Pyrovus
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Potassium carbonate from potassium sulphate
I've got a 2.5 kg bag of K2SO4, and I've been trying to think of a way to convert it into K2CO3, which is much more useful. Unfortunately
the carbonate is much more soluble than the sulphate, and therefore any attempted direct metathesis reactions will just precipitate the sulphate.
I've tried looking through solubility tables for any combinations that could work, but haven't had any success, which is rather frustrating.
Any ideas? Any help will be appreciated.
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Polverone
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the roundabout way
Mix it with a solution of a carboxylic acid whose potassium salt is only slightly soluble (oxalic acid, for example). Collect the carboxylate, ignite
it, and crush the porous carbon left over in hot water. Recover K2CO3 from the filtrate. Ta-da!
Okay, not a great method, but it was once used for the preparation of pure K2CO3.
Boil it with calcium hydroxide solution, filter out the CaSO4, and treat the KOH solution with carbon dioxide.
Heat it very strongly with powdered charcoal, to reduce it to K2S, then treat the solution with CO2. Beware of the H2S byproduct!
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Friedrich Wöhler
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Heureka !!! 
To Polverone:
Often I thought about getting KOH from K2SO4 and Ca(OH)2. But I see the problem in that nearly solubility of CaSO4 and Ca(OH)2. It must be a difficult
process. (?)
Today I remembered a fertilizer containing pottasium phosphate! ("Thomaskali", (German), I don't know the English trivial or trading
name). I believe, calcium phosphate is much more unsoluble than hydroxide and sulfate. So the convertion with K-phosphate must be higher quantitative
than with K-sulfate. Is it correct? Is it wrong?
If my mind work correctly here, it must be a famous cheaply method getting a lot of KOH-solution.
Kein Schwanz ist so hart wie das Leben.
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Polverone
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You are right. I was thinking that calcium sulfate was extremely insoluble, like barium sulfate. The exchange between Ca(OH)2 and K2SO4 would probably
not work very well. The calcium phosphate is less soluble. But in any case, I don't think this is a great method, because of the large volumes of
solution that would need to be evaporatively concentrated (without excessive exposure to air) to yield KOH in a convenient form.
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Friedrich Wöhler
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Ca(OH)2 = 1,28 g in 1000 ml H2O at 18°C
0,78 g in 1000 ml H2O at 100°C
CaSO4 = 2 g in 100 ml H2O at 18°C
1,62 g in 1000 ml H2O at 100°C
For Ca-phosphate I must still search, but I'm sure, it's extremely unsoluble like Ca-carbonate!
I don't believe, you need such lot of water, but a lot of time only.
(For shaking anyone for a long time you can take a plastic-can in a washing mashine (locked between clothes) and connect its motor for continued
running.) 
Well, I bought 5 kg of KOH (Merck) in a lab-shop, but it remains always actually how to get K-salts. Really the onliest sources I know are fertilizer
(-phosphate, -sulfate, -chloride) I want to safe my KOH for melting reactions, where dry KOH is recommended. I would be stupid, using dry KOH creating
a solution. Easier G.W.Bush or Bin Laden will get the nobel prize of peace(*), than heating KOH untill water free.
* (not impossable in this crazy world)
I found a further method (Engel and Precht) getting K2CO3, but it will be known already: K-salts, MgCO3 and CO2 build a unsoluble double salt
KHMg(CO3)2-tetrahydrate, decomposing in water at 120°C in CO2, K2CO3 and MgCO3. In Germany Na-carbonate is easy to get, but its a problem converting
with any K-salts: the solubilities of Na-sulfate/K-sulfate or NaCl/KCl or N-carbonate/K-carbonate are too similar.
Kein Schwanz ist so hart wie das Leben.
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