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When $\ce{Ag2SO4}$ dissociates it becomes $\ce{2Ag^+}$ and $\ce{SO4^2-}$

Why doesn't the $\ce{SO4^2-}$ dissociate further into oxygen and sulfur?

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The answer to your question is very simple. When a chemical species undergoes the process of dissociation in an aqueous solution, it is broken up into its constituent ions. In this case, silver (I) sulfate becomes silver (I) ions and sulfate ions. Dissociation into the constituent ions takes place because the enthalpy of hydration of the silver (I) ions and sulfate ions is greater than the lattice entahlpy of silver (I) sulfate.

However, for the sulfate ions, they can no longer break up into any other chemical species as there are no further strong-enough interactions that take place between the solvent and the sulfate ions to break the bonds between the atoms in the sulfate ion. This is most importantly due to the strong covalent bonds between the sulfur and oxygen atoms in the sulfate ion.

Now, you may ask how the supposedly-strong covalent O-H bonds are broken to allow many oxoacids to dissociate into their conjugate bases and the hydrogen ions. This is because of the strong solvent-solute interactions (i.e. hydrogen bonding) in the solvent, which make the extraction of the hydrogen atom from the oxoacid possible.

With a similar line of reasoning, the supposedly-strong bonds between hydrogen and the halogens in hydrohalic acids (e.g. HI, HCl, HBr) are also easily broken to make them effective acids in solution.

In essence, not everything can dissociate in solution... The energetics of the solvent-solute interactions need to be carefully considered. Usually, ionic bonds are easily broken in solution to give the constituent ions of the ionic solid because the ion-dipole interactions are very strong between solute and solvent species. On the other hand, molecular species with covalent bonds do not easily dissociate because of the strong covalent bonds between atoms of the molecule. Thus, we (to my knowledge) never see the dissociation of molecular species into their constituent atoms.

Hope this answers your query!

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