For centuries, both brewers and drinkers (as well as tax collectors and government regulators) have been concerned about the strength of their beers.
Beer strength measurements include such values as specific gravity, extract, attenuation, alcohol content and nutritional value. Brewing software can perform the calculations based on the original gravity (OG) and final gravity (FG) of the wort and beer, but this article will provide an understanding of the concepts behind them and may help increase your knowledge of what is occurring and its effect on your beer.
Matters of gravity
Many of the calculations are based on weight, both of the wort or beer itself and on the substances dissolved in it. Measurement by weight is not always convenient; in some cases it may be necessary to weigh a solution, evaporate the water or alcohol, and weigh the remainder. This also can require a high degree of precision in order to achieve meaningful results. In many cases it is much easier to measure the specific gravity (SG), that is, the density of a solution relative to that of pure water.
The most common device for measuring specific gravity is a hydrometer. If yours reads very close to 1.000 in distilled water at its reference temperature, it is sufficiently accurate for our purposes.
The density of water varies with temperature. Water has the unusual property of being most dense at 39 °F (4 °C); its volume increases both above and below this point. Because the temperatures we encounter in our daily environment are usually higher, brewing hydrometers are often calibrated at a reference temperature of either 60 °F (15 °C) or 68 °F (20 °C).
Because the wort and beer being measured are frequently at other than the reference temperature, hydrometer readings need to be corrected. This also requires a reasonably accurate thermometer. There is probably a table with your hydrometer that contains the values to add or subtract from the readings for various temperatures; brewing software often includes this function as well.
An accurate (over the range from 32–212 °F/0–100 °C) formula for determining the specific gravity correction factor for the temperature T (in degrees Fahrenheit) is:
SG correction factor = 0.00130346 - (1.34722124 * 10-4 * T) + (2.04052596 * 10-6 * T2) - (2.32820948 * 10-9 * T3)
Once the correction factor is known, the hydrometer reading needs to be adjusted by adding the factor:
Corrected SG = SG reading + SG correction factor
For example, a hydrometer reading of 1.046 at 75 °F and a reference temperature of 60 °F result in a correction factor of +0.0017 and a corrected specific gravity of 1.048 (1.0477).
One common brewing measure is called the “extract,” that is, the amount of materials (primarily sugars) extracted from the mash and dissolved in the wort or beer. This is usually reported by weight; for example, 10 pounds (4.5 kg) of extract dissolved in a total solution of water weighing 100 pounds (45 kg) would be described as 10 percent extract by weight. This also could be stated as 10 degrees Plato.
In 1843, after making up numerous reference solutions and weighing them, brewing scientist Carl Balling realized that the percentage by weight of all the dissolved solids in wort was essentially the same as if they were entirely sucrose, or table sugar. He called each percentage point a degree Balling (sometimes also called a degree Brix), that is, the number of grams of dissolved sugar per 100 grams of solution. Eventually, about 1900, Dr. Fritz Plato discovered some small errors in Balling’s tables and corrected them. The revised unit of measure is now known as a degree Plato, but it represents the same principle.
Extract is usually stated in degrees Plato and these are related to specific gravity units. Because Plato’s tables represent empirical measurements, the formula for converting the values reflects its ability to represent the curve described by the data. As such, there are small errors, especially at the extremes of the degrees Plato and specific gravity scales, but curve-fitting software has emulated the tables with very good accuracy.
One of the more accurate formulas for converting extract (E) in degrees Plato to specific gravity units (SG) (for specific gravities from SG 1.000–1.144/0–33 °Plato) is:
SG = 1.00001 + (0.0038661 * E) + (1.3488 * 10-5 * E2) + (4.3074 * 10-8 * E3)
And for the corresponding conversion of SG units to °Plato:
E = -668.962 + (1262.45 * SG) - (776.43 * SG2) + (182.94 * SG3)
Much simpler versions of the formulas sufficiently accurate near the center of the degrees Plato/specific gravity units scale and for values typically used by most homebrewers (4–16 Plato, or 1.012–1.064 SG) are:
E = (1000 * (SG - 1)) / 4
SG = 1 + ((E * 4) / 1000)
In other words, to determine degrees Plato, divide specific gravity “points” (the portion of the specific gravity to the right of the decimal point multiplied by 1000) by 4. For example, 1.048 is 48 SG points or 12 °Plato. To convert °Plato to SG points, multiply by 4. That is, 12 °Plato is 48 SG points.
Using the more accurate versions of the formulas above, a specific gravity of 1.048 converts to 11.90 degrees Plato, and 12 degrees Plato converts to 1.0484 specific gravity; however, the differences are minor for the purposes of homebrewing calculations. The odds are that the discrepancy is equal to or smaller than the resolution of the instruments you use.
Making it real
As the wort ferments, an important change occurs. The beer is no longer merely a solution of solids in water; there is also alcohol (and carbon dioxide) present. The specific gravity of ethanol at 59 °F (15 °C ) is 0.794, and is less dense than water. As the alcohol content increases, hydrometer readings are correspondingly lowered due to its presence.
This produces two values, the so-called “original extract” (OE) present before fermentation begins, and the “apparent extract” (AE), the reading after alcohol is present. The alcohol determines yet another value, the “real extract” (RE), which accounts for this change and difference in density. Imagine that the alcohol were removed and replaced with an equal volume of water. The RE is the reading that would result.
You could remove the alcohol by the relatively difficult process of distillation, but Balling developed a formula for calculating the real extract value. Here is a version accurate for most original extracts (it uses an “attenuation coefficient” based on an OE of
RE = (0.8114 * AE) + (0.1886 * OE)
Let’s assume a sample beer with an OG of 1.048 and an FG of 1.012. Using the simple formula (SG points divided by 4) to convert to degrees Plato results in an OE of 12 (11.90 using the more exact formula) and an AE of 3 (3.07). The resulting RE is 4.69 (4.73), which converts to SG 1.018 (1.019).
Two, four, six, eight — how do we attenuate?
Yeast strains sometimes are classified by their attenuation, that is, by how completely they ferment the sugars in the wort into alcohol and carbon dioxide. Attenuation also is used to describe the relative dryness or sweetness of a beer; less attenuated beers have a sweeter finish. Moreover, this is the commonly used indicator for fermentation and yeast performance.
Like the final extract values, attenuation comes in two forms, apparent and real. Apparent attenuation is merely the proportional difference in percent between the original and apparent extract values:
Apparent attenuation (AA) = ((OE - AE) / OE) * 100
These are the values published by yeast suppliers and typically vary from 60 to 80 percent or more, with 75 being a common “ballpark” assumption, although many factors can affect attenuation.
Real attenuation is the proportional difference in percent between the original and real extract values:
Real attenuation (RA) = ((OE - RE) / OE) * 100
RA values are typically about 15 percent lower than the corresponding AA value and may represent a better estimate of a beer’s perceived sweetness, with the lower values (below 60 percent) indicating sweeter beers.
Apparent and real attenuation are sometimes also referred to as apparent and real degree of fermentation (ADF and RDF).
You can substitute specific gravity points for the extract values and achieve essentially the same attenuation values. For example, our beer with an OG of 1.048 and FG of 1.012 represents an AA of 75 percent and RA of 61 percent, respectively.
Last call for alcohol
An important value, for a variety of reasons including legal requirements for commercial brewers and health factors for all beer drinkers, is the alcohol content. During fermentation, a molecule of glucose (the simplest sugar with a molecular weight of 180 grams per mole) is converted to two molecules of ethanol (molecular weight 46 grams per mole) and two molecules of carbon dioxide (molecular weight 44 grams per mole).
If this were the only reaction, the math would be straightforward and the results would be directly proportional to the difference between the original and real extract. However, fermentation is a biological process that produces other byproducts (for example, biomass due to yeast reproduction) and additional side reactions with intermediate compounds. Accordingly, the actual alcohol produced is somewhat less than a simple proportional formula would indicate.
In the laboratory, the alcohol content is measured by carefully weighing the beer after fermentation and then driving off the alcohol by distillation. The difference in the weight is the alcohol produced. Balling measured this empirically and constructed tables of the alcohol content based on the original and real extract. These are the basis for those published by the American Society of Brewing Chemists (ASBC) and used by many commercial breweries. Fortunately, a reasonably accurate formula can emulate the values in these tables and can be used by homebrewers:
Alcohol percentage by weight (ABW) = (OE - RE) / (2.0665 - (0.010665 * OE))
Employing the simple formula for OE (SG points divided by 4) and RE, a slightly simplified version of the above formula using OG and FG is:
ABW = (76.08 * (OG - FG)) / (1.775 - OG)
As has been mentioned, alcohol is less dense than water, so it is a relatively simple matter to use the specific gravity of ethanol (0.794) to convert the ABW values to alcohol percent by volume (ABV), which is more commonly used by homebrewers and legal entities and displayed on some beer labels:
Alcohol percentage by volume (ABV) = ABW * (FG / 0.794)
Our example beer with an OG of 1.048 and FG of 1.012 results in nearly identical ABW values of 3.77 percent using either of the above ABW formulas, and an ABV value of 4.80 percent.
For quick approximations of alcohol content, the following formulas also may be helpful:
ABW = (OG points - FG points) * 0.105
ABV = (OG points - FG points) * 0.132
To convert ABV to ABW, multiply the value by 0.794; to convert ABW to ABV, multiply by 1.259 (the inverse of 0.794).
Using the example beer with these formulas results in an ABW and ABV of 3.8 and 4.8 percent, respectively.
Watching the waistline
Calories in beer come from three sources: the residual extract (unfermented sugars), the alcohol and a small amount of protein from the malt. Therefore it is possible to calculate the Calories in a beer based on the residual extract, the alcohol content and a factor for the malt protein. Sugar (we assume all the residual sugars in the beer have the same composition) has a caloric value of 3.8 Calories per gram, while ethanol has a higher value of 7.1 Calories per gram. As for the protein, the amount varies with the beer style and malt varieties (the total protein content of beer has been measured at from 5 to 10 percent of the sugar content), but this percentage is small and an average value of 7 percent of the sugar is close enough for our purposes. Protein has a nutritional value of 4.0 Calories per gram. To determine the calories in a beer, add the contribution from each of these sources, using the following formula:
Calories per 12 US oz. (355 mL) bottle = 3.55 * ((3.8 * RE) + (7.1 * ABW) + (4.0 * 0.07 * RE))
The multiplier of 3.55 is 12 US oz. converted to milliliters and divided by 100 (to account for the implicit percent). For the calories in a US pint (16 oz.), multiply by a factor of 4.73.
Combining the formula above with the earlier simple formulas for original, apparent and real extract, as well as alcohol by weight, yields this equation based on the original and final specific gravity of a beer:
Calories per 12 US oz. (355 mL) bottle = 3621 * FG * (((0.8114 * FG) + (0.1886 * OG) - 1) + (0.53 * ((OG - FG) / (1.775 - OG))))
Using this equation, our beer with an OG of 1.048 and FG of 1.012 has 165 calories per 12 US oz. (355 mL) bottle, slightly more than the 163 calories the first formula predicted.
It can be seen that sugar gives up very few calories by being converted to alcohol. The 3.8 calories in a gram of sugar become 3.63 calories (7.1 * 92 / 180, based on the molecular weights) of alcohol. This means that the calories in beer are mostly determined by the original extract. Using a number of assumptions, such as 75 percent apparent attenuation and a moderate OG of 1.050, this results in the following approximation:
Calories per 12 oz. (355 mL) bottle = 851 * (OG - 1) * (OG + 3)
In our example, the approximate calories would be 165, essentially the same as the calculated value of 165 using the second equation.
Bill Pierce wishes to acknowledge the work of homebrewer and scientist Dr. Michael L. Hall, which has appeared in various online and printed sources.