# Shrinking volume of CO2 when cold-crashing - Physics question

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#### jesseb

##### Member
Hi there,

I've been brewing an ale last week and next step is to cold-crash. I tried collecting some CO2 in a keg (mostly to flush the keg of O2) and rubber glove to compensate for the suck-back when cold-crashing. But it got me worried it might create such a vacuüm, it actually can bend/pop-in my band-new stainless fermentor. How much, volume wise, does CO2 (and beer I guess) shrink, when going from fermenting temp to near freezing? What is the formula to calculate this?

The setup:

The numbers:

Total volume of fermentor: 35L
Beer volume: 19L
CO2 volume in fermentor: 16L
CO2 volume in keg: 10L
CO2 volume in glove: about 1L

The glove was connected at the end of fermentation, the tube was dipped in starsan before. It was a bit too late, it only swollen a bit, my guess is about a liter.
To make it a bit more complex, only the fermentor is in the fridge, so from the total volume of 27 liters of CO2, only 16L will go from 20C to 2C. The remaining 11L (outside the fridge) stays at 20C. So the pressure reduction is spread out over the total volume.

Many thanks.

#### balrog

##### Supporting Member
HBT Supporter
I would have thought volumes would be the ratio of T1/T2 (in Kelvin, so 290/270 roughly) so 3L to 3.25L, but PV=nRT and n is changing so I have to defer to @doug293cz for all things calculational.

#### doug293cz

##### BIABer, Beer Math Nerd, ePanel Designer, Pilot
Staff member
Mod
HBT Supporter
Hi there,​

I've been brewing an ale last week and next step is to cold-crash. I tried collecting some CO2 in a keg (mostly to flush the keg of O2) and rubber glove to compensate for the suck-back when cold-crashing. But it got me worried it might create such a vacuüm, it actually can bend/pop-in my band-new stainless fermentor. How much, volume wise, does CO2 (and beer I guess) shrink, when going from fermenting temp to near freezing? What is the formula to calculate this?

The setup:

View attachment 815825

The numbers:

Total volume of fermentor: 35L
Beer volume: 19L
CO2 volume in fermentor: 16L
CO2 volume in keg: 10L
CO2 volume in glove: about 1L

The glove was connected at the end of fermentation, the tube was dipped in starsan before. It was a bit too late, it only swollen a bit, my guess is about a liter.
To make it a bit more complex, only the fermentor is in the fridge, so from the total volume of 27 liters of CO2, only 16L will go from 20C to 2C. The remaining 11L (outside the fridge) stays at 20C. So the pressure reduction is spread out over the total volume.

Many thanks.
We can write the ideal gas law as: PV/T = C, where C is a constant if no gas is gained or lost, and T is absolute temperature (either °K or °R) and P is absolute pressure, not gauge pressure.

So, to start out we have:

14.7 * 16 / 293 + 14.7 * 10 / 293 + 14.7 * 1 / 293 = 14.7 * (16 + 10 + 1) / 293 = 1.3546​
After cold crashing, the glove volume drops to 0, and the pressure in the fermenter and keg will be the equal, so we have to solve the following for P:

P * 16 / 275 + P * 10 / 293 = 1.3546​
P * (0.058182 + 0.034130) = 1.3546​
P = 1.3546 / 0.092312 = 14.67​
So, the pressure drop due to just the temperature change would be about 0.03 psi, and the gauge pressure would be -0.03 psi. not a big deal.

However, when you cool down the beer, it will absorb more CO2 from the headspace. And it turns out this is more significant than the pressure drop due to temp change. In the case of just the fermenter, the additional pressure drop from CO2 reabsorption would be about 4.4 psi, so the gauge pressure would be - 4.43 psi. My spreadsheet for reabsorption doesn't handle the configuration that you have, but the pressure drop should be a bit less, maybe 3.5 - 4.0 psi (educated guess.)

Now, the CO2 reabsorption takes time (~3 weeks to near equilibrium) but occurs faster at the beginning and slows down as more CO2 is absorbed. An estimate of pressure drop vs. time is:

 Days of Cold Crashing Pressure Drop (PSI) 1 0.5 2 1.0 3 1.4 4 1.7 5 2.0 6 2.3 7 2.5

Brew on

OP
OP
J

#### jesseb

##### Member
Wow very detailed calculations! This is very helpfull!

So, the pressure drop due to just the temperature change would be about 0.03 psi, and the gauge pressure would be -0.03 psi. not a big deal.

Indeed this is not much at all. It took about 12 hours to cool it down to cold-crash temperatures, and the glove was nearly empty at that point but it was still not pulling a vacuüm.

Very interesting part about the CO2 absorption. It confuses me a bit though. The beer and CO2 above it are trying to get to a equilibrium, right? So if the pressure drops in the CO2 by cooling, instead of absorbing the CO2, it pulls from the beer, correct?

#### tracer bullet

##### Supporting Member
HBT Supporter
We can write the ideal gas law as: PV/T = C, where C is a constant if no gas is gained or lost, and T is absolute temperature (either °K or °R) and P is absolute pressure, not gauge pressure.
I feel like this gets asked every few weeks and we all wait for you to answer. It might be worthy of a sticky we can point to?

#### doug293cz

##### BIABer, Beer Math Nerd, ePanel Designer, Pilot
Staff member
Mod
HBT Supporter
Very interesting part about the CO2 absorption. It confuses me a bit though. The beer and CO2 above it are trying to get to a equilibrium, right? So if the pressure drops in the CO2 by cooling, instead of absorbing the CO2, it pulls from the beer, correct?
Not correct. As the temperature drops, the equilibrium amount of CO2 in solution goes up (at a constant CO2 partial pressure.) For example, @20°C (68°F) and 14.7 psi CO2 partial pressure (0 psi gauge pressure, if the headspace is 100% CO2), the equilibrium dissolved CO2 is 0.84 volumes (or 1.66 g/L.) At 0°C (32°F) and the same CO2 partial pressure, the equilibrium dissolved CO2 is 1.59 volumes (or 3.14 g/L.)

As we saw, just the temp drop does not lower the pressure much, so to maintain equilibrium, more CO2 has to be absorbed from the headspace into the beer. As more CO2 is absorbed from the headspace, the pressure drops, which reduces the amount of CO2 that will eventually be absorbed at equilibrium. You have to solve a couple of non-linear equations to figure out what the final pressure and dissolved CO2 level will be (but that's easy using the spreadsheet's "Goal Seek" tool.) In any case, you do get a net increase in the dissolved CO2, and a decrease in CO2 partial pressure.

Brew on

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