Displaying all posts in the series entitled Mathematics.

Alcohol by Volume vs. Alcohol by Weight

— Arvada, Colorado UNITED STATES

Let's talk about math. If math scares you, don't worry: we're really going to talk about beer.

Beer (or any alcoholic drink for that matter) is mostly just water with some ethyl alcohol thrown in for good measure and trace amounts of other stuff that provides the flavor. Typically, about 5% of a beer is alcohol, so a 12 floz serving of beer will have 0.6 floz of alcohol. Now, I said that a beer is typically 5% alcohol: beer does come in many different strengths. For example, last night, at the Wynkoop Brewing Company, I had a beer that was 8.2% alcohol, and, because the content was so high, the bar gave me a “short pour” which means that they gave me a brandy snifter's worth of beer.

One common strength of beer that is seen especially in the retail area is 3.2% beer. In Colorado, where I live, beer can only be sold in a grocery store if its strength is less than 3.2%. If you're fine with drinking 3.2% beer, you can pick up your watered-down, mass-marketed, American pale lager at the same time you buy your milk and eggs. If you prefer wine, whisky or beer that's stronger than 3.2%, you need to make a special trip to the liquor store.

So, if we do the simple math on this and divide 3.2 by 5, 3.2% beer has only 64% percent of the amount of alcohol that a normal 5% beer has. That math would be fine except that it's completely wrong.

For some legal or historical oddity that I think has to do with taxation laws (but I am not sure), when a beer advertises itself as 3.2%, what it means is 3.2% by weight as in 3.2% of the total weight of the liquid is alcohol. Just about every other time you look at an alcoholic beverage label, the alcohol content is listed by volume… and they are most definitely not the same. Well, they are the same, but only in exactly two instances: 0% alcohol by weight is 0% by volume, and 100% by weight is also 100% by volume. However, between those two extremes, the relationship is not linear.

Most calculators that I come across on the Internet, however, do assume that the relationship is linear with the common assumption that the alcohol by weight (ABW) is 80% of the alcohol by volume (ABV). This may be close enough for the range of alcohol that covers most beers, but “close enough” only counts in horseshoes and handgrenades. We can do better.

So, let's think: how is alcohol by volume (or ABV) defined? Well, it's a ratio of the volume of alcohol divided by the total volume. Now, the total volume of an alcoholic beverage is alcohol and water… and some other stuff that provides the taste, but for our purposes, let's just ignore that other stuff—alcohol, water and nothing else. When we put it together, we get something like this:

$$ABV = \frac{VolumeOfAlcohol}{VolumeOfAlcohol + VolumeOfWater}$$

Okay, but we're dealing in weights—not volumes. Since a volume of something is equal to the weight of that something divided by its density, we can replace the volumes with weights, so let's consult Wikipedia (the most peer reviewed publication on the planet). If we look up the appropriate articles, we'd find that ethyl alcohol (the type used in alcoholic beverages) has a density of 0.78945 kilograms per liter at 20°C (68°F). At that same temperature, water has a density of 0.9982336 kilograms per liter. For this exercise, let's round it at 0.99823 kilograms per liter. If we replace the volumes in our equation, we end up with:

$$ABV = \frac{\frac{WeightOfAlcohol}{0.78945 \frac{\mathrm{kg}}{\mathrm{l}}}}{\frac{WeightOfAlcohol}{0.78945 \frac{\mathrm{kg}}{\mathrm{l}}} + \frac{WeightOfWater}{0.99823 \frac{\mathrm{kg}}{\mathrm{l}}}}$$

…and if we simplify that, do the dimensional analysis and eliminate the decimals, we get:

$$ABV = \frac{99823 \cdot WeightOfAlcohol}{99823 \cdot WeightOfAlcohol + 78945 \cdot WeightOfWater}$$

Our units cancel out in the dimensional analysis, which, since the ABV is a ratio, makes sense. Now, remember when I said that we're dealing with weights? I lied. What we actually know is the alcohol by weight (ABW), and we know the ABW is the ratio of the weight of the alcohol to the total weight. This is very similar to the equation that we started with for the volumes:

$$ABW = \frac{WeightOfAlcohol}{WeightOfAlcohol + WeightOfWater}$$

Well, we could try to solve this all as a system of equations. Let's solve our ABW equation for either the weight of the alcohol or the weight of the water. (It won't matter which one in the end, but let's pick weight of water.) When we go through the steps to solve for the weight of water, we get:

$$WeightOfWater = \frac{WeightOfAlcohol \cdot (1 - ABW)}{ABW}$$

Let's plug that into tho ABV equation and simplify. If you've been following along on your TI-89 from high school, you should get:

$$ABV = \frac{99823 \cdot ABW}{20878 \cdot ABW + 78945}$$

…and there's our formula. What's interesting is that when we plug in the weight of water from the ABW equation, the weight of the alcohol gets canceled out. If we would have plugged in the weight of alcohol instead, we get the same results.

If you need to convert the other way around (i.e., obtain the ABW from an ABV), you just need to solve the above formula for ABW:

$$ABW = \frac{-78945 \cdot ABV}{20878 \cdot ABV - 99823}$$

So, when we put 3.2% into the equation for the ABW, we come within a rounding error of 4.0%. Remember how we did the math and we said that 3.2% beer is 64% as strong as regular beer? Well, it turns out that, since 3.2% beer should really be marketed as 4.0% beer, it's really 80% as strong as regular beer. Keep that in mind before you consider chugging down another 3.2% bottle after you've already had a few: it may be more potent that you think.

Tags: Beer, Mathematics

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