[BlueTheCow]

This situation is rather hypothetical at the moment, as I want to invest as little money as possible before I plan most of it through. It is also simplified as some of the intermediate steps are not relevant.

I start with a solution of NaOH (sodium hydroxide), SiO₂ (silicon dioxide), and KOH (potassium hydroxide) in water at room temperature (20-25ºC). The specific amounts of each are unknown, though it is probable that the proportions of NaOH and SiO₂ are comparable and that the KOH accounts for less than 10% of the three dissolved solids (all comparisons by weight).

Before I go further, I want to be sure this is possible - silicon dioxide is not normally soluble in water (0.012g in 100g water, from Wikipedia), but after some searching it seems that it can dissolve in the alkaline solution formed by the sodium hydroxide.
Is this correct? Can someone link me to a reliable source explaining why, specifically, this is (this second part is not as important, but more to satiate my curiosity)?

[The rest of this post assumes that the situation is indeed possible.]
This solution is then mixed with carbon dioxide. It is my understanding that this will trigger a reaction with both the NaOH and KOH, producing Na₂CO₃ (sodium carbonate) and K₂CO₃ (potassium carbonate).
Because I don't understand why the SiO₂ dissolved in the NaOH/KOH solution, I am unsure of whether it will precipitate after this change. I think it might, considering that the NaOH, which added to the alkalinity of the solution before, now contributes to the acidity.¹

My next goal is separating out the SiO₂, Na₂CO₃, and K₂CO₃. If the SiO₂ precipitated, then I will filter it out.
I cannot decide how to best separate the Na₂CO₃ and K₂CO₃, and this is my main reason for posting. Actually, it's not a matter of indecision; rather, I just don't know enough about either substance to identify the physical or chemical difference(s) that would allow me to separate them. I will continue to search for more information while I wait for a reply, but I haven't found anything promising yet.


Post Script:
Firstly: Is there a good database or book or something that contains information about various inorganic reactions? This problem might be a bit more straightforward if I knew, for example, all the possible reactions in which Na₂CO₃ is a reactant.
On another note: I considered posting this in the materials forum, but I wasn't entirely sure it belonged there. I am of high school age and took high school chemistry a little while back so this forum seemed fitting.
Lastly, I thank you for reading this far. I really appreciate the effort.

Edits:
¹I read my source for this bit of information incorrectly. Sodium carbonate is actually alkaline.

[Mitch]

Can't you just try to use only KOH or NaOH. That way you don't have to separate counter ions which is nearly impossible to get done in one shot.

[Borek]

Look for CRC handbook or smething similar and compare K₂CO₃ and Na₂CO₃ solubilities. That could be a good starting point.

[enahs]

SiS and SiC are soluble in alkaline solution, with different results. I do not think SiO₂ would be soluble in an alkaline solution. In fact, I am fairly certain it is not to much extent, since most glassware is ~70% SiO₂, and I have never had my glassware dissolve away with a solution at pH of 13, for a couple of hours.


I am not even sure why you are adding the Silicon Dioxide in the first place? Your goal seems to make Sodium and Potassium Carbonate.


Potassium Carbonate is insoluble in 95%+ Ethanol, while I believe Sodium Carbonate is soluble.
I think if I am remembering correctly, you might do as Borek said and look those up in a CRC handbook and verify.

Though I am not sure you will get a very large yields with this method of the production and seperation.

[Borek]

I don't think he targets production, I suppose it is part of some larger project. That's why I have not commented anything else

[AWK]

Na2CO3 and K2CO3 show different solubility at lower temperatures. Hence at least some relatively pure K2CO3 can be obtained by a fractional crystallization but this a a tedious task.

[BlueTheCow]

Thank you all for your replies.

I don't think he targets production, I suppose it is part of some larger project. That's why I have not commented anything else 

Right.

Can't you just try to use only KOH or NaOH. That way you don't have to separate counter ions which is nearly impossible to get done in one shot.

Sadly, it is inevitable that KOH is part of the solution (see first quote).

Look for CRC handbook or smething similar and compare K2CO3 and Na2CO3 solubilities. That could be a good starting point.

I've ordered the CRC Handbook of Chemistry and Physics (88th Edition) online as it seems very useful (to this project and others). Unfortunately I could not find it in any book stores - I was really surprised by the lack of chemistry sections (Barnes & Noble had about 7 books squeezed in on one of the Physics shelves).

[see his post - too long to repeat]

After a bit more research, I found that the SiO₂ reacts with molten (or hot aqueous, as in my source's case) NaOH to form Na₂SiO₃ (sodium silicate). This explains why you never experienced symptoms of vanishing glassware.
While it is not my desire to produce any Na₂SiO₃, I believe I can change it back to SiO₂ as I state below.

I have altered my process such that instead of having the single solution of water, SiO₂ (which I now know is actually Na₂SiO₃), Na₂CO₃ and K₂CO₃, I have two solutions - one containing water, most of the NaCO₃, and all the Na₂SiO₃, and the other containing water, the rest of the Na₂CO₃, and all the K₂CO₃. Originally, the ratio of Na₂CO₃ to K₂CO₃ was high, whereas in the new second solution their quantities by weight are comparable.
HCl Hydrochloric acid can be added to the first solution, reacting with the NaOH to form NaCl and water and with the Na₂SiO₃ to form H₂SiO₃ (metasilicic acid) and NaCl; the composition of the solution would then be water, HCl, NaCl, and H₂SiO₃. I think the hydrochloric acid solution can be boiled away (or distilled for reuse - not sure if it's legal or even cost effective to vent it).
I'm a little bit sketchy on the separation of NaCl and H₂SiO₃, but fairly sure that a bit of experimentation will fix that. Generally, I'll heat it (I think dry, but I'm not sure), changing the H₂SiO₃ to SiO₂ in the form of silica gel or powder. Adding water, the NaCl will dissolve whereas the SiO₂ will not. Filtration divides the two, both of which should probably be heated to remove moisture.
This is all fine and dandy, but I still need to separate the carbonates in the second solution.

To all three posts about solubilities: all sounds good, but I need to do a bit more research (I still await my new reference book in the mail) and possibly some experimentation. I'll post back with any updates.
Again, thank you.

(Lastly, I don't have much time to edit out any mistakes, so please forgive me for typos etc.)

[Borek]

I'm a little bit sketchy on the separation of NaCl and H₂SiO₃, but fairly sure that a bit of experimentation will fix that.

Simple filtration, if the solution is acidic there is no such thing as H₂SiO₃, rather some hydrated form of SiO₂.

Looks to me like you should read some good book on inorganic chemistry. General chemistry book won't hurt either, as most of the reactions you are writing about is not simple & straightforward, these are rather more or less complicated multistage equilibra.