The Math and Science of Carbonation

Natural Conditioning (Homebrew Professionals please skip this small intro)

Conditioning is the process of carbonating one's beer. This can be done naturally with the yeast already in your homebrew. In the first 36 hours of a yeast cell's life, its Aerobic cycle wherein the yeast store Oxygen for the fermentation cycle the yeast is already producing CO2. After this time, the yeast enters its anaerobic cycle; this is where CO2 (and alcohol) is generated most vigorously. Finally the sedimentation period ensues. During this phase the yeast becomes pseudo-domant. The yeast may be reactivated by the brewer adding more sugar to his batch, starting the conditioning process.

To bottle condition one's beer they need only add sugar; the brewer may add sugar to the fermenter, and quickly fill the bottles from the mixture, or add the sugar directly to the bottles, then fill and cap directly after. By capping the beer and sealing it in the bottles, the gas is not allowed to escape; this in turn pressurizes the beer and traps the CO2 produced by the yeast and the sugar in the final product. A good mid point to start your CO2 conditioning experiments is to dissolve 180ml (3/4 cup) of corn sugar in 400ml of water (per 5 gallon batch) and boil the solution for around 5 minutes, allow the solution to cool in an ice bath then pitch into your bottling bucket, stirring SLOWLY (so as not to oxidize your batch). You could substitute the dried malt extract for corn sugar but remember that 1 cup of corn sugar equals 1 and 2/3 cups of dried malt extract.

There is also a method for the snobbiest of homebrewers called kraeusening. Kraeusening is the name given to the process of priming (conditioning) the beer with Gyle (unfermented wort set aside until carbonation time). This keeps the beer adjunct free. So how much Gyle do you need? Charlie Papazian, the author of The Complete Joy of Home Brewing came up with a handy formula for just that question. This formula assumes that the brewer wants the equivalent of 3/4 cup of corn sugar worth of carbonation. All the formula is doing is figuring out how much sugar is in your wort (I know you knew that).

Formula

Quarts of Gyle needed= (12*gallons of wort to be carbonated) / ((specific gravity - 1)*1000) Example for 5 gallons of beer with the wort at a gravity of 1.040: Quarts of Gyle = (12*5) / ((1.040 - 1)*1000)

Force Carbonating

It's neat, it's easy, it's the only way I will ever carbonate beer again. Most of the following math I learned from George Fix's (Ph.D) book Principles of Brewing Science.

First the basics:

1 Atmosphere (atm) is equal to 14.695 Pounds per inch squared (PSI).

Air is around 21% Oxygen and 78% Nitrogen, the remainder being trace gases.

Gas consists of large numbers of small particles (molecules). Every molecule moves at a constant speed in a straight line. Each molecule may have a different velocity.

Gas molecules are so small that our beer math will not break down if we assume they have no volume. Because of this, the volume of the gas is the volume of the container it is in.

The total amount of kinetic energy of gas is conserved. The molecules of gas behave like billiard balls in that they collide and change directions and speed as they transmit energy to each other, but the total amount of motion (energy) remains the same.

The following has many different sources, listed and not. But is essentially an idiots guide to Dr. Fix's Principles of Brewing Science.

FIX pg. 157 - GASES Ideal Gas - "The absolute temperature (which is measured from absolute zero [-273.15 C] in kelvins, K) is proportional to the average kinetic energy of the ensemble of gas molecules. This statement means that as the temperature increases, the gas particles move faster. The gas molecules also have collisions with the container walls, and the result of these collisions constitutes the pressure in the system. The greater the number of collisions per unit time, the greater the pressure. The average pressure of the atmosphere at sea level is referred to as one atmosphere (1 atm). It is equal to 14.7 psi."

If the same amount of gas in a balloon were to be heated then, the pressure would increase because of the increased molecular collisions, thus increasing the volume of the balloon until equilibrium. Pressure, Volume and Temperature are all connected.

This is all one needs to understand, and the following math becomes quite easy. When operating with gas laws, the home brewer needs to understand the mole. The mole (mol) is simply a large collection of molecules or atoms, of any substance, carbon, hydrogen, CO2, etc. 1 mol is equal to 6.02*10^23 atoms of an element or 6.02*10^23 molecules for any compound. This number is called Avogadro's number. It is simply a count of how many molecules of CO2 (or any substance) are present, counted in 6.02*10^23 segments.

A compounds MOLECULAR WEIGHT is the mass of 1 mol of that compound. For example 1mol of CO2 has a mass of 44 grams, 1mol of N2 (Nitrogen) has a mass of 28g (grams), 1mol of O2 has a mass of 32g. A compounds DENSITY is the ratio of mass to volume. Water's density is 1g/ml (milliliter), the density of ethanol (alcohol) is .7893 g/ml. FIX pg 158 -" Thus, there is a relationship between the number (n) of moles(the amount of a substance), the mass (m) of the substance in grams, and the molecular weight (MW) of the substance":

n=m/MW

Example: How many moles of CO2 are in 60g of CO2?

As noted above, CO2 has a molecular weight of 44g/mol so n= (60g of CO2 ) / (44g/mol) = 1.36 mol of CO2

This also means that 1.36 moles of CO2 has a mass of: m = (1.36mol) * (44g/mol) = 60g

A brewer may also realize that 44g of CO2, 28g of N2, and 32g of O2, all have the same number of molecules because 1 mol is Avogadro's number and these are these gases molecular weights.

In the realm of brewing all of our pressures and temperatures of gases behave as ideal gases. FIX pg. 159 " For these, volume V(in liters, L), pressure P(in atmospheres, atm), absolute temperature T(in kelvins, K), and number of moles n (in moles, mol) satisfy the relationship:

(PV)/(nT)=R

R is the Universal gas constant. 0.0821 (L*atm)/(K*mol)

This is transformed into the ideal gas law:

PV=nRT

Lets try it out. A 23 L tank holds 55 grams of CO2, and it is 10 degrees C or Celsius (Centigrade is synonymous with Celsius) What is the pressure?

First let us rearrange the equation into P=(nRT)/V

Next we need to figure out n, the number of moles of CO2 that is in the tank. We know that 1 mol of CO2 is 44 grams so (55 grams) / (44 grams/mol) = 1.25 mols of CO2 in the tank (n= 1.25)

, Next we need to get the temperature in kelvins; this is easy because the gradations of Celsius and kelvins are the same, they just start at different places, C at the freezing point of water, and K at the point of which there is NO molecular movement or absolute zero as it is called. So to get kelvins from Celsius you simply add 273 to what ever your C number is. For example T = 10C + 273 = 283K

So now we have everything we need to solve this formula P=(nRT)/V : P = 1.25mol * 0.0821R * 283K / 23L = 1.26 atm. Further (1.26 atm * 14.7 = 18.52psi) NEAT!

This math can be further simplified by dropping the n and the R , this is because P, V, and T are all in ratio with each other. If one of these variables is constant and the other is known, then the remainder can be figured out.

The Ideal Gas Laws

Boyle's law is P1*V1=P2*V2 where T1=T2

Charles' law is V1/T1=V2/T2 where P1=P2

Gay-Lussac's law is P1/T1=P2/T2 where V1=V2

This means if the temperature in our example was raised 10 degrees C, or to 20 C (or to 293K) we could use Gay-lussac's law. Our pressure as calculated at 10C was 1.26atm so this means if we rearrange the equation to (P1/T1)*T2 = P2 then P2 = 1.26/283*293 = 1.30atm

In beer, we need only worry about three major gases CO2, O2, and N2. Dalton's law of partial pressures states that each gas has it's own independent, PV=nRT. This means that even as a mixture of these gases together, each gas can be factored as it's own PV=nRT, then the sum of all the gases is added together! That is, the partial pressure of the different gases, is added for the total pressure. Thus P = P(co2) + p(n2) + p(o2)

Volumes of CO2

Brewers use VOLUMES of CO2 dissolved in beer to measure the amount of CO2 in beer. 1Vol. = 1 Liter of CO2 per Liter of beer.

A chart from Principles of Brewing Science (Dr. Fix) indicates the following: Real ale contains between 1 and 1.75 volumes of CO2, Draft Lager contains between 2.2 and 2.4 volumes of CO2, U.K. beers contain between 2.4 and 2.6 volumes, U.S. lagers contain between 2.6 and 2.8, and Wheat beers contain 3 volumes and higher of CO2.

Volumes Chart for Type of Beer(click)

So how do we get Volumes of CO2? Easy. First we must understand Standard Temperature and Pressure (STP). Standard Temperature is 273K or O Centigrade. Standard Pressure is 1 atm, which is 14.7 psi, or 760mm of mercury.

Finally this gets us to Standard Molar Volume; this is the VOLUME occupied by 1 mole of a gas where the temperature is 273k and the pressure is 1 atm (Standard Temperature and Pressure) This is easy to remember because it is the same for ALL gases! 22.4 Liters or 22,400 ml at STP.

Molar Volumes chart here (click)

Gas Problems

This means if CO2 is 44 grams per mole and we want to know how much 1 liter of CO2 weights at STP we can do this: 1mol CO2 at STP = 22.4 Liters as it is for all gases, and we know that CO2 = 44Grams/mol (weight) so 44 / 22.4 = 1.964 grams per liter. Therefore the density of CO2 at Standard Pressure and Temperature is 1.964g/l.

Just a few more to make sure you have it: Methane has a molecular weight of 16 (grams per mole). Calculate the volume occupied by 3.6 grams of Methane at STP. This means that 3.6 grams / (divided by) 16 grams per mole means that we're working with .225 mols of gas. Then we know that at STP the volume occupied by one mole is 22.4 liters so 22.4 * .225 = 5.04 liters.

Now that we have some what of a grasp of the relationships that gases posses, the formula that gives us the amount of pressure needed to get the volumes of CO2 required is slightly more complicated because of all the constants required, but the most popular short cut formula is as follows.

Pressure = F(Temperature, Volume) Temperature is is Fahrenheit, Volumes is in Volumes of CO2 required. Please remember your order of operations! And to convert Celsius to Fahrenheit the formula is C = (F - 32) x (5 / 9) . The way to do it in your head is: Centigrade to Fahrenheit double the number, subtract 10%, then add 32.

P = -16.6999 - 0.0101059 T + 0.00116512 T^2 + 0.173354 T V + 4.24267 V - 0.0684226 V^2

Formula and Carbonation Chart.

For a few final comments, remember to burp your keg, meaning that when you put the green (uncarbonated) beer in the keg fill it will CO2 at 14.1 PSI then release all the co2 from the purge valve. Then repeat a couple of times, this replaces the O2 with CO2 because compound pressure compounds oxidation! If you are naturally conditioning your brew, put 5 psi in the keg first to assure that the keg is sealed. Finally, always add .2 volumes of CO2 to the recommended volumes chart to make up for the carbonation lost at the tap.

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