## Monday, August 17, 2015

### Determining Recovery Times After Drinking Alcohol

One can use the formula for BAC as a function of time to estimate the recovery time after drinking a quantity of alcohol. As an example suppose an 86 kg man drinks 2 liters of 5.5% beer. Using the data below...

...we can compute a BAC curve as a function of a parameter x = λt for a more general result and compare the results with the limiting BAC value of 0.08 gm/100ml. The equation that we need to solve is derived as follows. γ is the BAC converted to mass using the individual's effective fluid volume, Vfl, divided by the mass of alcohol consumed, Am.

Mathcad can easily solve the transendental equation for x given γ as follows. The advantage of using x is the curve is the same no matter what the individual's λ value is.

We can use the same method to compute a table to for the peak BAC value for given body mass and quantity of drink. For the 5.5% beer the peak BAC table using pounds and fluid ounces is,

Solving the corresponding BAC curves for right intersection point gives the recovery parameter x values for the BACs above the BAC limit. One may have to use Newton's method for approximating the zeros of a set of equations for each pair of mass and drink quantity.

If λ = 0.33/hr, the value for nondrinkers, the corresponding number of hours that it will take for an individual's BAC to drop below BAC limit are as follows. The zeros in the tables are for BACs below the BAC limit so there are no "recovery times."

For comparison note that three 12 fl.oz. bottles of beer is 36 fl.oz. and a gallon is 128 fluid ounces.

Supplemental (Aug 18): The tables above were computed for men with a Widmark ρW of 0.7 liter/kg. Women can use the same tables if they use a body mass reduced by 6/7 = 0.86. A gender neutral table would replace the body mass with the effective fluid volume Vfl.

Supplemental (Aug 20): The fit to Schweisheimer's data was fairly good considering the simple model assumed but all the subjects consumed the same amount of alcohol and therefore we can't really say that Widmark's formula for the peak BAC was validated. The amount of data was also limited and as a result we didn't have accurate knowledge of true shape of the curve to be fit. Consequently, we shouldn't be surprised if our estimates of the recovery times are off somewhat.