Al --> AlCl3 --> Al$OH$3 --> Na3AlO3 --> Al2$SO4$3:

Aluminum $Al$ reacts with chlorine $Cl2$ to form aluminum chloride $AlCl3$.
2 Al + 3 Cl2 → 2 AlCl3

Aluminum chloride $AlCl3$ reacts with water $H2O$ to form aluminum hydroxide $Al(OH)3$.
AlCl3 + 3 H2O → Al$OH$3 + 3 HCl

Aluminum hydroxide $Al(OH)3$ reacts with sodium hydroxide $NaOH$ to form sodium aluminate $Na3AlO3$.
Al$OH$3 + 3 NaOH → Na3AlO3 + 3 H2O

Sodium aluminate $Na3AlO3$ reacts with sulfuric acid $H2SO4$ to form aluminum sulfate $Al2(SO4)3$.
Na3AlO3 + 3 H2SO4 → Al2$SO4$3 + 3 Na2SO4

S --> H2S --> SO2 --> H2SO4 --> Na2SO4 --> BaSO4:

Sulfur $S$ reacts with hydrogen gas $H2$ to form hydrogen sulfide gas $H2S$.
S + 2 H2 → 2 H2S

Hydrogen sulfide gas $H2S$ reacts with oxygen gas $O2$ to form sulfur dioxide gas $SO2$.
2 H2S + 3 O2 → 2 SO2 + 2 H2O

Sulfur dioxide gas $SO2$ reacts with water $H2O$ to form sulfuric acid $H2SO4$.
SO2 + H2O + 1/2 O2 → H2SO4

Sulfuric acid $H2SO4$ reacts with sodium hydroxide $NaOH$ to form sodium sulfate $Na2SO4$.
H2SO4 + 2 NaOH → Na2SO4 + 2 H2O

Sodium sulfate $Na2SO4$ reacts with barium chloride $BaCl2$ to form barium sulfate $BaSO4$.
Na2SO4 + BaCl2 → BaSO4 + 2 NaCl