The following reaction occurs in acidic medium: $KMnO_4 + 8 H^{+} + 5 e^{-} \longrightarrow K^{+} + Mn^{2+} + 4 H_2O$. What is the equivalent weight of $KMnO_4$? (Molecular weight of $KMnO_4 = 158$)

After balancing the equation,$C_2O_4^{2-} + H^+ + MnO_4^- \longrightarrow CO_2 + Mn^{2+} + H_2O$,the coefficient of $CO_2$ is:

$2MnO_4^{-} + 5H_2O_2 + 6H^{+} \to 2Z + 5O_2 + 8H_2O$. In this reaction $Z$ is

Manganese ions $(Mn^{2+})$ can be oxidised by persulphate ions $(S_2O_8^{2-})$ according to the following half-equations:
$S_2O_8^{2-} + 2e^- \longrightarrow 2SO_4^{2-}$
$Mn^{2+} + 4H_2O \longrightarrow MnO_4^- + 8H^+ + 5e^-$
How many moles of $S_2O_8^{2-}$ are required to oxidise $1 \ mole$ of $Mn^{2+}$?

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.

Phenolphthalein does not act as an indicator for the titration between