Balancing Chemical Equations and Understanding Reaction

I have a 1 liter solution of unknown weak acid in water of which the pH is 4.0. I have 1 known compound that will be added to this solution --- that being alum (aluminum sulfate = Al2(SO4)3. The objective is to add either sodium carbonate or calcium carbonate to the solution to adjust the pH to 5.0. The desired output reaction is to produce aluminum hydroxide (Al(OH)3) solids but the aluminum solids are supposed to form at a pH in which those solids would exhibit the lowest level of solubility!!!!! Also, to the extent that any sodium or calcium preciptate occurs, the desired result is once again to have the most hydrophobic of sodium or calcium precipitate ---- I suspect that is calcium???

On the aluminum side of the equation, I can reason the following:

Al2(SO4)3 + 6H2O = 2Al(OH)3 + 3H2SO4 -------- or preferably:

Al2(SO4)3 + 6HCO3- (from water)  = 2AL(OH)3 + 6CO2 + 3SO4--  (this reaction would seem preferable)


On the carbonate side of the equation, I can reason the following:

Na2CO3 + H2O = H2CO3 + Na2O; or,

CaCO3 + H2O = H2CO3 + CaO

If my desire is to have the least soluble of the carbonates, I believe CaCO3 and its by-product CaO would seemingly be the least soluble? Do you agree or disagree?

Now what happens when you combine these equations:

xCaCO3 + yAl2(SO4)3 + zH2O ---------------- again if my desire is to produce the most amount of least soluble aluminum hydroxide and I guess presumably calcium hydroxide as well ------- what would the right hand side of this equation look like??

My guess is that it will be yAl(OH)3 + xCaSO4 + CO2 + H+ ????