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Is there any way in which a solution can have more negative charge than positive charge, or vice versa? For example, in the context of acid-base chemistry, is there any scenario in which $[\ce{H+}]\ne[\ce{A-}]$? I have never heard of a solution with a non-neutral overall charge, but is it possible in some extreme scenario?

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    $\begingroup$ Well, a trivial and unhelpful answer would be to consider a solution of $\ce{MgCl2}$: there are twice as many negative ions as there are positive ions. A solution that is electrically neutral does not necessarily have the same number of negative ions and positive ions. $\endgroup$ – orthocresol May 27 '16 at 19:39
  • $\begingroup$ You're right, of course. Question edited to say charge, not ions. $\endgroup$ – Marcel May 28 '16 at 16:28
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The only point that $[\ce{H+}]=[\ce{A-}]$ is when the $\mathrm{pH}=\mathrm{p}K_\mathrm{a}$ of a given acid, so indeed in most instances $[\ce{H+}] \neq [\ce{A-}]$. In addition to $\ce{A-}$, the solution also contains negatively charged $\ce{OH-}$ that balances the positive charge of the $\ce{H+}$ ions. If the acid is titrated with a strong base (e.g. $\ce{NaOH}$), the positively charged $\ce{Na+}$ ions balance the negative charge of $\ce{OH-}$ and $\ce{A-}$.

To answer the question in your title though, yes, charges are always balanced. To quote another good post on this topic:

While it is true that solutions are not always locally electroneutral, that non-neutrality is small, and does not extend to larger scale. Thus, it does not impact chemical concentrations: you cannot separate positive and negative charges in a solution at a macroscopic level.

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