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n the following article I will develop

equations to describe many different

facets of homebrewing. In each section, I first give

an equation that is as accurate as possible, including the ef-

fects of all of the important parameters. This equation is some-

times rather complicated, but the idea is to program it into a

computer spreadsheet and never look at it again. Then, where pos-

sible, I simplify the equations using appropriate assumptions so that

calculations can be made on the fly in the brewery or kitchen.

One thing I should explain at the outset is curve fitting. There

are many occasions when a functional relationship for a set of da-

ta is needed but not available. Curve fitting (also called regression

or least-squares fitting) is nothing more than a mathematical way to

draw the best curve through a set of experimental data points. In

this article I use quadratic (a + bx + cx2) and cubic (a + bx + cx2 +

dx3) functions to draw the curves.

Specific Gravity

Specific gravity is defined as the density of a substance relative

to the density of water. This is not quite sufficient as a definition be-

cause the density of water varies with temperature. Above 39 de-

grees F, water expands as it heats up, making it necessary to spec-

ify the temperature when indicating a specific gravity.

The most common way for a homebrewer to measure specific

gravity is to use a hydrometer. A hydrometer measures the den-

sity of a fluid in units of the density of water at some reference

temperature (the temperature at which the hydrometer is cali-

brated). In other words, the measurement you get from your hy-

drometer is really

Measured SG =

density of wort at temperature T

density of water at reference temperature

Z Y M U R G Y S u m m e r 1 9 9 5

54

Brew By the

Numbers —

Add Up What’s

in Your Beer

Have you ever wondered just how much wallop your favorite homemade

beverage packs, alcoholwise and caloriewise? Have you ever heard your

brewing buddies talk about apparent extract and real attenuation and wondered

what all the hubbub was about? Itʼs not as hard to understand as you might think.

I

Corrected SG =

density of wort at T

density of water at T

=

density of wort at T

density of water at 60˚F

density of water at 60˚F

density of water at T

= (Measured SG at T) (SG correction factor)

This correction factor is solely a function of the density of water at

various temperatures, which is well-known. I made a curve fit to

some data from a couple of sources (De Clerck, Weast) that yield-

ed the SG correction factor as a function of measuring temperature

(T in Fahrenheit):

SG correction factor = 1.00130346 – 1.34722124 x 10-4 T

+ 2.04052596 x 10-6 T2 – 2.32820948 x 10-9 T3

For most of the hydrometers available to homebrewers the reference

temperature is 60 degrees F, which is the value that will be used in

the rest of this section.

The measured specific gravity value must be temperature cor-

rected before it is meaningful. The ideal way to do this would be to

convert the density of the wort to the reference temperature, like this:

Ideal SG =

density of wort at 60˚F

density of water at 60˚F

Modifying the wort density to account for temperature is not that easy,

however, because we don’t have tabulated data for the variation of

wort density with temperature for every possible wort composition. For-

tunately, the temperature coefficients of expansion for wort and water

are practically the same, making the density ratio roughly invariant to

temperature. Making this assumption leads us to this equation:

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Z Y M U R G Y S u m m e r 1 9 9 5

55

(

)

)(

by

Michael L .

Hall , Ph.D.

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ROADSCAPE PHOTO BY MICHAEL LICHTER PHOTOGRAPHY

BEER BOTTLE PHOTO BY GALEN NATHANSON

manner as if the dissolved materials were entirely sucrose. He could

do all of his experiments with pure sucrose and water, and they

would be a good approximation to beer worts. After making up sug-

ar solutions, Balling developed a table relating the density of the so-

lution (at 17.5 degrees C) to the weight percent of sugar. He mea-

sured the weight percent in degrees Balling (˚B), defined as the num-

ber of grams of sugar per 100 grams of wort. Several years later,

around 1900, Dr. Fritz Plato corrected some slight mistakes and de-

veloped his own set of tables, calling the corrected unit a degree Pla-

to (˚P). To convert back and forth between extract (E) in degrees Pla-

to and specific gravity (SG), you can use these equations:

E = –668.962 + 1262.45 SG – 776.43 SG2 + 182.94 SG3

SG = 1.00001 + 0.0038661 E + 1.3488 x 10-5 E2 + 4.3074 x 10-8 E3

These equations are accurate cubic fits that I did to Plato’s data

(Timmermans) over the range of 0 to 33 ˚P, which covers specific

gravities between 1.000 and 1.144. A simpler formula that is good

for most applications is:

E = 1000 (SG – 1) / 4

In other words, just take the number of specific gravity points and

divide by four to get the extract value in degrees Plato. For exam-

ple, if you measured the specific gravity of your wort to be 1.065,

then the simple formula (Equation 9) gives an extract of 16.25 ˚P,

while the cubic formula (Equation 7) gives an answer of 15.88 ˚P.

But there’s still a little more to extract than this. If you mea-

sure the specific gravity before fermentation starts and convert

This equation is accurate over the entire region from 32 degrees F

to 212 degrees F.

If you’re putting this equation in a spreadsheet, the most accu-

rate way to do it would be to multiply your measured specific grav-

ity by the correction factor evaluated at the temperature at which

the SG measurement was made, as shown in Equation 4. Howev-

er, for the times when it is easier to just add or subtract a point or

two, we can make the following approximation. Both the measured

SG and the correction factor are numerically very close to one. If we

represent them by one plus a small number (eSG or eCF), then

Corrected SG = (Measured SG at T) (SG correction factor)

= (1 + eSG) (1+ eCF)

= 1 + eSG + eCF + eSG eCF

Because both e terms represent small quantities, multiplying them

together makes a number that is very small with respect to the oth-

er terms and can be neglected. This yields

Corrected SG ≅ 1 + eSG + eCF

= (Measured SG at T) + (SG correction factor – 1)

So, instead of multiplying by the correction factor, you can just add

the correction factor minus one. Table 1 gives the additive correc-

tion factors (in points, see SG in glossary for explanation) as a func-

tion of the measuring temperature. Using an additive correction fac-

tor is not as accurate as the multiplicative correction factor, but it is

easier and is adequate for most circumstances.

Extract

Since the early days of brewing, brewers (and drinkers) have been

concerned about the strength of their concoctions. One measure of wort

strength is how much material has been extracted from the malted grains

into solution by the mashing and lautering processes. The term “extract”

refers to the weight percent of dissolved materials in the wort. Weight

percent is percent by weight. For example, if you have five pounds of

sugar in a solution that weighs 100 pounds total, the solution is 5 weight

percent sugar. (Incidentally, this is also 5 ˚Plato.)

The only difficulty with this is that there is not a good way to

measure the amount of dissolved materials, short of evaporating

your wort until you have dry malt extract, measuring the weight of

the powder and then dividing by the weight of the original solution.

One thing that you can measure is the specific gravity, which is the

density of the solution relative to the density of water. But there is

still a problem: the dissolved materials are made up of fermentable

sugars, non-fermentable sugars, proteins and other goodies, with

proportions that vary from wort to wort. How can the relationship

between weight percent of solids and specific gravity be determined

if the identity of the solids isn’t even known?

In 1843, Carl Joseph Napoleon Balling determined a way around

this problem. He noticed that the specific gravity of a wort increased

with the weight percent of dissolved materials in almost the same

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Z Y M U R G Y S u m m e r 1 9 9 5

56

Measuring

Addition to Specific

Temperature (F)

Gravity in points

32

–0.83

40

–0.97

45

–0.84

50

–0.69

55

–0.38

60

0.00

65

0.53

70

1.05

75

1.69

80

2.39

85

3.17

90

4.01

95

5.01

100

5.91

Measuring

Addition to Specific

Temperature (F)

Gravity in points

105

6.96

110

8.08

115

9.26

120

10.50

125

11.80

130

13.16

140

16.07

150

19.15

160

22.45

170

25.93

180

29.59

190

33.40

200

37.35

212

42.42

NOTE: Subjecting a room-temperature hydrometer to 212-degree-F

temperatures can result in a broken hydrometer.

TABLE 1: SPECIFIC GRAVITY

TEMPERATURE CORRECTION

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Z Y M U R G Y S u m m e r 1 9 9 5

57

that number to degrees Plato, that’s called the original extract

(OE). If you take a measurement after fermentation is finished,

that’s called the apparent extract (AE). The reason it isn’t the

real extract is that your beer is no longer just a solution of solids

and water. Now you have alcohol in there too, and alcohol is less

dense than water (specific gravity of 0.794 at 15 degrees C) so it

changes all the neat equations that we’ve just come up with. Just

as before, the hard way to determine the real extract (RE) of your

beer would be to boil it until all of the alcohol is boiled off, re-

place the volume with water and then measure the specific grav-

ity. Fortunately, Balling once again comes to the rescue with an

empirical relationship between the real extract, the apparent ex-

tract and the original extract (De Clerck):

q = 0.22 + 0.001 OE

RE = (q OE + AE) / (1 + q)

The variable q is called the attenuation coefficient, but you can just

think of it as an intermediate value. Many sources quote this equa-

tion in the form RE = 0.8192 AE + 0.1808 OE, but this form assumes

that q is calculated at an OE of zero, which is not very accurate. For

our simplified version, we’ll calculate q assuming a reasonable OE

of 12.5 ˚P, which gives us

RE = 0.8114 AE + 0.1886 OE

Let’s say our beer has finished fermenting and we have measured

the specific gravity to be 1.014. The cubic formula for extract (Equa-

tion 7) gives an apparent extract of 3.57 ˚P, but the simple formula

does very well and gives a value of 3.50 ˚P. Calculating the real ex-

tract using the most accurate RE formula (Equation 10) and the cu-

bic extract formula gives 5.92 ˚P. Making the same calculation with

the simple version of both formulas (Equations 9 and 11) gives 5.90

˚P. The difference in real extract calculated by these two methods

is small, but the two methods show greater differences in subsequent

calculations. If you’re not interested in extreme accuracy, then the

simple formulas are probably adequate. This is especially true of

beers with original gravities less than 1.070, because the two for-

mulas only diverge significantly for high specific gravities. You might

want to go through all the complicated calculations for a barleywine

or a mead (or if you’re putting all of this in a spreadsheet), but for

regular beers the simple divide-by-four rule is all you need.

The simple formulas will be used in the rest of the examples,

with the results from the more accurate formulas in parentheses for

comparison.

Attenuation

A closely related term that is often used to describe a yeast strain

or the dryness of a beer is attenuation. Attenuation is simply the

percentage of sugar that has been converted to alcohol. Attenuation

comes in two forms, just like the final extract value. Apparent at-

tenuation (AA) is calculated using the apparent extract value:

(OE – AE)

AA =

x 100%

OE

Real attenuation (RA) is calculated using the real extract value:

(OE – RE)

RA =

x 100%

OE

For our sample beer, the apparent attenuation is 78.5 percent (77.5

percent) and the real attenuation is 63.7 percent (62.7 percent). Most

beers will have a real attenuation that falls in the 60 to 80 percent range.

Alcohol Content

One might think that the alcohol percentage of the final beer

would be directly proportional to the difference in the original and

final (real) extract values. After all, the chemical equation for the

conversion of a monosaccharide to ethanol is

C6H12O6 2C2H5OH + 2CO2

so the weight of the sugar molecule (180 amu) should be converted

into the weight of two ethanol molecules (92 amu) and two carbon

dioxide molecules (88 amu). This would give us an equation for the

alcohol percent by weight (A%w) of:

A%w = (OE – RE)

92

=

(OE – RE)

180

1.9565

The fly in the ointment here is that fermentation is not that simple. It’s

a biological process with all kinds of intermediate products and side re-

actions that don’t lead to our desired result. Balling comes through for

us again with an empirical formula for the alcohol content (De Clerck):

A%w =

OE – RE

2.0665 – 0.010665 OE

If we insert the simple extract equation (Equation 9) and the sim-

plified version of Balling’s equation for real extract (Equation 11),

we can derive a relationship for the alcohol content as a function of

the original and final specific gravities:

A%w =

76.08 (OG – FG)

1.775 – OG

Either of these equations can be converted to alcohol percent by

volume (A%v) by the following formula:

A%v = A%w (FG / 0.794)

where 0.794 is the specific gravity of ethanol. For our sample brew,

the alcohol percentages are 5.46 percent (5.25 percent) by weight

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Z Y M U R G Y S u m m e r 1 9 9 5

58

TABLE 2: ALCOHOL PERCENT BY WEIGHT (USING MOST ACCURATE EQUATIONS)

Original

Specific

Final Specific Gravity

Gravity

0.990

0.995

1.000

1.005

1.010

1.015

1.020

1.025

1.030

1.035

1.040

1.045

1.050

1.030

4.17

3.63

3.10

2.57

2.05

1.53

1.01

0.51

0.00

—

—

—

—

1.035

4.69

4.15

3.62

3.09

2.56

2.04

1.52

1.01

0.50

0.00

—

—

—

1.040

5.22

4.68

4.14

3.61

3.08

2.55

2.03

1.52

1.01

0.50

0.00

—

—

1.045

5.75

5.21

4.67

4.13

3.60

3.07

2.55

2.03

1.51

1.01

0.50

0.00

—

1.050

6.28

5.73

5.19

4.65

4.12

3.59

3.06

2.54

2.02

1.51

1.00

0.50

0.00

1.055

6.82

6.26

5.72

5.17

4.64

4.10

3.58

3.05

2.53

2.02

1.51

1.00

0.50

1.060

7.35

6.80

6.25

5.70

5.16

4.62

4.09

3.56

3.04

2.52

2.01

1.50

1.00

1.065

7.89

7.33

6.78

6.23

5.68

5.14

4.61

4.08

3.55

3.03

2.52

2.01

1.50

1.070

8.42

7.86

7.31

6.75

6.21

5.67

5.13

4.60

4.07

3.54

3.02

2.51

2.00

1.075

8.96

8.40

7.84

7.28

6.73

6.19

5.65

5.11

4.58

4.06

3.53

3.02

2.50

1.080

9.50

8.94

8.37

7.82

7.26

6.72

6.17

5.63

5.10

4.57

4.04

3.52

3.01

1.085

10.05

9.48

8.91

8.35

7.79

7.24

6.70

6.15

5.62

5.08

4.56

4.03

3.51

1.090

10.59

10.02

9.45

8.88

8.33

7.77

7.22

6.68

6.14

5.60

5.07

4.54

4.02

1.095

11.14

10.56

9.99

9.42

8.86

8.30

7.75

7.20

6.66

6.12

5.59

5.06

4.53

1.100

11.69

11.11

10.53

9.96

9.39

8.83

8.28

7.73

7.18

6.64

6.10

5.57

5.04

1.105

12.24

11.65

11.07

10.50

9.93

9.37

8.81

8.26

7.71

7.16

6.62

6.09

5.56

1.110

12.79

12.20

11.62

11.04

10.47

9.90

9.34

8.78

8.23

7.69

7.14

6.60

6.07

Original

Specific

Final Specific Gravity

Gravity

0.990

0.995

1.000

1.005

1.010

1.015

1.020

1.025

1.030

1.035

1.040

1.045

1.050

1.030

93.6

96.0

98.3

100.7

103.1

105.5

107.9

110.2

112.6

—

—

—

—

1.035

110.1

112.5

114.8

117.2

119.6

122.0

124.3

126.7

129.1

131.5

—

—

—

1.040

126.7

129.0

131.4

133.8

136.1

138.5

140.9

143.2

145.6

148.0

150.4

—

—

1.045

143.3

145.6

148.0

150.3

152.7

155.0

157.4

159.8

162.1

164.5

166.9

169.3

—

1.050

159.9

162.2

164.6

166.9

169.3

171.6

174.0

176.4

178.7

181.1

183.4

185.8

188.2

1.055

176.6

178.9

181.2

183.6

185.9

188.3

190.6

193.0

195.3

197.7

200.0

202.4

204.7

1.060

193.3

195.6

197.9

200.3

202.6

204.9

207.3

209.6

212.0

214.3

216.6

219.0

221.3

1.065

210.0

212.3

214.7

217.0

219.3

221.6

224.0

226.3

228.6

231.0

233.3

235.6

238.0

1.070

226.8

229.1

231.4

233.7

236.1

238.4

240.7

243.0

245.3

247.7

250.0

252.3

254.7

1.075

243.6

245.9

248.2

250.5

252.8

255.2

257.5

259.8

262.1

264.4

266.7

269.1

271.4

1.080

260.5

262.8

265.1

267.4

269.7

272.0

274.3

276.6

278.9

281.2

283.5

285.8

288.1

1.085

277.3

279.6

281.9

284.2

286.5

288.8

291.1

293.4

295.7

298.0

300.3

302.6

304.9

1.090

294.3

296.6

298.8

301.1

303.4

305.7

308.0

310.3

312.6

314.9

317.2

319.5

321.7

1.095

311.2

313.5

315.8

318.1

320.4

322.6

324.9

327.2

329.5

331.8

334.0

336.3

338.6

1.100

328.3

330.5

332.8

335.1

337.3

339.6

341.9

344.2

346.4

348.7

351.0

353.3

355.5

1.105

345.3

347.6

349.8

352.1

354.4

356.6

358.9

361.1

363.4

365.7

367.9

370.2

372.5

1.110

362.4

364.7

366.9

369.2

371.4

373.7

375.9

378.2

380.4

382.7

385.0

387.2

389.5

TABLE 3: CALORIES PER 12-OUNCE BOTTLE OF BEER (USING MOST ACCURATE EQUATIONS)

and 6.98 percent (6.71 percent) by volume. Table 2 shows the al-

cohol percent by weight as a function of original and final specific

gravities calculated with the most accurate equations (Equations 7,

10 and 16) above.

Calorie Content

Before all that alcohol goes to your head, let’s calculate how your

homebrew adds to your beer belly. The following equations all give

calories (C) per one 12-ounce bottle of beer. First, there’s a contri-

bution due to the residual sugar (extract) in the beer:

Cext = 3.55 FG (3.8) RE

The 3.8 factor is the number of calories per gram of sugar, the final

specific gravity converts from grams of solution to grams of water,

and the factor of 3.55 is the number of grams of water in a 12-ounce

bottle, divided by 100 to cancel with the implicit percent of the real

extract. There’s also a contribution because of the alcohol present:

Calc = 3.55 FG (7.1) A%w

The 7.1 factor here is indicative of alcohol’s higher number of calo-

ries per gram. Finally, there’s a small contribution because of the

protein in the beer:

Cpro = 3.55 FG (4.0) (0.07) RE

The 4.0 factor represents the calories per gram of protein, and the

percentage of protein has been estimated at 7 percent of the per-

centage of sugar. This estimate is a median value for protein esti-

mates I have seen in the literature, which range from 5 to 10 per-

cent of the real extract value. The calorie per gram values are from

De Clerck. The total number of calories in your homebrew is then:

C = Cext + Calc + Cpro

= 3.55 FG [3.8 RE + 7.1 A%w + (4.0) (0.07) RE]

= 3.55 FG (4.08 RE + 7.1 A%w)

If we convert the calorie equation to a function of specific gravities

using both of Balling’s approximations and the simple equation for

extract (Equations 9, 11 and 17) we get:

C = 3621 FG (0.8114 FG + 0.1886 OG – 1) + 0.53 OG – FG

1.775 – OG

For our example beer this gives us a calorie count of 226.5 (221.2)

per 12-ounce bottle. Table 3 shows the calorie content as a function

of original and final specific gravities calculated with the most ac-

curate equations (Equations 7, 10, 16 and 22) above.

Okay, now, hang on tight for some fast and furious approxima-

tions. First, note that a gram of sugar gives 3.8 calories, but when it

converts to alcohol it gives roughly 7.1 (92 / 180) = 3.63 calories.

Z Y M U R G Y S u m m e r 1 9 9 5

59

This means that the sugar doesn’t lose a lot of calories by con-

verting to alcohol, and therefore the calorie count is primarily a

function of the original specific gravity. We can take advantage of

this fact by calculating the calories of the unfermented wort (set-

ting the FG = OG), and realizing that our estimate will be a little

high. Better yet, we can make an educated guess about the final

specific gravity, setting it equal to one-fourth of the point value of

the original specific gravity:

FG =

OG – 1

+ 1

4

Second, that (1.775 – OG) term in the denominator of Equation 23

is going to give us trouble, so let’s set that particular OG to a mid-

range value of 1.050. Making those approximations and fiddling

around with the numbers a bit yields:

C = 851 (OG – 1) (OG + 3)

For our example beer this gives a calorie count of 224.8 (221.2) per

12-ounce bottle. For most beers, this equation will be a reasonable

approximation of the number of calories.

Carbonation Level

If you are a homebrew bottler instead of a homebrew kegger,

you’re probably a little bit jealous of all the control that a kegger

has over carbonation levels. If you’re a kegger, it’s relatively easy:

you just put your beer (whether or not it has finished ferment-

ing) into a keg and adjust the temperature to the desired serving

temperature and adjust your pressure so that the carbonation

level is what you want. You determine the desired carbonation

level by reading the “volumes of CO2” off a chart as a function

of the temperature and pressure. You can always readjust things

if they are not to your liking. (Okay, it’s not really that simple,

but you get my point.)

A bottler has a more difficult life when it comes to carbonation.

Once the cap is on, everything is fixed. If you overcarbonate you

have to chill all your bottles down and drink them as fast as you

can. If you undercarbonate, your beer suffers from a lack of aroma,

lack of tingly mouthfeel and overall aesthetics suffer. Clearly, this

situation could benefit from a little more control.

First, what exactly is a “volume of CO2”? The number of vol-

umes of CO2 is a measure of the amount of dissolved carbon diox-

ide. It is equal to the volume occupied by the carbon dioxide if it

were taken out of solution and put at standard temperature and

pressure (STP = 32˚F or 0 C and 1 atmosphere) divided by the vol-

ume of the beer. In other words, if you took all the carbon dioxide

out of your five-gallon batch of beer, changed it to STP and got 10

gallons of gas, the CO2 level in the original beer was 2 volumes.

The amount of carbon dioxide that will dissolve is a fixed quanti-

ty which depends on temperature and pressure (and also the oth-

er things in solution, but that effect is negligible for our purposes).

[

(19)

(20)

(21)

(22)

(23)

(25)

(24)

]

This formula makes use of the simplistic chemical equation of

fermentation (Equation 14) and assumes that 100 percent of the

sugar is fermented. This assumption is valid for corn sugar and

table sugar, but if you’re priming with something else, like hon-

ey or dry malt extract, you should include a factor for the frac-

tion of the priming substance that is fermentable, and increase

your priming rate accordingly.

The total amount of carbon dioxide in our primed and condi-

tioned beer is then:

CD = CDgen + CDinit

= 6.5811 x 10-2 PS + 3.0378 – 5.0062 x 10-2 T + 2.6555 x 10-4 T2

VB

Inverting this formula we can get an equation for the weight of prim-

ing sugar:

PS = 15.195 VB (CD – 3.0378 + 5.0062 x 10-2 T – 2.6555 x 10-4 T2)

So there’s the complicated formula. Table 5 gives values from this

formula for various combinations of carbonation level and tem-

perature. What would an “average” case look like? Let’s assume

that you have five gallons of fully fermented beer at 65 degrees F

and you want to have 2.5 volumes of CO2 in the final product. Run-

ning the numbers gives a priming sugar weight of 121 grams. If you

insist on having a volume of priming sugar to work with, this comes

out to 0.8 cups (using the density for corn sugar that I measured).

Lest you think that it always comes out close to three-quarters of a

cup, here’s another example: This time we’ll assume that you’ve

been making a lager and your beer is waiting to be primed at 40 de-

grees F, with everything else the same. The result for this case is 79

grams of priming sugar, considerably less than before.

I should mention here the caveat that if your beer hasn’t fin-

ished fermenting, none of this analysis is accurate and you may

have significant amounts of sugar left that will create large amounts

of carbon dioxide. The amount of priming sugar that is normally

used will only raise the specific gravity by 0.002, so you can see

that it is imperative that there is not an extra bit of fermentable sug-

ar hanging around. In practice, termination of fermentation is easy

to discern by watching your fermentation lock or taking specific

gravity readings.

Conclusion

I hope that these explanations have made the confusing world

of real extracts and apparent attenuations a little bit clearer. Us-

ing the equations in this article (and keeping good records) should

enable you to duplicate your beer successes and to avoid rein-

venting your beer duds. And now that you’ve been armed with in-

formation about how your homebrew affects your head and body,

you will be able to make a better decision about whether or not to

have that next beer.

Z Y M U R G Y S u m m e r 1 9 9 5

60

A desirable carbonation level for most beers is 2.2 to 2.6 volumes

of CO2. Miller recommends 1.8 to 2.2 volumes for British ales, 2.5

volumes for lagers and German ales, 2.6 to 2.8 volumes for Amer-

ican beers and 3.0 volumes for wheat beers and fruit ales.

Now that our target carbonation level is set, we need to deter-

mine our starting point. If your beer has been bubbling away in a

carboy since the last time you racked it, it has an overpressure of

carbon dioxide at one atmosphere. This means that a considerable

amount of carbon dioxide is already in solution in your beer. The

amount that is in solution (CDinit in volumes) is a function of tem-

perature (T, in Fahrenheit). I fit a function to some empirical data

(Linke) to give this relationship:

CDinit = 3.0378 – 5.0062 x 10

-2 T + 2.6555 x 10-4 T2

Table 4 shows that this function varies from 1.71 to 0.73 volumes

over the range of possible bottling temperatures, which indicates

that determining the amount of dissolved CO2 at the start of bottling

is crucial.

The next step is calculating how much sugar is necessary to get

from the initial carbon dioxide level to the target level. I want to

point out here that measuring your priming sugar by volume (the

old three-fourths cup per five-gallon batch rule) is not very accurate.

We’re going to need a certain mass of sugar to get the desired mass

of CO2, so if you measure by volume you’ll need to know the den-

sity to determine the mass. I’ve seen density estimates for corn sug-

ar that varied between 133 and 193 grams per cup. I measured it

myself to be 151 grams per cup. In the calculations that follow, it is

assumed that the sugar is measured by weight instead of volume,

and I strongly recommend that you measure it that way too.

Given a weight of priming sugar (PS) in grams and the volume

of beer (VB) in gallons, we can estimate the amount of carbon diox-

ide generated through fermentation (CDgen):

CDgen = PS

88g CO2

1

1

180g C6H12O6

7.4287g CO2 / gallon at STP

VB

= 6.5811 x 10-2

PS volumes

VB

Volumes of CO2

Temperature (F)

in Solution

32

1.71

35

1.61

40

1.46

45

1.32

50

1.20

55

1.09

Volumes of CO2

Temperature (F)

in Solution

60

0.99

65

0.91

70

0.83

75

0.78

80

0.73

TABLE 4: CARBON DIOXIDE

EQUILIBRIUM CONCENTRATION

(26)

(27)

(29)

(28)

(

)(

)(

)

( )

Z Y M U R G Y S u m m e r 1 9 9 5

61

Nomenclature/ Glossary

A%w – Alcohol percent by weight.

A%v

– Alcohol percent by volume.

AA

– Apparent attenuation (%), apparent percentage of sugar

that converted to alcohol.

AE

– Apparent extract (degrees Plato), the apparent weight per-

cent of dissolved solids in the beer, before correcting for

the lower density of the alcohol.

amu

– Atomic mass unit.

C

– Calories in a single 12-ounce beer.

Calc

– Calories in a single 12-ounce beer attributed to alcohol.

Cext

– Calories in a single 12-ounce beer attributed to extract

(residual sugar).

Cpro

– Calories in a single 12-ounce beer attributed to protein.

CD

– Total carbon dioxide concentration in the conditioned beer

(volumes).

CDinit – Initial carbon dioxide concentration in the beer before

priming (volumes).

CDgen – Incremental carbon dioxide concentration caused by the

fermentation of the priming sugar (in volumes).

C2H5OH – Chemical formula for ethanol, the primary alcohol in

beer.

C6H12O6 – Chemical formula for a monosaccharide sugar (glucose).

CO2

– Chemical formula for carbon dioxide.

E

– Extract (degrees Plato), the weight percent of dissolved

materials in the wort.

FG

– Final specific gravity.

OE

– Original extract (degrees Plato).

OG

– Original specific gravity.

PS

– Weight of the priming sugar (grams).

Temp.

Desired Carbonation Level (Volumes of CO2)

of beer

1.8

2.0

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

3.0

32

7.0

22.2

29.8

37.4

45.0

52.6

60.2

67.8

75.4

83.0

98.2

35

14.4

29.6

37.2

44.8

52.4

60.0

67.6

75.1

82.7

90.3

105.5

40

25.8

41.0

48.6

56.2

63.8

71.4

79.0

86.6

94.2

101.8

117.0

45

36.3

51.5

59.1

66.7

74.3

81.9

89.4

97.0

104.6

112.2

127.4

50

45.7

60.9

68.5

76.1

83.7

91.3

98.9

106.5

114.1

121.7

136.9

55

54.1

69.3

76.9

84.5

92.1

99.7

107.3

114.9

122.5

130.1

145.3

60

61.5

76.7

84.3

91.9

99.5

107.1

114.7

122.3

129.9

137.5

152.7

65

68.0

83.2

90.8

98.3

105.9

113.5

121.1

128.7

136.3

143.9

159.1

70

73.4

88.6

96.2

103.8

111.3

118.9

126.5

134.1

141.7

149.3

164.5

75

77.8

92.9

100.5

108.1

115.7

123.3

130.9

138.5

146.1

153.7

168.9

80

81.1

96.3

103.9

111.5

119.1

126.7

134.3

141.9

149.5

157.1

172.3

TABLE 5: PRIMING SUGAR WEIGHT (IN GRAMS) FOR A FIVE-GALLON BATCH OF BEER

RA

– Real attenuation (%), real percentage of sugar that is con-

verted to alcohol.

RE

– Real extract (degrees Plato), the real weight percent of dis-

solved solids in the beer, after correcting for the lower den-

sity of the alcohol.

SG

– Specific gravity (density relative to water). Specific gravi-

ty in points is equal to 1000 (SG - 1).

STP

– Standard temperature (0 degrees C or 32 degrees F) and

pressure (1 atmosphere).

T

– Temperature of the beer (Fahrenheit).

VB

– Volume of the beer (gallons).

References

De Clerck, Jean, A Textbook Of Brewing, Chapman & Hall Ltd., 1958.

Miller, Dave, Brewing the World’s Great Beers, Storey Communi-

cations Inc., 1992.

Linke, William F., Solubilities of Inorganic and Metal Organic Com-

pounds, American Chemical Society, 1958.

Timmermans, Jean, The Physico-Chemical Constants of Binary

Systems in Concentrated Solutions, Vol. 4: Systems with In-

organic and Organic or Inorganic Compounds, Interscience

Publishers, 1960.

Weast, Robert C., CRC Handbook of Chemistry and Physics, CRC

Press, Inc., 1980.

Michael L. Hall, Ph.D., is a computational physicist at Los Alamos

National Laboratory in New Mexico. Mike has been brewing for five

years and is a Certified judge in the BJCP. He was one of the founding

members of the Los Alamos Atom Mashers and has worn many hats in

the club (newsletter editor, treasurer, librarian, organizer). Mike can

be reached via the Internet at: hall@lanl.gov.