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Think of it like this:
You are given a white powder sample and you are asked to confirm whether there is a sulphate present in the sample. If you add some barium chloride solution , You get a white precipitate. So you conclude that a sulphate is present because of the following reaction:
BaCl2(aq) + SO4 2-(aq) → BaSO4(s) + 2Cl-(aq)
But is this correct? What would happen if the sample had actually been a carbonate? You add a solution of BaCl2 and you get a precipitate because of the following reaction:
BaCl2(aq) + CO3 2-(aq) → BaCO3(s) + 2Cl-(aq)
Now the only conclusion you can arrive at is that the BaCl2 test is not conclusive for sulphate - it will also give a precilitate if there is a carbonate present . What to do? If there is some way that you can eliminate the carbonate ( if present) then you can be sure that any precipitate comes from the presence is the sulphate only.
This elimination of the carbonate is done by reacting any carbonate present with an acid. The usual acid of choice is hydrochloric acid , Therefore you add some HCl to the solution of your unknown sample. If a carbonate is present the following reaction occurs:
2HCl(aq) + CO3 2- (aq) → H2O(l) + CO2(g) + 2Cl-(aq)
After the reaction with the HCl , there is no more carbonate in the test solution , so adding BaCl2 will not produce a precipitate.
If a precipitate is produced after the HCl reaction , you can be sure that this is only because a sulphate is present in the original sample.
The CO3 2- ion is called an interfering ion in the sulphate test , and it must be eliminated before BaCl2 is added.
You will note that if the SO4 2- ion is present in the original sample , it does not react with the HCl.
To answer your second question: I seem to remember that when I was learning these basic chemical test , we always used nitric acid to eliminate the possible carbonates , followed by BaCl2 to precipitate the sulphates. Likewise Ba(NO3)2 could be used in place of BaCl2
I cannot see any reason why any combination is HCl/HNO3 and BaCl2 / Ba(NO3)2 cannot be used. What is not permitted is to use H2SO4 as the acid - you are wanting to test for the sulphate ion - so you must not add this to the test solution.
I think that HCl / BaCl2 are used because they are both cheap and readily available in most laboratories.