(a) (i) Calculate the number of moles of HCl (g) produced by the absorption of one Joule of radiant energy of wavelength 480 nm…

(a)(i) Moles of HCl formed by 1 J of 480 nm light (φ = 1.0 × 10⁶)

Energy of one photon

E = hc/λ = (6.626 × 10⁻³⁴ J s × 2.998 × 10⁸ m s⁻¹)/(480 × 10⁻⁹ m) = 4.138 × 10⁻¹⁹ J

Per mole of photons (one einstein) this is 4.138 × 10⁻¹⁹ × 6.022 × 10²³ = 249 kJ mol⁻¹. That is just above the Cl–Cl bond energy (242 kJ mol⁻¹), so 480 nm light can dissociate Cl₂.

Photons in 1 J

• N = 1 J/4.138 × 10⁻¹⁹ J = 2.416 × 10¹⁸ photons
• In moles: 2.416 × 10¹⁸/6.022 × 10²³ = 4.01 × 10⁻⁶ einstein

Applying the quantum yield. φ = (molecules that react)/(photons absorbed) = 1.0 × 10⁶.

• Amount of H₂ (and of Cl₂) reacting = 4.01 × 10⁻⁶ × 1.0 × 10⁶ = 4.01 mol
• Since H₂ + Cl₂ → 2HCl, moles of HCl = 2 × 4.01 = 8.0 mol
• In one line: n(HCl) = 2φλQ/(hcN_A), with Q = 1 J

(If the quantum yield is instead quoted per molecule of HCl formed, the answer is 4.0 mol. The definition used must therefore be stated.)

Why φ is so large: a chain reaction

• Initiation: Cl₂ + hν → 2Cl·
• Propagation: Cl· + H₂ → HCl + H·, then H· + Cl₂ → HCl + Cl·
• Termination: 2Cl· + M → Cl₂ + M (on the walls or with a third body)

Each photon starts a chain that regenerates Cl·, so a single photon can produce around 10⁵–10⁶ HCl molecules.

(a)(ii) Molar extinction coefficient

Beer–Lambert law: A = log₁₀(I₀/I) = εcl.

• I/I₀ = 0.50, so A = log₁₀ 2 = 0.301
• ε = A/(cl) = 0.301/(0.04 mol L⁻¹ × 2 cm) = 3.76 L mol⁻¹ cm⁻¹ (0.376 m² mol⁻¹ in SI units)

This assumes monochromatic light, a dilute non-associating solute, and negligible reflection or scattering. The small value shows a weak absorption band. If ε is defined with natural logarithms (the Napierian coefficient), it is 2.303 times larger, 8.66 L mol⁻¹ cm⁻¹.

(b) Heterogeneous catalysis; Langmuir kinetics of PH₃ decomposition on W

Heterogeneous catalysis means the catalyst is in a different phase from the reactants: usually a solid (metal or oxide) with gaseous or liquid reactants. The reaction occurs at active sites on the catalyst surface. Examples are the Haber process (Fe), the contact process (V₂O₅) and catalytic converters (Pt/Pd/Rh).

Consecutive steps

1. Diffusion of reactants from the bulk fluid to the external surface of the catalyst.
2. Diffusion through the pores to the active sites.
3. Adsorption (usually chemisorption) of the reactant(s) on the sites.
4. Surface reaction of the adsorbed species. This may be between two adsorbed species (Langmuir–Hinshelwood mechanism) or between an adsorbed species and a gas molecule (Eley–Rideal mechanism).
5. Desorption of the products.
6. Diffusion of the products out of the pores and back to the bulk fluid.

The slowest of these steps controls the overall rate. For PH₃ on W, the surface reaction (step 4) is rate-determining.

Langmuir treatment

The assumptions are:

• a fixed number of equivalent sites;
• at most a monolayer;
• no interaction between adsorbed molecules;
• adsorption–desorption equilibrium is fast compared with the surface reaction.

At equilibrium, k_a P(1 − θ) = k_d θ. With K = k_a/k_d this gives:

θ = KP/(1 + KP)

The adsorbed PH₃ decomposes unimolecularly (4PH₃ → P₄ + 6H₂ overall), so the rate is proportional to the covered fraction:

rate = kθ = kKP/(1 + KP)

• Low pressure (KP ≪ 1): θ ≈ KP, so rate ≈ kKP. The rate is proportional to P, which is first order. Only a small fraction of the sites is covered, so doubling P doubles the coverage and the rate.
• High pressure (KP ≫ 1): θ → 1, so rate ≈ k, independent of P, which is zero order. The surface is saturated, and extra gas molecules simply have no free sites.
• Intermediate pressures: the rate follows kKP/(1 + KP), with a fractional order between 0 and 1.

This is exactly what Hinshelwood observed for PH₃ on tungsten. A further consequence: at low pressure the rate constant is kK, so the apparent activation energy is E_app = E_true + ΔH_ads. Since adsorption is exothermic, E_app is lower than the true activation energy of the surface step. (NH₃ on W, which covers the surface strongly, is zero order even at low pressure.)

(c)(i) The sodium pump

The sodium pump is the Na⁺/K⁺-ATPase, an integral P-type ATPase of the plasma membrane of all animal cells. J. C. Skou discovered it in 1957 and received the Nobel Prize for it in 1997.

Mechanism: active transport (the Post–Albers E1/E2 cycle)

1. In the E1 form the pump binds 3 Na⁺ from the cytosol.
2. ATP phosphorylates an aspartate residue (with Mg²⁺ required).
3. The pump changes to the E2 form, which opens outwards and releases the 3 Na⁺ outside.
4. E2-P binds 2 K⁺ from outside.
5. Dephosphorylation returns the pump to E1, which releases the 2 K⁺ inside.

Net per ATP hydrolysed: 3 Na⁺ out and 2 K⁺ in, both against their concentration gradients (outside Na⁺ ≈ 145 mM and K⁺ ≈ 4 mM; inside Na⁺ ≈ 12 mM and K⁺ ≈ 140 mM). Because one net positive charge leaves per cycle, the pump is electrogenic.

Roles in biological systems

1. Membrane potential and excitability. The K⁺ gradient it creates, together with K⁺ leak channels, generates the resting potential of about −70 mV, and the pump adds a few millivolts directly. After action potentials it restores the Na⁺ and K⁺ gradients, so nerve impulses and muscle contraction depend on it.
2. Cell volume. Pumping 3 ions out for every 2 in counteracts the osmotic influx of water (the pump–leak balance). When the pump is inhibited, cells swell and burst.
3. Secondary active transport. The inward Na⁺ gradient drives:

• uptake of glucose (SGLT1 in the intestine and kidney) and amino acids;
• Na⁺/H⁺ exchange (intracellular pH);
• Na⁺/Ca²⁺ exchange (removal of Ca²⁺ from heart muscle).

4. Kidney function. The basolateral pumps of the tubules power the reabsorption of Na⁺, water and solutes.
5. Energy cost. The pump consumes about 20–25% of the resting ATP of the body, and far more in the brain.
6. Pharmacology. Cardiac glycosides (ouabain, digoxin) inhibit the pump. The resulting rise in intracellular Na⁺ slows Na⁺/Ca²⁺ exchange, raises intracellular Ca²⁺ and strengthens heart contraction, which is how they treat heart failure.

(c)(ii) The Bohr effect; the role of cytochrome P450

Bohr effect (C. Bohr, 1904). A fall in pH, or a rise in pCO₂, lowers the O₂ affinity of haemoglobin and shifts the O₂-dissociation curve to the right (P₅₀ increases).

• Mechanism. H⁺ protonates His146β, which then forms a salt bridge with Asp94β. CO₂ reacts with the N-terminal amino groups to form carbamates. Both effects stabilise the low-affinity T (deoxy) state.
• Physiological role. Actively respiring tissues produce CO₂ and lactic acid, so Hb there releases more O₂. At 40 mmHg, saturation drops from about 75% to about 59% in the example shown, delivering roughly 16% more of haemoglobin's O₂ capacity. In the lungs, loss of CO₂ raises the pH, and O₂ loading is favoured.
• The Haldane effect is the reciprocal: deoxygenated Hb binds more H⁺ and CO₂. Together the two effects couple O₂ delivery to CO₂ removal.

Cytochrome P450 (CYP) is a superfamily of heme-thiolate monooxygenases: the heme iron carries a cysteine thiolate as its axial ligand. Humans have 57 CYP genes.

• Location: mainly the endoplasmic reticulum (the microsomal fraction) of liver cells, and the mitochondria of the adrenal cortex and gonads.
• Name: the reduced Fe(II)–CO complex absorbs strongly at 450 nm (Omura and Sato, 1964).

Reaction (monooxygenation):

RH + O₂ + NADPH + H⁺ → ROH + H₂O + NADP⁺

One O atom of O₂ goes into the substrate and the other into water. Electrons come from NADPH through NADPH–cytochrome P450 reductase.

Catalytic cycle

1. The substrate binds, and the heme Fe(III) is reduced to Fe(II).
2. O₂ binds, and a second electron and protons are added.
3. The O–O bond breaks and water leaves, giving Compound I, a high-valent Fe(IV)=O porphyrin radical cation.
4. Compound I abstracts an H atom from the substrate, and "oxygen rebound" gives ROH.

Roles

1. Phase I metabolism of drugs and xenobiotics. P450 carries out hydroxylation, N- and O-dealkylation, epoxidation and similar reactions. These make molecules more polar, ready for conjugation and excretion. CYP3A4 alone handles about half of all prescribed drugs.
2. Examples:

• codeine → morphine (CYP2D6);
• paracetamol → the toxic metabolite NAPQI in overdose (CYP2E1);
• activation of pro-carcinogens such as benzo[a]pyrene (CYP1A1).

3. Drug interactions: rifampicin and phenobarbital induce CYPs, while grapefruit juice and ketoconazole inhibit CYP3A4. Blood levels of co-administered drugs change as a result.
4. Biosynthesis:

• steroid hormones (cholesterol side-chain cleavage by CYP11A1; aromatase, CYP19, which makes oestrogens);
• bile acids;
• activation of vitamin D (25- and 1α-hydroxylation);
• eicosanoid metabolism.