I did some calculations based on the idea that the lids would influence the gas concentration in the aquarium. I think I can easily set up experiments to test the CO2 kinetics in this model system. For the O2 part, I would need to convince myself and my wife that I really need that O2 meter...
So here is the model:
Let's take two systems; in one, we have 120 liters of water; in the other, 100 liters of water + 20 l headspace. For the sake of simpler calculation, I did not calculate with any gas exchange, as if the aquariums were sealed air-tight. Theoretically, the k diffusion values could be adjusted in both models to reach the below steady states, in which the produced CO2 would equal the removed CO2. I will not complicate the model with kinetics just yet.
I used the following constants, calculated with 25°C temperature:
Gas solubilities in water:
CO2_solubility = 1449 mg/l at 1 atm
O2_solubility = 40 mg/l at 1 atm
Gas densities in the air:
CO2_density = 1780 mg/l at 1 atm
O2_density = 1.291 mg/l at 1 atm
In an air/water system, the gases are distributed between the two compartments. In equilibrium, the partial pressures of the two compartments should be the same, so the concentration ratios between the air and the water are:
concentration_ratio = gas_density / gas_solubility
The ratios of the gas amounts (since we have different water and gas volumes here) for any gas should be:
mass_ratio = (gas_density * headspace_volume) / (gas_solubility * water_volume)
That is: mass_ratio = concentration_ratio * headspace_volume/water_volume
With all this above, we can calculate the actual concentrations with given gas amounts, both in the water and the headspace. I uploaded the python code for calculations to
github if someone is interested, but I think the following is logical even without knowing the exact calculations. So, let's see how the gas concentrations look like if we have the two systems equilibrated with atmospheric air:
As we can see, there is much more O2 in the system with headspace; the headspace functions as a reservoir for O2. Now let's put some biomass into the systems which use O2 and produce CO2. The ratio of produced CO2 and used-up O2 is described with the
respiratory quotient (RQ). I don't know what the RQ should be in an aquarium system, but if carbohydrates are burned into energy + CO2, the ratio will be 1. In the case of burning proteins (e.g. from fish food), this will be ~0.8. I don't know the RQ values for plants and microbes, they depend on their energy source and type of metabolism, but I calculated simply with an RQ of 1. However, whatever it is, we should get similar differences between the two systems, except that the absolute slopes would be different (see the slopes later).
So let's assume the systems are closed, and the biomass uses O2 to produce CO2. After we used up 700 mg O2 (and produced the same molar amounts of CO2), this will be the new state of the two systems:
As we can see, the CO2 concentration in the water is slightly lower in the system with headspace, but the O2 concentration is much higher because of the reservoir. The no-lid system has no reservoir; all the used-up O2 is depleted from the water.
We can do the same calculations for a range of used O2 for both systems and check the O2/CO2 levels:
As we can see, we can go much higher with CO2 accumulation with a smaller O2 drop in the headspace system. These closed model systems are different from real-world aquariums in a couple of things, though:
1. Aquariums, even with a lid, are not closed systems; there is always some gas exchange between water and air. Therefore there will be CO2 loss and O2 resupplying. But as I mentioned above, theoretically, one could control the headspace/air gas exchange (ventilation) to adjust the right levels. I have no idea, though, how much the ventilation is realistically restricted with a lid. But I could check that in my aquariums or using model experiments.
2. The CO2 and O2 diffusion rates are different, so the gas exchange rate will not be the same for the two gasses. This would probably slightly modify the CO2/O2 ratios in the water in a dynamic system.
3. There is photosynthesis in planted aquariums, which also removes CO2 while producing O2
4. RQ could be very different depending on O2 availability
Nevertheless, I think that we can still conclude that adding headspace volume while limiting the ventilation (between the headspace and surrounding air) could help accumulate the CO2 in the water with less O2 depletion as in a no-lid system. I'll run some experiments to see how much the lid matters and how much ventilation restriction is needed to have a significantly higher CO2 concentration in a model system.