Cain 1: a) mathrm(Al)+mathrm(HCl) longrightarrow mathrm(ACCl)_(3)+mathrm(H)_(2) b) mathrm(KClO)_(3) longrightarrow mathrm(KCl)+mathrm(O)_(2) mathrm(O)_(2) c) mathrm(Fe)(mathrm(OH))_(3) longrightarrow mathrm(Fe)_(2) mathrm(O)_(3)+mathrm(H)_(2) mathrm(O) d) mathrm(Al)_(2)(mathrm(SO)_(4))_(3)+mathrm(BaCl)_(2) longrightarrow mathrm(BaSO)_(4)+mathrm(ACCl)_(3)

This question consists of three separate chemical reactions that need to be balanced. <br /><br />a) The first reaction is between Aluminum trichloride ($AlCl_{3}$) and Hydrogen chloride (HCl) to form Aluminum trichloride ($AlCl_{3}$). This reaction is already balanced as there are equal numbers of each type of atom on both sides of the equation.<br /><br />b) The second reaction is the decomposition of Potassium chlorate ($KClO_{3}$) into Potassium chloride (KCl) and Potassium peroxide ($K_{2}O_{2}$). To balance this reaction, we need to ensure that the number of atoms of each element on the reactant side (left) is equal to the number of atoms of each element on the product side (right).<br /><br />c) The third reaction is a bit more complex. It involves the conversion of Iron(II) methylpentene ($Fe_{2}(CH_{4})_{3}$) to Carbon dioxide ($CO_{2}$), then to Iron(III) carbide ($Fe_{2}C_{3}$), and finally to Sulfuric acid ($H_{2}SO_{4}$) and Hydrogen chloride (HCl). This reaction is not straightforward and may require additional steps or intermediates to achieve balance. <br /><br />Please note that the provided reaction equations might contain errors or inaccuracies, as the formation of $Fe_{2}C_{3}$ and $HeCF$ from $Fe_{2}(CH_{43}$ and $H_{2}SO_{4}$ is not a recognized or common chemical reaction.