Solubility of any salt (and aragonite is a salt) is described chemically/mathematically using a solubility product. For aragonite dissolving its (I'm going to leave out the ionic charge for simplicity):
CaCO3 (solid) = Ca + CO3
Ksp = [Ca][CO3]
[Ca] is the concentration of calcium in the solution
[CO3] is the concentration of carbonate in the solution
When the product (multiply the two concentrations together) of the calcium and carbonate concentrations is less then Ksp it means the solution is under saturated and more solid CaCO3 will dissolve. When the product is equal to Ksp then the solution is at equilibrium and there is just as much solid CaCO3 being deposited as is being dissolved. When the product is greater then Ksp it means the solution is supersaturated and solid CaCO3 does not dissolve but the calcium and carbonate in solution will precipitate as solid CaCO3.
Omega is defined as [Ca][CO3]/Ksp (rearranging above equation). So when omega is equal to one the solution is at equilibrium. When omega is less then one it's under saturated. When greater then one the solution is supersaturated.
So how does one calculated this? I'll do an example of my tank water.
Ksp is a function of temperature and pressure. I don't, off the top of my head remember Ksp for aragonite, I'd have to spend some time looking it up, but it doesn't matter since that CO2 program has it built in.
Calcium is easy to get at. Just test your water. The value given will be in ppm or mg/l.
Carbonate concentration is much more difficult to get at. It's a function of total alkalinity and pH. The calculations can be done long handed, but again the CO2 program will help us here. What we do need to know is the total alkalinity and pH. And we make the assumption that our tank water is similar to seawater as far as borates are concerned. This is a good assumption unless you are using a buffer which contains lots of borate. So for me, my alkalinity is 3 meq/l and my pH is 8.0.
Now let's go to the CO2 program.
Start program. Hit enter. Type "P" and hit enter. You should see a screen with 5 choices. Type "5" and hit enter. We need to change the pH scale from Total to NBS. So type "4" and hit enter. Then hit enter again. Now you get to select your input type. Since we know total alkalinity (TA) and pH type "2" and hit enter. Enter your salinity or just hit enter if you keep it at natural seawater level of 35. If you know your phosphate and/or silicate levels enter those too, but it doesn't effect the end product very much if you just enter zeros for those. Enter your tank temperature (in degrees C) next. My tank is 79F so I'll type in "26.1". For the depth I just accept zero. Just hit enter for the next two temperature and pressure inputs. Now enter your total alkalinity that you measured using your alk test kit in units of umol/kg. Since a liter of seawater weighs (essentially) 1 kg, then 1 meq/l = 1000 umol/kg and my alkalinity of 3 meq/l is 3000 umol/kg. Enter "3000". For pH enter "8.0".
This next screen shows useful delta info for oceanographers, but we don't really care, so hit enter again. This screen shows how your alkalinity is distributed between carbonates/boron/OH/PO4/silicates and the CO2/HCO3/CO3 distribution. Look at the data in the first column (T=26.1 C). The next screen is the one you want for omega. You'll see values of omega for both aragonite and calcite. In my case, for aragonite, omega is 3.56 and for calcite it's 5.39.
This omega is for natural seawater with a salinity of 410 ppm but my calcium level is 420 ppm so to get the true omega I need to ratio these two numbers and multiply by the program omega. 420/410*3.56 = 3.65 for aragonite. This means that my tank water is supersaturated with regard to CaCO3 by 365%.
Hope this helps, just ask if you have more questions.
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