For the given redox reaction: $a MnO_4^{-} + b S_2 O_3^{2-} + H_2 O \longrightarrow x MnO_2 + y SO_4^{2-} + z OH^-$,the values of $a$ and $y$ respectively are:

Explain the balancing of a redox reaction by the half-reaction method with a suitable example.

How many grams of $Mg$ is required to completely reduce $100 \ mL$ of $0.1 \ M$ $NO_3^{-}$ solution using the following reaction:
$NO_3^{-} + Mg \longrightarrow Mg^{2+} + NH_3$

$ClO_2$ oxidizes $H_2O_2$ to $O_2$ in basic medium and is itself reduced to $Cl^-$. How many moles of $H_2O_2$ will be oxidized by one mole of $ClO_2$ (in $.5$)?

Balance the following equations in basic medium by ion-electron method and oxidation number methods and identify the oxidising agent and the reducing agent.
$(a)$ $P_{4(s)} + OH_{(aq)}^{-} \to PH_{3(g)} + HPO_{2(aq)}^{-}$
$(b)$ $N_{2}H_{4(l)} + ClO_{3(aq)}^{-} \to NO_{(g)} + Cl_{(g)}^{-}$
$(c)$ $Cl_{2}O_{7(g)} + H_{2}O_{2(aq)} \to ClO_{2(aq)}^{-} + O_{2(g)} + H^{+}$

The $Mn^{3+}$ ion is unstable in solution and undergoes disproportionation to give $Mn^{2+}$,$MnO_2$ and $H^{+}$ ion. Write a balanced ionic equation for the reaction.