(a) (i) Transition metals show variable oxidation states, therefore, they and their compounds act as catalyst.
(ii) Oxygen can form double bond, therefore, it can form Mn₂O₇ , whereas 'F' cannot form double bonds, so, it can form MnF₄ .
(b) \(3 \mathrm{MnO}_{4}^{2-}+4 \mathrm{H}^{+} \longrightarrow \mathrm{MnO}_{2}+2 \mathrm{MnO}_{4}^{-}+2 \mathrm{H}_{2} \mathrm{O}\) 

(i) The black powder acts as a catalyst. The catalysts accelerate the speed of reactions without itself undergoing any permanent changes.
(ii) Black powder is manganese dioxide (MnO₂).
(iii) Black powder cannot be used for all decomposition reactions because catalysts are highly specific in nature. A catalyst which can catalyze one reaction may have no effect on another reaction even, if that reaction is very similar.
(iv) Glycerol slows down the decomposition of H₂O₂ and it is called negative.catalyst.

This is because of ability to form multiple bonds.

The variability in oxidation states of transition metals is due to the incomplete filling of d-orbitals in such a way that their oxidation states differ from each other by unity, e.g., Fe⁺ and Fe³⁺, Cu⁺ and Cu²⁺ etc. In the case of non-transition elements, the oxidation states differ by units of two, e.g., Pb²⁺ and Pb⁴⁺, S²⁺ and S⁴⁺ etc. Moreover, in transition elements, the higher oxidation states are more stable for heavier elements in a group. For example, in group 6, Mo (VI) and W (VI) are more stable than Cr (VI). In P block elements, the lower oxidation states are more stable for heavier members due to inert pair effect, e.g., in group 16, Pb (II) is more stable than Pb (IV).

\({ Cr }^{ 2+ }\) is a stronger reducing agent than \({ Fe }^{ 2+ }\). This is because the configuration of \({ Cr }^{ 2+ }\) changes from \({ d }^{ 4 }\) to \({ d }^{ 3 }\) configuration is stable \(({ t }_{ 2g }^{ 3 })\) being half-filled \({ t }_{ 2g }\) level.

The high enthalpies of atomization of transition elements are due to the participation of electrons (n-1) d-orbitals in addition to ns electrons in the interatomic metallic bonding. In the case of zinc, no electrons from 3d-orbitals are involved in the formation of metallic bonds. On the other hand, in all other metals of 3d series electrons from d-orbitals are S involved in the formation of metallic bonds.

Oxidising behavior of KMnO ₄  depends on the pH of the solution.
In acidic medium, it gives colorless \({ Mn }^{ 2+ }\) ions
 \({ MnO }_{ 4 }^{ - }+{ 8H }^{ + }+{ 5e }^{ - }\)\(\rightarrow \)  \({ Mn }^{ 2+ }\)\(+\) \({ 4H }_{ 2 }{ O }\)
In alkaline medium, green colored  \({ MnO }_{ 4 }^{ 2- }\) ions are formed
 \({ MnO }_{ 4 }^{ 2- }\)\(+\)\({ e }^{ - }\rightarrow { MnO }_{ 4 }^{ 2- }\)
In neutral medium, brown colored \({ MnO }_{ 2 }\) are formed
 \({ MnO }_{ 2 }^{ - }+{ 2H }_{ 2 }{ O }+{ 3e }^{ - }\rightarrow { MnO }_{ 2 }+{ 4OH }^{ - }\)

(a) Cr₂O₇^2-+14H⁺+6e^-\(\longrightarrow \)2Cr³⁺ +7H₂O
(b) CrO₄^2-+2H⁺\(\rightleftharpoons \)2HCrO₄^-\(\rightleftharpoons \)Cr₂O₇^2-+H₂O
(c) MnO₄^-+2H₂O+3e^- \(\overrightarrow { medium } \)MnO₂ +4OH^-

Cr²⁺ has electronic configuration 3d⁴ Chromium also shows +3 and +6 oxidation states. Fe²⁺ has electronic configuration 3d⁶. Iron has oxidation states +2 and +3.

(i) F is more electronegative than CI and Br, therefore, fluorides are ionic; whereas chlorides and bromides are covalent.
(ii) It is due to poor shielding effect off-electrons, effective nuclear charge increases, so, ionic size decreases.
(iii) It is due to similar ionic size which is due to lanthanoid contraction, they resemble in their properties.

(i) It is due to the presence of vacant d-orbitals of suitable energy, the smaller size of cautions and higher charge.
(ii) It is due to increase in effective nuclear charge gradually because unpaired electrons increase in the beginning and then decreases. There is repulsion between paired electrons.
(iii) It is due to absence of unpaired electrons, they do not absorb light from visible region and do not radiate color.

(i) It is because Mn²⁺ is more stable than Mn³⁺ due to stable half filled 3d5 configuration, whereas Cr³⁺(t_2g ³) and Fe³ (3d⁵) are more stable than Cr²⁺ sg and Fe²⁺ respectively.
(ii) It is because oxygen and fluorine are strong oxidising agents, highly electronegative, small size and can provide energy for formation of transition metal ion in higher oxidation state.
(iii) It is due to lanthanoid contraction which is due to poor shielding effect off-electrons.

Elements (+ 2 state ) ₂₁Sc²⁺'₂₂Ti²⁺ ₂₃V²⁺ ₂₄ Cr²⁺ and ₂₅Mn²⁺have more stable +2 oxidation state and the outer electronic configuration are 3d¹ , 3d² , 3d³ , 3d⁴ and 3d⁵, respectively. In all the elements listed, the removal of two 4s electrons (in Cr²⁺, 1e^- from 4s and 1e^- from 3d), the 3d-orbitals get gradually occupied. Since, the number of empty d-orbitals decreases or the number of unpaired electrons in 3d-orbitals increases with increase in atomic number of cations, so the stability of the cations (M ²⁺) increases from Sc²⁺ to Mn.

'A' is iron chromite (FeCr2 04 ), '8' is sodium chromate (Na₂CrO₄ ), 'C' is sodium dichromate (Na₂CrO₇) and 'D' is potassium dichromate (K₂Cr₂O₇). 
\(4 \mathrm{FeCr}_{2} \mathrm{O}_{4}+8 \mathrm{Na}_{2} \mathrm{CO}_{3}+7 \mathrm{O}_{2} \longrightarrow 8 \mathrm{Na}_{2} \mathrm{CrO}_{4}\) + 2Fe₂O₃ + 8CO₂ (g)
    'A'                                                                'B'
(Chromite ore)                                            (Yellow solution)
\(2 \mathrm{Na}_{2} \mathrm{CrO}_{4}+\mathrm{H}_{2} \mathrm{SO}_{4} \longrightarrow \mathrm{Na}_{2} \mathrm{Cr}_{2} \mathrm{O}_{7}+\mathrm{Na}_{2} \mathrm{SO}_{4}+\mathrm{H}_{2} \mathrm{O}\) \(\mathrm{Na}_{2} \mathrm{Cr}_{2} \mathrm{O}_{7}+2 \mathrm{KCl} \longrightarrow 2 \mathrm{NaCl}+\mathrm{K}_{2} \mathrm{Cr}_{2} \mathrm{O}_{7}\) 
                                                     'C'                                                      'C'                                                                   'D'
                                                                                                                                                                                (Orange crystals)