Final answer:
The reddish-brown residue (q) is likely a transition metal oxide, with iron(III) oxide, Fe2O3, being a strong candidate given the color and reactions described.
Step-by-step explanation:
The question suggests that a green crystalline salt (p) undergoes thermal decomposition to give a reddish-brown residue (q) and gases (r) and (s). Given that the gases have the same elements, it is likely that they are oxygen and a nitrogen oxide, common byproducts of heating nitrates. The salt (p) also reacts with Na2O2 (sodium peroxide) to form a reddish-brown precipitate (t), which indicates the possible presence of a transition metal capable of forming hydroxides when treated with a reactive oxygen species, as described in the provided hyroxides information snippet.
Considering the information and examples provided, where heating a hydroxide leads to the formation of an oxide, such as aluminum hydroxide heating to form aluminum oxide, and transition metals forming colored precipitates, (q) is likely to be a transition metal oxide. Moreover, the color red-brown suggests the presence of iron, possibly hinting at iron(III) oxide, Fe2O3, as iron compounds are known to impart such colors.