

\( 3\text{NiS} + 2\text{HNO}_3 + 6\text{HCl} \rightarrow 3\text{NiCl}_2 + 2\text{NO} + 3\text{S} + 4\text{H}_2\text{O} \)
The problem asks for the number of protons that are not involved in hydrogen bonding in the final product B, which is formed through a sequence of two reactions.
1. Redox Reaction: The first reaction involves Nickel(II) sulfide (NiS) reacting with nitric acid (HNO₃) and hydrochloric acid (HCl). This is a redox reaction where nitric acid acts as an oxidizing agent. The products are determined by the oxidation of sulfide and the reduction of nitrate.
2. Coordination Chemistry (Qualitative Analysis): The second reaction is a characteristic test for the Nickel(II) ion (\(Ni^{2+}\)). \(Ni^{2+}\) reacts with dimethylglyoxime (DMG), \(H_3C-C(=NOH)-C(=NOH)-CH_3\), in an ammoniacal (basic) medium to form a stable, brightly colored coordination complex.
3. Hydrogen Bonding: This is an electrostatic attraction between a hydrogen atom covalently bonded to a highly electronegative atom (like O, N, F) and another nearby electronegative atom. Protons attached to carbon atoms (C-H bonds) are generally considered non-polar and do not participate in conventional hydrogen bonding.
4. Structure of Nickel(II) dimethylglyoximate: The complex formed, \(Ni(DMG)_2\), has a specific square planar geometry with intramolecular hydrogen bonds, which must be analyzed to count the different types of protons.
Step 1: Identify the product A from the first reaction.
The first reaction is:
\[ NiS + HNO_3 + HCl \longrightarrow A + NO + S + H_2O \]In this reaction, the sulfide ion (\(S^{2-}\)) in NiS is oxidized to elemental sulfur (S). The nitrate ion (\(NO_3^-\)) from nitric acid is reduced to nitric oxide (NO). The nickel ion remains in its +2 oxidation state and, in the presence of HCl, forms nickel(II) chloride. The balanced reaction is:
\[ 3NiS + 2HNO_3 + 6HCl \longrightarrow 3NiCl_2 + 2NO + 3S + 4H_2O \]Therefore, the compound A is Nickel(II) chloride, \(NiCl_2\).
Step 2: Identify the product B from the second reaction.
The second reaction is:
\[ A + NH_4OH + \text{Dimethylglyoxime} \longrightarrow B + NH_4Cl + H_2O \]Substituting A = \(NiCl_2\), we have the reaction of Nickel(II) chloride with dimethylglyoxime (DMG) in an ammoniacal solution (\(NH_4OH\)). This is a classic confirmatory test for \(Ni^{2+}\), which forms a stable, bright red precipitate of Nickel(II) dimethylglyoximate. The overall reaction is:
\[ NiCl_2 + 2(C_4H_8N_2O_2) + 2NH_4OH \longrightarrow Ni(C_4H_7N_2O_2)_2 + 2NH_4Cl + 2H_2O \]Thus, the product B is the complex Nickel(II) dimethylglyoximate, often written as \(Ni(DMG)_2\). Its chemical formula is \(C_8H_{14}N_4NiO_4\).
Step 3: Analyze the structure of product B, \(Ni(DMG)_2\).
In the complex \(Ni(DMG)_2\), the central \(Ni^{2+}\) ion is coordinated to two dimethylglyoximate ligands. Each ligand is bidentate, bonding to the nickel ion through its two nitrogen atoms. This results in a square planar geometry around the nickel atom. The structure is stabilized by two strong intramolecular O-H···O hydrogen bonds between the two ligands.
Step 4: Identify and count the different types of protons in the structure of B.
The structure of \(Ni(DMG)_2\) contains two types of hydrogen atoms (protons):
We need to find the number of protons that do not involve in hydrogen bonding.
Total number of methyl protons = (Number of methyl groups) × (Protons per group)
\[ \text{Number of protons not involved in H-bonding} = 4 \times 3 = 12 \]The number of protons that do not involve in hydrogen bonding in the product B is 12.

Let \( f : \mathbb{R} \to \mathbb{R} \) be a twice differentiable function such that \[ (\sin x \cos y)(f(2x + 2y) - f(2x - 2y)) = (\cos x \sin y)(f(2x + 2y) + f(2x - 2y)), \] for all \( x, y \in \mathbb{R}. \)
If \( f'(0) = \frac{1}{2} \), then the value of \( 24f''\left( \frac{5\pi}{3} \right) \) is: