Handy Homebrew Equation

These days, most homebrewers rely on recipe formulation software to help them calculate the relevant numbers (expected OG, IBU, SRM, etc.) for most brewing situations. There is, however, a simple equation that frequently comes in handy that may or may not be included with your brewing software. And in either case, it’s so simple and handy that you should just memorize it. 

The equation I’m talking about is the one that allows you to calculate changes in concentration due to changes in volume and vice versa. There are three common uses for this in homebrewing, calculating the post-boil OG from pre-boil specific gravity and volume, calculating alcohol content (% ABV), IBUs, and FG when diluting a beer (as in high-gravity brewing), and calculating alcohol levels when fortifying a beer with distilled spirits.

The equation is simple — C₁V₁=C₂V₂, where:
C₁ = the initial concentration of the substance in question

V₁ = the initial volume of the solution

C₂ = the adjusted concentration of the substance in question

V₂ = the adjusted volume of the solution

Original Gravity (OG) From Pre-Boil Wort Volume, Gravity

Let’s say you were planning to brew 5.0 gallons (19 L) of beer with an original gravity (OG) of 14 °Plato (SG 1.056).You collect 6.5 gallons (25 L) of wort with a gravity of 11 ° Plato (SF 1.044) — what will your specific gravity be when you boil this down to 5.0 gallons (19 L)?

Your pre-boil numbers will give you C₁ and V₁. Use °Plato as the unit for concentration, or the decimal part of the specific gravity. (For example, 1.044 would be 44.) Use any unit (gallons, liters, quarts, etc) that is convenient for volume.

You know your target post-boil volume, so that will be V₂. This leaves you with 6.5 gallons * 11 °Plato = 5.0 gallons * X °Plato. Divide both sides of the equation by 5.0 gallons to solve for X. In this case, it’s 14.3 °Plato (roughly SG 1.057). Notice that if you use the decimal of specific gravity for C, you have 6.5 * 44 = 5.0 * X, you get X = 57 — in other words, the same answer.

For this purpose, you can also remember the equation in rearranged form as C₂ = (C₁ * V₁)/V₂.

One assumption you make when using this equation is that everything soluble and contributing to the specific gravity of the pre-boil wort will remain soluble and contributing to the specific gravity of the post-boil wort. And, we know this isn’t true. Some soluble proteins will coagulate during the boil, forming the hot break. Although they were contributing to the “thickness” of the pre-boil wort, they won’t be in the post-boil wort. Therefore, your estimation of your post-boil OG may be slightly higher than it turns out to be. In practice, however, the difference is always small because the overwhelming majority of the pre-boil specific gravity is due to dissolved sugar, not protein.

High-Gravity Brewing

High-gravity brewing is a process in which a brewer brews a beer stronger than his target beer, then dilutes it post-fermentation to it’s intended strength. In commercial breweries, this lets brewers produce more beer than they have fermenter space for (within reason). (Here’s how to do this in your homebrewery.)

Let’s say you have 5.0 gallons (19 L) of a cream ale you’ve brewed to have 6% ABV, 25 IBUs and an FG of 3 °Plato (SG 1.012). You plan to dilute it to 6.0 gallons (23 L) — what will the ABV, IBUs, and FG be?

For the alcohol content, you’ll have 6% ABV * 5 gallons = X% ABV * 6 gallons. Solving for X yields 5, or 5% ABV. Likewise 25 IBU * 5 gallons = X IBU * 6 gallons results in x = 20.8, or roughly 21 IBUs. And, an FG of 3 °Plato become an FG of 2.5 °Plato (roughly FG 1.010).

In this case, everything should scale exactly as the equation estimates.

When planning a high-gravity brew, you can reverse the equation to figure out the target OG, IBUs, and ABV of your strong beer from the specifications of your target beer.

Fortifying Beer

In some cases, you may want to fortify a beer by adding distilled liquor to it. My winter warmer is one example of that. You can use the “CV” equation to calculate the boost in alcohol percentage when doing this.

Let’s say you have 5.0 gallons (19 L) of a beer that’s 7.0% ABV, and you have a 1 L bottle of liquor that’s 80 proof (or 40% ABV). Since we have mixed units, let’s switch to metric to make it easier. If you combine the two, you’d have 20 L of fortified beer. To figure out the percentage alcohol, you’d start with 40% ABV * 1 L = X% ABV * 20 L. Notice that we’re ignoring the alcohol in the beer for now. Solving for X, we get 2. This means the alcohol content due to the liquor alone is 2%. Now, you have two options, you can simply add 2.0% to the 7% alcohol the of the beer to get 9% ABV. Or you can notice that the 7% alcohol from the beer is going to be slightly diluted by the volume of the liquor. In this case 7% ABV * 19 L = X% ABV * 20 L. Solving X yields 6.65. So the alcohol coming strictly from the beer is 6.65% and the overall alcohol content is 2% higher than this, or 8.65%.

If you are a winemaker, you may know that there is another way to do this calculation by using the  Pearson Square. The Pearson Square is very straightforward, more so than using the CV equation twice, as above. Either way yields the same result, however, and if you’re already familiar with the “CV” equation, it may be easier to use it.

Conclusion

Checking the specific gravity of your pre-boil wort, and calculating the projected post-boil specific gravity can give you a heads up as to whether you are likely to hit your target or not. It only takes a few seconds to do the calculation, and — since the boil has yet to begin — you have options if an adjustment needs to be made.

If you project that your OG is going to be significantly higher than you expected, you have a couple options. The easiest is simply to accept higher gravity and brew a stronger beer. Another option is shorten the boil and quit boiling when you hit your target OG. You will need to adjust your hop additions upward in this case. In this case, you will also need to consider if boil is sufficiently long. If not, adding water to your pre-boil wort and boiling it down to your OG over a reasonable amount of time is another option. In either case, using the CV equation, you can figure out the volume at which your target OG is reached and adjust your recipe accordingly.

If your OG is going to be low, you can add malt extract or boil the wort longer to reach the target OG, but yield less beer. The volume you’d need to reach can, of course, be calculated by the CV equation. In this case, you’d need to adjust your hop additions downward to reflect the smaller volume of beer.

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