The Cl- ion is in general a poor oxidizer (despite the fact that Cl2 is a very strong oxidizer). When HCl dissolves common metals, it is a simple
single replacement reaction between hydrogen and the reactive metal (it is not a redox reaction). Aqua regia works on noble metals because a redox
reaction occurs with the nitric acid (nitric acid is a strong oxidizer which does the dirty work of oxidizing the noble metal to the noble metal
cation), while the hydrochloric acid is merely present to provide a constant stream of Cl- ions for the dissolved noble metal ions to bind with
(nitrates do not form because they are reduced into nitrogen oxides which exit the solution as a gas).
Adding sodium carbonate to your mixture would form sodium chloride and sodium nitrate, but these will have no effect on the solubility of the PdCl2.
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