2 molecules of water

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DingDongD

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I have a question, why does not a reaction of hcl and naoh give off 2 molecules of water.

This is what I was thinking:
HCl + NaOH -> H2O + NaCl
HCl + H2O -> H3O+ + Cl-
Naoh -> Na+ and OH-
H3O+ + Cl- + OH- + Na+ -> Na+ + Cl- + H2O
But, H3O+ and OH- does not equal one mole of H2O.

So, I am just a bit puzzled. I can see that H+ instead will give H2O, by why wouldn't this work?

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