(a) The rate Law for the reaction N₂O₂(g) → 2NO(g) is of first order in the concentration of N₂O₂. Derive an expression for the…

(a) Let the concentration of N₂O₂ at time t be [N₂O₂] = x, and initially [N₂O₂]₀ = a. The given first-order rate law is

Rate = −d[N₂O₂]/dt = k[N₂O₂]

So,

−dx/dt = kx

Separating variables:

dx/x = −k dt

Integrating from x = a at t = 0 to x at time t:

∫(a to x) dx/x = −k ∫(0 to t) dt

ln(x/a) = −kt

x = a e^(−kt)

Thus, [N₂O₂] = a e^(−kt).

From the stoichiometry N₂O₂ → 2NO, the amount of N₂O₂ consumed is a − x, so the amount of NO formed is 2(a − x). If initially [NO]₀ = b,

[NO] = b + 2(a − a e^(−kt))

If initially no NO is present, b = 0, so:

[NO] = 2a(1 − e^(−kt))

As t → ∞, [NO] → 2a. This is valid for constant volume, isothermal conditions, no reverse reaction, and when the given first-order law holds. The unit of k is s⁻¹.

(b) In a Jablonski diagram, the vertical axis represents energy and the horizontal axis represents spin multiplicity. S₀, S₁, S₂ are singlet states; T₁, T₂ are triplet states. Radiative transitions are shown by straight arrows and non-radiative transitions by wavy arrows.

• (A) Allowed absorption: An upward straight arrow from S₀ to S₁ or S₂. It is a spin-allowed singlet → singlet transition, ΔS = 0. It is very fast, about 10⁻¹⁵ s, and gives strong absorption.

• (B) Fluorescence: A downward straight arrow from S₁ to S₀. It is radiative emission between states of same multiplicity, singlet → singlet. It occurs after vibrational relaxation and internal conversion to the lowest vibrational level of S₁. Lifetime is typically 10⁻⁹–10⁻⁷ s.

• (C) Phosphorescence: A downward straight arrow from T₁ to S₀. It is radiative emission between different multiplicities, triplet → singlet, hence spin-forbidden. It is slow, often 10⁻³–10 s, and is favoured in rigid media, at low temperature, or when heavy atoms increase spin-orbit coupling.

• (D) Internal Conversion (IC): A wavy arrow between electronic states of the same spin multiplicity, e.g. S₂ → S₁ or S₁ → S₀. It is non-radiative and often very fast. It converts electronic energy into vibrational energy.

• (E) Inter System Crossing (ISC): A wavy arrow between states of different multiplicity, usually S₁ → T₁. It is non-radiative and spin-forbidden, but becomes efficient when spin-orbit coupling is large.

For photochemical reaction, the favourable mode is usually absorption to an excited singlet, followed by internal conversion/vibrational relaxation to S₁, and then ISC to T₁ if spin-orbit coupling is significant. The triplet T₁ is long-lived, so it can undergo energy transfer, electron transfer, or bond-breaking before phosphorescence. If the reaction is very fast, the singlet S₁ itself may be the reactive state.

(c) Allosteric effect is the change in the activity of a protein at one site caused by the binding of a ligand at a different, non-overlapping site. The binding causes a conformational change that alters the affinity or catalytic activity of the active site. It is usually non-covalent, reversible, and important in regulation of enzymes and transport proteins.

An example of an allosteric protein is hemoglobin, which transports oxygen. Binding of O₂ to one heme increases the affinity of the remaining hemes for O₂, giving a sigmoidal oxygen-binding curve.

Homotropic allosteric modulators are modulators that are identical to the normal ligand or substrate. Their binding at one site affects binding at other identical sites. Usually they produce positive cooperativity. Examples:

• In hemoglobin, O₂ itself is a homotropic positive modulator: binding of O₂ to one subunit increases O₂ affinity of other subunits.
• In aspartate transcarbamoylase, aspartate is a homotropic positive modulator for its own binding and reaction.
• In phosphofructokinase-1, fructose-6-phosphate shows homotropic positive cooperativity; ATP can act as a negative homotropic modulator at high concentration.
• Negative homotropic effects are less common and appear as substrate inhibition at high substrate concentration.

(d) IUPAC names:

• [Ni(CO)₄]: tetracarbonylnickel(0)
• [Ni(CN)₄]⁴⁻: tetracyanonickelate(0)

For [Ni(CO)₄], CO is neutral, so Ni is in oxidation state zero. Ni(0) has d¹⁰ configuration. With four ligands, hybridization is sp³, giving a tetrahedral shape. The complex is diamagnetic. Structure: Ni at the centre bonded to four carbonyl carbon atoms, each Ni–C≡O group being linear at carbon.

For [Ni(CN)₄]⁴⁻, each CN⁻ has charge −1. Total ligand charge = −4, so:

x + 4(−1) = −4

x = 0

Thus Ni is again in oxidation state zero, d¹⁰. It is also sp³ hybridized and tetrahedral. Structure: Ni at the centre bonded to four cyanide carbon atoms, each Ni–C≡N linear at carbon. It is diamagnetic.

Perspective description: in both complexes, the four ligand donor atoms occupy the corners of a tetrahedron around Ni. Two ligands lie roughly in the plane, one projects toward the viewer, and one projects away. If the cyanide complex were [Ni(CN)₄]²⁻, Ni would be Ni(II), d⁸, and the shape would be square planar dsp²; however, for the species written as [Ni(CN)₄]⁴⁻, Ni is Ni(0), d¹⁰, and the shape is tetrahedral.

(e) The general synthetic route to lower cyclosiloxanes such as cyclic dimethylsiloxane trimer, (Me₂SiO)₃, is controlled hydrolysis and condensation of dimethyldichlorosilane.

Procedure:

1. Take Me₂SiCl₂ in an inert solvent such as dry ether or benzene.
2. Add water or dilute hydrochloric acid slowly with vigorous stirring at low temperature, about 0–10 °C.
3. Add a base such as pyridine or ammonium hydroxide to neutralize the HCl produced.
4. Hydrolysis gives dimethylsilanediol:

Me₂SiCl₂ + 2H₂O → Me₂Si(OH)₂ + 2HCl

5. The silanediol condenses with loss of water:

n Me₂Si(OH)₂ → (Me₂SiO)ₙ + n H₂O

Overall:

n Me₂SiCl₂ + n H₂O → (Me₂SiO)ₙ + 2n HCl

For the trimer, n = 3.

The crude product is a mixture of linear and cyclic dimethylsiloxanes, mainly cyclic trimer D₃, cyclic tetramer D₄, and higher cyclics. The lower cyclics are separated by fractional distillation or recrystallization. To increase the yield of lower cyclics, high dilution, low temperature, and an acid acceptor are used. Linear siloxanes can also be cracked with KOH at elevated temperature and redistributed to give more cyclic trimer and tetramer.

Structure of (Me₂SiO)₃: it is a six-membered ring with alternating Si and O atoms:

Si1–O–Si2–O–Si3–O–Si1

Each silicon atom carries two methyl groups. Thus the ring contains three Si–O–Si linkages and three dimethylsilyl units, giving the cyclic trimer. The ring is puckered rather than planar.

MeSiCl₃ is not used as the starting material for this synthesis because it is trifunctional. Hydrolysis of MeSiCl₃ gives MeSi(OH)₃, which can condense in three directions. This leads to branching, cross-linking, and three-dimensional silsesquioxane or resin-type networks of formula (MeSiO₁.₅)ₙ. It does not give discrete lower cyclosiloxanes like (Me₂SiO)₃. For cyclic dimethylsiloxane trimer, the required monomer is difunctional Me₂SiCl₂, so that each silicon has two methyl groups and forms only two siloxane bonds, allowing ring formation without cross-linking.