When I look at various carbonation tools, I noticed that the amount of bottling sugar was MUCH reduced when naturally carbonating in a keg:
It would seem to me that a certain amount of sugar would provide a certain amount of CO2 dissolved in the beer. Apparently something else is going on.
Why would it take 1/2 as much sugar to create the same amount of carbonation if the batch is naturally carbonated in a keg versus bottles?
I suspect it has to do with the ratio of liquid volume to gas (head) volume. When bottling each bottle has a certain amount of headspace, while when kegging the amount of headspace relative to the liquid vloume is much smaller since there is only one vessel.
Now, I am not talking about the speed of carbonation, just the final result (the equilibrium):
http://byo.com/stories/issue/item/56-3-ways-to-carbonate-your-keg-techniques
"The rate of carbonation, or the time it takes for beer and headspace pressure to equilibrate, depends on three primary factors: (1) the ratio of gas surface to beer volume, (2) the difference in carbon-dioxide concentration between the gas and beer phase and (3) time. The topic of gas diffusion can be made very complicated by mathematicians and engineers, but the basic concepts are fairly simple. ... Rate is by definition dependent on time. A system will equilibrate given enough time. In relation to carbonation, beer will eventually equilibrate with headspace pressure. Fortunately for homebrewers, batch size is small enough for time to be on our sides."
Basically, the end result of carbonation in the liquid will find an equilibrium dependent on the final headspace pressure - the higher the headspace pressure, the higher the carbonation.
As stated in the link above, one way is to ensure the pressure by fixing it using a carbon dioxide source. The only variable left is the time necessary to achieve equilibrium (until then the carbonation will climb with time until stopping at equilibrium).
If using sugar, like bottle priming, the headspace pressure is built via fermentation. The ideal gas law states that the number of gas molecules necessary for a given temperature is proportional to the pressure and to the volume:
P*V = n*R*T
The only source of the molecules (n) is the fermentation - determined by the amount of sugar provided for fermentation.
All else being equal (temperature and pressure required for target carbonation) the volume determines the amount of sugar.
Since the headspace volume in a keg is smaller that that of all the bottles for a brew, more sugar is needed for bottle conditioning.
In my opinion, the specifics of the conditioning will vary depending on the actual liquid volume to headspace volume ratio.
