What does my distilled water contain

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I had this distilled battery water, it fizzes when I throw it on rock and it turns copper hydroxide into a greenish solution. does it have hydrochloric acid?
 
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When you say battery water, are you talking about the liquid that is inside of a lead-acid battery? That is a mixture of distilled water and battery acid (sulfuric acid, H2SO4)

Copper Hydroxide [ Cu(OH)2 ] reacts with diluted Sulfuric Acid [ H2SO4 ] to form water and Copper Sulfate [ CuSO4 ].

Edit: Though, why it's green I couldn't tell you...I'm not super good with chemistry, but this reaction is pretty much the way to create Blue Vitriol.
 
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Battery "water" is sulfuric acid (H2SO4).
It turns green when it reacts copper hydroxide because copper (II) ions are blue in solution.
From the reaction that should occur:
Cu(OH)2 (s) + H2SO4 (aq) = CuSO4 (aq) + H2O (l)
Copper (II) sulfate should be produced. In solution, this is a deep blue color. The fact that it is green as opposed to blue means that there is something else in your H2SO4 that the copper is forming a complex with. For instance, in citric acid, copper (II) ions form a light green complex. So, you've got H2SO4 with some sort of contamination. I'm not sure what it could be.