The electronic configuration of Hg(I), i.e., Hg⁺ is [Xe] 4f¹⁴5d¹⁰ 6s¹ and thus has one electron in the valence 6s-orbital. If this were so, all Hg (I) compounds should be paramagnetic but actually they are diamagnetic. This behaviour can be explained if we assume that the singly filled 6s-orbitals of the two Hg⁺ ions overlap to form a Hg-Hg covalent bond. Thus, Hg+ions exist as dimeric species, i.e., Hg_2²⁺. In contrast, the electronic configuration of Cu(l) ion, i.e., Cu⁺ is [Ar] 3d¹⁰.Therefore, it has no unpaired b electrons to form dimeric species, i.e., Cu₂²⁺ and hence it always exists as Cu⁺ ion.

As we proceed along a transition series, the nuclear charge increases which tends to decrease the size but the addition of electrons in the d-subshell increases the screening effect which tends to counterbalance the effect of the increased nuclear charge

Lanthanoids primarily show three oxidation states (+2, +3, +4). Among these oxidation states, +3 state is the most common. Lanthanoids display a limited number of oxidation states because the energy difference between 4f, 5d, and 6s orbitals is quite large. On the other hand, the energy difference between 5f, 6d, and 7s orbitals is very less. Hence, actinoids display a large number of oxidation states. For example, uranium and plutonium display +3, +4, +5, and +6 oxidation states while neptunium displays +3, +4, +5, and +7. The most common oxidation state in case of actinoids is also +3.

In transition elements penultimate d-orbitals are progressively filled whereas in non-transition elements outermost s or p-orbitals are progressively filled.

Mn exhibits largest number of oxidation states because it has 7 electrons in 's' as well as 'd' orbitals which can take part in bond formation.

It is due to increase in ionisation enthalpy, tendency to lose electron decreases, therefore, reactivity decreases.

Scandium (Z = 21) does not exhibit variable oxidation states.

(i) Electronic configuration
Lanthanoids = [Xe] 4f^0-14 5d^0-1 6s²
Actinoids = [Rn] 5f^0-14 6d^0-1 7s²
(ii) Oxidation states In lanthanoids, +3 oxidation state is most common along with + 2 and + 4. While in actinoids, there is a greater range of oxidation states because 5f, 6d and 7s levels are of comparable energies.They show + 2, + 3, + 4, + 5, + 6 and + 7 oxidation states. Common oxidation state in actinoids is + 3.
(iii) Chemical reactivity Lanthanoids are less reactive than actinoids. Actually, earlier members of lanthanoids are quite reactive similar to calcium but with increasing atomic number, they behave more like aluminium. Lanthanoids react with dilute acids to liberate H₂ gas while actinoids react with boiling water and gives a mixture of oxide and hydride.

Reaction between acidified KMnO₄ and iron (II) ions KMnO₄ oxidises ferrous salts to ferric salt.
5Fe²⁺ + Mn\(O_4^-\) + 8H⁺ ⟶ Mn²⁺ + 4H₂O + 5Fe³⁺

Because of large number of unpaired electrons in their atoms they have stronger interatomic interaction and hence stronger bonding between atoms resulting in higher enthalpies of atomisation.

(a)\((i) Cr_2O_7^{2-} + 14 H^+ + 6Fe^{2+}\longrightarrow\ 2Cr^{3+} + 6Fe^{3+} + 7H_2O\)
\((ii) 2 MnO_4^- + 10 I^- + 16 H^+\longrightarrow 2Mn^{2+} + 8H_2O + 5I_2\)
(b) Mn²⁺: 3d⁵4s⁰ has S unpaired electrons.It is highly paramagnetic and it is attracted by magnet.

(i) (a) Due to the comparatively smaller size of the metal ions, their high ionic charges and the availability of vacant d-orbitals for bond formation, transition metals form a  large number of complex compounds.
(b) As oxidation number (or oxidation state) of an element increases ionic character decreases. In general, the oxides in lower oxidation states of metals are basic and in their higher oxidation state, the oxides are amphoteric.
In lower oxidation state of the metal, some of the valence electrons of the metal atom are not involved in bonding. Hence, it can donate electrons and behave as a base. In higher oxidation state, valence electrons are involved in bonding and hence, electrons are not available for donation. Instead, their effective nuclear charge is high and hence they behave as acids.
(c) Mn³⁺(3d⁴) is less stable than Mn²⁺(3d⁵) because Mn²⁺ has stable half-filled configuration. Cr³⁺ has stable 3d³(t³_2g) configuration, therefore, Cr³⁺ cannot be reduced to Cr²⁺. That's why, EO value for the Mn³⁺ / Mn²⁺ couple is much more positive than Cr³⁺ /Cr²⁺. In other words, Mn³⁺ is a strong oxidising agent.
(ii) Similarity Both lanthanoids and actinoids exhibit +3 oxidation state predominantly.Difference Lanthanoids have less tendency towards complex formation while actinoids have greater tendency towards complex formation.

\((a) Ce(58)-[Xe]{ 4f }^{ 1 }{ 5d }^{ 1 }{ 6s }^{ 2 }\) 
\({ Ce }^{ 3+ }(58)-[Xe]{ 4f }^{ 1 }\) 
Spin only formula:
Magnetic moment 
\(=\sqrt { 4S(S+1) } =\sqrt { 4\times \frac { 1 }{ 2 } (\frac { 1 }{ 2 } +1) }\)  
\( \\ =\sqrt { 2\times \frac { 3 }{ 2 } } =\sqrt { 3 } =1.732B.M.\)
(i) It is because of strong metallic bonds due to a large number of unpaired electrons in d-orbitals.
(ii) It is because of transition metal i.e. 1 lowest oxidation state are more metallic and in higher oxidation state are least metallic, therefore oxides in lower oxidation state are basic whereas in higher oxidation state are amphoteric acidic.
(iii) Strong oxidizing agents provide energy for loss of one more electron from CO²⁺