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Is it correct to say that $\ce{H2SO_4}$ acts as an acid in the following reaction?

$\ce{2Ag + H2SO_4} \rightarrow \ce{Ag_2SO_4 + 2H_2O + SO_2}$

I know it acts as an oxidizing agent, but is it correct to say it shows acidic properties?

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Do you get the same reaction with sodium sulfate?

I do not think so, at least under "normal" conditions.

When the sulfuric acid acts as an oxidizing agent giving off $\ce{SO_2}$, the excess oxygen comes off as oxide ions which must be somehow put into more stable form. The protons from the sulphuric acid do that by turning them to water.

Getting the reaction to go requires the sulfuric acid to act as both an acid and an oxidizing agent.

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