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Primary Fermentation: 50*F/10*C       2 weeks
Diacetyl Rest:        65*F/18.3*C     2 weeks
Lager:                31*F/-0.5*C     4 weeks

I am reading contradictory information regarding the amount of priming sugar to add for carbonation to a lager.

Some sources suggest using the highest temperature (in my case, 65*F/18.3*C) while other sources suggest using the lowest temperature (31*F/-0.5*C) assuming it was held "for a while", and other sources suggest using the current temperature as the input to calculating the total priming sugar.

For example, using the calculators from Brewer's Friend and Northern Brewer, for 2.2 volumes of CO2 for 5 gallons of beer:

Brewer's Friend  33*F  1.5oz  of corn sugar
Brewer's Friend  65*F  3.8oz  of corn sugar
Northern Brewer  33*F  1.57oz of corn sugar
Northern Brewer  65*F  3.5oz  of corn sugar

What about the fact that when I take my carboy out of the freezer into room temperature, it takes me no less than 5 minutes to siphon it into a bottling bucket, and no less than 30 minutes to bottle 5 gallons (operating at my peak efficiency)?

Additionally, I fill 12 bottles at a time before capping them all. Not only is the bottling bucket warming at one rate, but each of the bottles are warming up at different rates based on the time they have been out.

What is relevant here? Is it "the temperature of the bottle at the moment it was sealed with a cap"?

Am I over thinking this or are these relevant issues? Should I just wait for the beer to warm up to the stable room temperature, calculate the priming sugar based on room temperature and go from there? Or, does carbon dioxide escape the solution based on a function of time and temperature, not just temperature, which would require me to leave the carboy at room temperature for some time before adding the priming sugar?

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What the calculators are trying to estimate is how much CO2 from fermentation is still dissolved in the beer. The yeast produce CO2 during fermentation, most of which escapes into the atmosphere, but some remains in solution in the beer. The amount that remains is influenced by temperature and pressure. Assuming the pressure is nominally atmospheric, we only need to consider temperature. At colder temperatures, more CO2 remains in solution which is why the priming sugar calculators indicate less sugar when you enter a lower temperature.

If you warm the beer up, CO2 will slowly escape into the atmosphere. Subsequently cooling the beer will not affect the amount of dissolved CO2. So, when calculating the amount of priming sugar, you should use the highest temperature the beer was stored at for a reasonably long time period after fermentation was complete.

I think for your purposes, the two week diacetyl rest at 65 F. seems like the right temperature to use in your calculation. Before the rest finished, fermentation should have been complete so no more CO2 was being produced. Furthermore, the beer stayed at that temperature long enough after fermentation for the dissolved CO2 to reach equilibrium with the atmosphere. The subsequent drop to lagering temperature would have no effect on the CO2 content, neither would the temperature change during bottling.

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Apparently these calculators are telling you how much sugar is currently in the beer, assuming the yeast have been working at the current temperature (and that the beer remains saturated with CO2).

If cold beer is put in a room temperature bottle, it will warm and lose some CO2 in a few seconds. So I'd say calculate based on temperature when the cap is put on. But to predict that means knowing the temperature of the beer, the bottle, the mass of the bottle, the specific heat of the bottle, the heat transfer through the hose, etc. And, also how temperatures might change during the process.

The simplest way is to have everything at the same temperature the whole time. If you are planning to bottle condition at room temp, just let the carboy warm up before bottling.

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