Yesterday I got into a discussion about what adjustments needed to be made to accurately determine the alcohol content of a sour beer. I’d always assumed that the souring process didn’t have much effect on the standard method, using a formula that takes into account the OG of the wort and the FG of the beer, but I’d never spent any time looking into it.
When we calculate the amount of alcohol in a beer we are judging the relative density of the wort before- and after fermentation. This reduction in density (specific gravity) comes from a couple sources (including yeast biomass growth), but most of it is from the conversion of sugars in the wort to ethanol and carbon dioxide by fermentation. I’d always read that the CO2 escaping into the atmosphere was the main cause of the reduction in density during fermentation.
During a clean fermentation a single molecule of glucose is fermented to create two molecules of CO2 and two molecules of ethanol. So our formula is only valid if one molecule of CO2 produced always means the creation of one molecule of ethanol. Luckily alcoholic fermentation is the same regardless of which brewing yeast (Saccharomyces or Brettanomyces) is responsible. However, these aren’t the only microbes at work in a sour beer.
The production of lactic acid by homofermentative strains of lactic acid bacteria utilizes sugar to make lactic acid and essentially nothing else (no CO2 is produced). This could theoretically lead to a higher FG than otherwise as these sugars would not be available for alcoholic fermentation, but it wouldn’t disrupt the calculation of ABV because these microbes don’t generate carbon dioxide as a byproduct. So our original assumption is still valid.
The heterofermentative production of lactic acid produces 1 molecule of lactic acid, plus 1 molecule each of CO2 and ethanol from each molecule of glucose. As a result you’ll still get the same gravity drop for each unit of alcohol produced. The molecule of lactic acid is incidental, similar to homofermentative lactic acid production.
This spurred a question about what was really causing the gravity of the wort to drop as it is fermented: the loss of CO2, or the addition of ethanol? To get a lower density we either need to decrease the mass, or increase the volume (because density is mass/volume). As I’ve never finished with more beer than the amount of wort I started with, I assumed that the loss of weight to gaseous CO2 made sense.
It turns out that fermentation converts sugar into 51.1% ethanol and 48.9% CO2 by weight (a mol of ethanol weighs 46.07 g, a mol of CO2 weighs 44.01 g). A 20 L batch with an OG of 1.050 weighs 21 kg (water weighs 1 kg/L, this wort is by definition 5% denser than water). If fermentation generated 5% alcohol by volume, this would be 1 L of ethanol (which has a density of .789 kg/L). So .789 kg of ethanol. This means that we've fermented 1.544 kg of sugar, and created .755 kg of CO2 in the process. If the total volume remained constant, we'd have 20 L of beer with a weight of 21-.755 = 20.245 kg, divided by the weight of 20 L of water (20 kg) gives us a gravity of 1.0123.
Plugging those numbers (OG=1.050, FG=1.0123) into an alcohol calculator I get 4.95% ABV. Which is almost exactly what we'd expect given that 5% ABV was one of the conditions I specified. This indicates that most the gravity lost is in fact a result of CO2 leaving the beer.
However, it brings up an interesting question, if we are creating 1 L of alcohol, why don’t we see a volume increase as a result of fermentation? When you add rum to a Coke, it certainly seems like it would have the same effect on the volume as adding an equal amount of water. It turns out that when you mix equal parts water and ethanol the volume increases only 1.92 fold. So we’ll only get 92% of the increase from adding the ethanol, .92 L in this case (close enough that we don’t notice a difference mixing spirits with water-based drinks).
If we simply added the same 1 L of pure ethanol to our 20 L of 1.050 wort, we'd have a resulting density of ((.789*1)+(1.05*20))/20.92 = 1.042. So the addition of alcohol is not itself enough to account for much of the drop in gravity we see during fermentation.
In a fermentation we aren’t just adding alcohol, so we’d also need to consider the reduction in volume from the sugar that is fermented (1.544 kg in our case - the sum of the CO2 and ethanol created by our hypothetical fermentation). I couldn’t find similar numbers for maltose, so I’ll treat this sugar as if it were sucrose. When you dissolve sucrose in water, approximately 54% of the volume is added to the volume of the solution. Sucrose weighs about .869 kg/L, so our 1.544 kg of sucrose would take up 1.778 L dry, the loss of which from the wort would result in a .960 L decrease in its volume).
So the loss of volume attributed to the destruction of sugar (.96 L) during fermentation completely cancels out the volume gained (.92 L) from the resulting ethanol, which is why we don’t see a noticeable volume increase (or decrease) during fermentation!
When trying to determine the ABV of a sour beer specifically, there are four other factors you might consider.
1. Highly acidic beers are around 1% lactic acid. As the formation of lactic acid doesn’t directly lead to anything entering or leaving the fermentor, we know the mass of the beer will remain constant regardless of how much lactic acid is created. Any change in the beer’s density would purely be the result of a change in the total volume.
1 molecule of glucose goes to create 2 molecules of lactic acid in homofermentative lactic acid production. Glucose has a density of 1.54 g/ml, and a molar mass of 180.16 g/mol. Lactic acid has a density of 1.2 g/ml and a molar mass exactly half that of glucose, 90.08 g/mol (which makes sense). In a 20 L batch of beer, 1% lactic acid by volume, would be 200 mL of lactic acid. The equivalent of 240 g of lactic acid, or 2.66 mol. This would require the fermentation of half that number of glucose molecules, 1.33 mol, which is 155.59 mL.
If we assume glucose behaves similarly to sucrose when dissolved in water, the loss of wort volume attributed to 155.59 mL of glucose converted to lactic acid would be 84.02 mL. I can’t find a number for what happens to volume when you mix lactic acid and water, but even if we assume the entire volume of lactic acid goes to increasing the beer’s volume, we’d be increasing the volume of the beer by 115.98 ml, just .58%. This is the equivalent of causing the 5% ABV calculated to be .03% too high (the larger volume with the same mass would indicate a more substantial reduction in gravity than would have been actually caused by alcoholic fermentation).
2. Acetic acid production by either Acetobacter or Brettanomyces consumes a molecule of ethanol along with a molecule of oxygen. This reduces the alcohol content, but also raises the mass slightly as the oxygen comes from the atmosphere. It seems like without the carbon atom of the CO2 released during fermentation the gravity would rise slightly less than what is lost during fermentation, causing an infinitesimally small over-estimate of the ABV. Acetic acid is legally limited to.15%, so it shouldn’t play a major role anyway (even if flavor wasn't a consideration).
3. Ester formation by Brettanomyces could have a small impact on alcohol content because esters are formed by combining an alcohol with an acid (e.g., ethyl lactate is a combo of ethanol and lactic acid). However, all of these are present in parts per million at the most, making their presence too low to substantially impact ABV calculations.
4. While not specific to sour beers, evaporation from a barrel reduces the amount of both water and ethanol a beer contains. As ethanol has a lower boiling point than water, it would tend to evaporate at a faster rate. The humidity in the room apparently plays a role in the relative evaporation rates as well (higher ambient humidity reduces the evaporation rate of water). At 70% humidity, the loss of alcohol and water is supposedly about equal.
In relation to these various areas, when compared to the errors in calibration and measurement I don't think it is necessary to make special considerations for the ABV calculations of mixed fermentation sour beers. This has been one of those things I spend hours thinking about only to come to the conclusion that I was better off ignoring the whole thing.
That said, this isn't exactly my area of specialty, if anyone sees any issues with my assumptions, data, or math, please let me know! I'm sure I've oversimplified some things, like the sugars, or how a water molecule is added when some polysaccharides are split, but I think I'm close enough to prove the point.
I have a complicated relationship with fruited sour beers. When I’m buying I shy away from them because the fruit tends to conceal the interesting flavors from the fermentation and barrel. Some of them are so delicious though, that I can’t help myself. When I drank Jester King’s Atrial Rubicite earlier this year it was one of those “wow” beers for me, saturated with concentrated raspberries (from hundreds of pounds of fruit in each barrel), without being sticky sweet like New Glarus Raspberry Tart. 
Intergalactic Raspberry Sour Brown
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