Step 1: Define the criterion for disproportionation.
Disproportionation requires a single element in a single reactant to be simultaneously oxidized (to a higher state) and reduced (to a lower state) in the same reaction.
Step 2: Analyze BaO2 + H2SO4 reaction.
BaO2 + H2SO4 $\rightarrow$ BaSO4 + H2O2. Assign oxidation states: In BaO2, Ba = +2, O = -1 (peroxide). In H2SO4, H = +1, S = +6, O = -2. In H2O2, H = +1, O = -1. In BaSO4, Ba = +2, S = +6, O = -2.
Step 3: Check each element for oxidation-state change.
Ba: +2 in BaO2 and +2 in BaSO4 - no change. O in BaO2: -1 in BaO2 and -1 in H2O2 - no change. H: +1 throughout. S: +6 throughout.
Step 4: Conclude whether redox occurs.
No element changes its oxidation state in this reaction. It is simply a double displacement (metathesis) reaction: the peroxide ion from BaO2 is transferred to form H2O2, and SO42- forms BaSO4.
Step 5: Confirm it is not disproportionation.
Disproportionation requires one element to both increase and decrease in oxidation state. Here, no element undergoes any change in oxidation state at all. This reaction is neither redox nor disproportionation.
Step 6: Final answer.
BaO2 + H2SO4 $\rightarrow$ BaSO4 + H2O2 is NOT a disproportionation reaction because no oxidation-state changes occur.
\[ \boxed{\text{Option 4: } BaO_2 + H_2SO_4 \rightarrow BaSO_4 + H_2O_2 \text{ is NOT disproportionation}} \]