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Operation and Experimentation
Measure out a desired quantity of Ca(OH)2 for your test and add it to a coffee filter that has already been weighed without Ca(OH)2 added to it. Place the coffee filter onto the HDX lid that is within the humidifier box. Close the lid and run the air pump for your desired amount of time. The powder will start to clump up and form cracks as humidification takes place.

Calculating CO2 Uptake

The coffee filter plus product should be thoroughly dried before weighing. Subtract the weight of the coffee filter and also the weight of the Ca(OH)2 that you added. That will give the amount of mass change that has taken place. This change in mass is equal to the grams of CO2 sequestered minus the grams of water that have evaporated.
The carbonation reaction once again is Ca(OH)2 + CO2 -> CaCO3 + H2O. Since both sides of the equation must have the same mass (mass balance), the following must also be true:
change in mass = x * (mol wt CaCO3 - mol wt Ca(OH)2) = x * (100.0869 g/mol - 74.09268 g/mol) = x * 25.9942
where x = moles of CaCO3 produced.
It also follows that the grams of CO2 captured is x * (mol wt CO2), which equals x * 44.0095 g/mol.
Thus the change in mass that is measured must be scaled by a factor of (44.0095/25.9942) = 1.693 to obtain the actual amount of CO2 taken up.
Example: You have added 10.00 g Ca(OH)2 to a coffee filter weighing 0.86 g. The total initial weight is 10.86 g.
After your run, you measure a weight of coffee filter + product of 12.11 g.
You subtract the weight of the coffee filter to get the weight of the product, which is 12.11 g - 0.86 g = 11.25 g.
The mass change from final to initial time points is thus 11.25 g - 10.00 g = 1.25 g.
You must then increase the mass change by a factor of 1.693 to get the weight of CO2 taken up. This is 1.25 g * 1.693 = 2.12 g.
The above was taken from the best experimental run so far. For approximately every 5x Ca(OH)2 inputted, 1x CO2 is taken up. Making a thin layer of Ca(OH)2 really seems to help.

Confirming the Presence of CO2

You can confirm the presence of CaCO3 by doing a 5% distilled white vinegar test. A positive test is the visible emission of bubbles (CO2) upon applying the vinegar to the product. Make sure you do the test after you have made sure your sample is completely dry and you have already quantified your CO2 uptake, as you will release some of your captured CO2 during this test.

Storage

The unit is fine to sit closed for long periods of time. If you leave the water inside, the water will very slowly evaporate. For longer-term storage, empty the water, dry, and store the equipment together. As for storage of the product, partially-reacted Ca(OH)2 stored in thin layers will fully react over time, and it would be educational to check the weight increase over time.

Disposal/Reuse

Calcium carbonate is a safe material that surrounds any unreacted Ca(OH)2. Over time, even the unreacted Ca(OH)2 will get carbonated. Disposal should be the same for typical soil materials. Calcium carbonate also has many uses. Note that just because this link includes many medical applications of CaCO3 does not mean you should ingest it - this rule applies for any product you might produce. However, it is fine to use it in the garden in small amounts as a treat for tomatoes and other alkaline-loving plants. It can also be mixed in a 1:1 mixture with cement to form figurines, tiles, bricks, and other creative works.
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Calculating CO2 Uptake
Confirming the Presence of CO2
Storage
Disposal/Reuse