Consider the reactions:
$(a)$ $6CO_{2(g)} + 6H_2O_{(l)} \to C_6H_{12}O_{6(aq)} + 6O_{2(g)}$
$(b)$ $O_{3(g)} + H_2O_{2(l)} \to H_2O_{(l)} + 2O_{2(g)}$
Why is it more appropriate to write these reactions as:
$(a)$ $6CO_{2(g)} + 12H_2O_{(l)} \to C_6H_{12}O_{6(aq)} + 6H_2O_{(l)} + 6O_{2(g)}$
$(b)$ $O_{3(g)} + H_2O_{2(l)} \to H_2O_{(l)} + O_{2(g)} + O_{2(g)}$
Also,suggest a technique to investigate the reaction mechanism (path) of the above $(a)$ and $(b)$ redox reactions.

(N/A) Photosynthesis is a complex process occurring in multiple steps. Experimental evidence using isotopic labeling shows that all the oxygen evolved in photosynthesis comes from water,not from carbon dioxide. Therefore,$12$ molecules of $H_2O$ are involved in the reaction,and $6$ molecules are released as a byproduct.
$(b)$ In the reaction between ozone $(O_3)$ and hydrogen peroxide $(H_2O_2)$,the two oxygen molecules produced originate from different reactants. Specifically,one $O_2$ molecule comes from $O_3$ and the other comes from $H_2O_2$. Writing it as $O_{2(g)} + O_{2(g)}$ highlights this distinct origin.
Technique: The path of these redox reactions can be investigated using isotopic labeling (e.g.,using $^{18}O$ isotope). For reaction $(a)$,water labeled with $^{18}O$ $(H_2^{18}O)$ is used to confirm that the evolved $O_2$ contains the $^{18}O$ isotope. For reaction $(b)$,labeling either $O_3$ or $H_2O_2$ with $^{18}O$ allows researchers to track the source of the oxygen gas produced.