that figure. Two 0.5 l beers (39.5 g) for the 80 kg man: about 0.73 ‰. Three 125 ml glasses of wine for the 60 kg woman: about 1.08 ‰. This is the theoretical reading if all the alcohol arrived at once and nothing had been cleared.

Step 4: elimination

The liver clears alcohol at a nearly constant rate. The commonly used average is 0.15 ‰ per hour; real people range from roughly 0.10 to 0.20. Subtract it for every hour since the first drink. The 80 kg man, three hours after starting: about 0.28 ‰. Clearing completely takes about 4:50–7:15 hours and minutes from the first drink.

Counting from the first drink is a simplification: alcohol takes time to be absorbed, and nothing is cleared until it arrives. For a session spread over several hours the simplification is reasonable. For a few drinks downed quickly on an empty stomach it slightly flatters the early part of the curve.

The Watson refinement

An average factor ignores the difference between a lean 80 kg athlete and an 80 kg office worker. In 1980 Watson and colleagues published equations that estimate total body water from sex, age, height and weight. Divide that by the water content of blood, about 0.8, and by body weight, and you get a factor for one particular body. Turn on height and age in the calculator to use it.

Watson narrows the error, it does not remove it. The equations themselves carry a standard deviation of several litres of body water. Published versions also differ in the third decimal of the male age coefficient — a difference far too small to change any figure you see, but a reminder that this is a model fitted to a population.

A worked example from start to finish

Put the steps together for one evening. A 75 kg man drinks two 0.5 l beers at 5 % and a 4 cl shot of 40 % spirit, starting at eight. The grams are 19.7 g twice plus 12.6 g, about 52.1 g in total. His factor times his weight is 0.68 × 75 = 51.0, so the peak is roughly 1.02 ‰. At eleven, three hours in, subtract 0.45 and he is near 0.57 ‰ — still over the 0.5 ‰ limits of most of Europe and, at that moment, far from zero. Every line of that arithmetic is what the calculator shows under its result when you open the working.

What Widmark cannot tell you

• How fast your own liver works today. Tiredness, illness, medication and drinking habits all change it.
• How much your food delayed absorption, and so whether you are still rising.
• What a breath tester will show, because each country's law sets its own breath figure.
• Whether you are fit to drive. Impairment begins well below every legal limit.

Forensic experts use Widmark to reconstruct a reading backwards from a blood test, with ranges rather than single numbers, and they argue about those ranges in court. A calculator gives you a single number because that is easier to read. Treat it as the middle of a range that may run 30 % either way.

Use the formula to understand how much you drank and how long it takes to clear — never to decide that you can drive. The only safe reading is 0.00.

Try it with your own numbers

Summary

Grams of alcohol divided by body weight times a distribution factor gives the peak; subtract about 0.15 ‰ per hour for elimination. That is the whole of Widmark. Every number on this site comes from that line, run through the same tested code, with the working shown under each result.