(a) (i) Determine the distance from the nucleus at which the electron is most expected in the hydrogen atom in its ground state…

(a)(i) Most probable distance of the electron in the H-atom ground state

For hydrogen, Z = 1 and n = 1, so ρ = 2r/a₀ and

R₁₀ = 2(1/a₀)^(3/2) e^(−r/a₀)

The probability of finding the electron in a thin spherical shell between r and r + dr is P(r) dr, where the radial probability density is P(r) = r²R₁₀². This is the same as 4πr²|ψ₁ₛ|², because ψ₁ₛ = R₁₀·Y₀₀ with Y₀₀ = 1/√(4π).

P(r) = (4/a₀³) r² e^(−2r/a₀)

For a maximum, dP/dr = 0:

dP/dr = (4/a₀³) [2r − (2r²/a₀)] e^(−2r/a₀) = (8r/a₀³)(1 − r/a₀) e^(−2r/a₀) = 0

• The solutions are r = 0 (where P = 0, a minimum), r → ∞ (P → 0), and r = a₀.
• P rises from zero, peaks and then decays, so r = a₀ is the maximum.

Most probable distance r = a₀ = 52.9 pm (0.529 Å), the radius of the first Bohr orbit.

Note the difference from Bohr's picture: the electron is not confined to this radius. It is only most likely to be found in a shell there. The average distance is ⟨r⟩ = 1.5 a₀, and |ψ|² itself is largest at the nucleus.

(a)(ii) Why NaBr and NaI have higher lattice energies than calculated

"Theoretical" lattice energies (Born–Landé or Born–Mayer, Kapustinskii) assume a purely ionic crystal of hard, spherical, non-polarisable point charges held only by electrostatic forces. The "experimental" values come from the Born–Haber cycle.

• NaF and NaCl: the two values agree closely, because these crystals are nearly ionic.

NaBr and NaI: the Born–Haber value is larger than the calculated one.

• Polarisation (Fajans' rules): Br⁻ and especially I⁻ are large, with diffuse, easily distorted electron clouds. The small Na⁺ cation distorts (polarises) them, drawing electron density into the region between the ions.
• Partial covalent character: this polarisation gives the bond some covalent character. The extra covalent bonding energy is not included in a purely electrostatic model, so the real lattice is held more strongly.
• Dispersion forces: London forces between the large, polarisable anions add a further (smaller) attraction that the simple ionic model omits.
• Trend: the discrepancy grows as the anion becomes larger and more polarisable (F⁻ < Cl⁻ < Br⁻ < I⁻). It is much larger still for salts of polarising cations such as Ag⁺ (AgI), where covalent character is strong.

(b)(i) Separation factor for one stage of UF₆ effusion

By Graham's law, the rate of effusion ∝ 1/√M.

• M(²³⁵UF₆) = 235.04 + 6(19.00) = 349.0 g mol⁻¹
• M(²³⁸UF₆) = 238.05 + 6(19.00) = 352.0 g mol⁻¹

The separation factor α is the ratio ²³⁵U/²³⁸U in the effused gas divided by the ratio in the feed:

α = rate(²³⁵UF₆)/rate(²³⁸UF₆) = √(352.0/349.0) = √1.00860 = 1.0043

So each stage enriches ²³⁵U only by about 0.43%. Fluorine has only one isotope (¹⁹F), so the whole mass difference comes from uranium, which is why UF₆ is used.

Number of stages: raising the abundance from 0.72% to 3% (reactor grade) needs about ln(4.26)/ln(1.0043), roughly 340 ideal stages. This is why gaseous-diffusion plants were enormous. Gas centrifuges, whose separation factor per stage is much larger, have now replaced them.

(b)(ii) Unit cell and the cubic Bravais lattices

A unit cell is the smallest repeating unit of a crystal lattice. Translating it along its three edges (a, b, c) reproduces the whole lattice. It is described by the edge lengths a, b, c and the interaxial angles α, β, γ. Of the 14 Bravais lattices, three belong to the cubic system (a = b = c; α = β = γ = 90°):

Simple (primitive) cubic, P:

• lattice points at the 8 corners only: 8 × 1/8 = 1 point per cell;
• coordination number 6; a = 2r; packing efficiency 52% (for example α-Po).

Body-centred cubic, I:

• corners plus one point at the body centre (½, ½, ½): 2 per cell;
• coordination number 8; √3a = 4r; packing efficiency 68% (for example Na, K, α-Fe).

Face-centred cubic, F:

• corners plus the centres of all six faces: 1 + 6 × ½ = 4 per cell;
• coordination number 12; √2a = 4r; packing efficiency 74% (for example Cu, Al, Au, NaCl lattice).

(c) Normal boiling point of mercury

Hg(l) ⇌ Hg(g). At the normal boiling point, liquid and vapour are in equilibrium at 1 atm (about 1 bar, the standard state), so ΔG°vap = 0 and T_b = ΔH°vap/ΔS°vap.

• ΔH°vap = ΔH°f(g) − ΔH°f(l) = 60.78 − 0 = 60.78 kJ mol⁻¹ = 60 780 J mol⁻¹
• ΔS°vap = S°(g) − S°(l) = 174.7 − 77.4 = 97.3 J K⁻¹ mol⁻¹
• T_b = 60 780/97.3 = 624.7 K ≈ 351.5 °C

The experimental normal boiling point is 629.9 K (356.7 °C), so the estimate is within about 1%.

Assumptions

1. ΔH°vap and ΔS°vap, taken from 298 K data, do not change with temperature up to about 625 K (that is, ΔCp ≈ 0). This is the main source of error.
2. Mercury vapour behaves as an ideal (monatomic) gas.
3. The liquid and vapour are pure, and the difference between 1 bar and 1 atm is negligible.

(A useful check: ΔS°vap = 97.3 J K⁻¹ mol⁻¹ is close to Trouton's value of about 88 J K⁻¹ mol⁻¹.)

(d)(i) Will CCl₂F₂ condense at 20 °C and 18 atm?

• Known points on the liquid–vapour line: the normal boiling point (−30 °C, 1 atm) and the critical point (112 °C, 40 atm). The line joins them.
• Estimating the vapour pressure at 20 °C (Clausius–Clapeyron): ln P is linear in 1/T. The slope is −ΔH/R = ln(40/1)/(1/385.15 − 1/243.15) ≈ −2430 K. At 293.15 K this gives ln P ≈ 1.71, so P ≈ 5.5 atm (the measured value is about 5.7 bar). ΔH/R ≈ 2430 K corresponds to ΔH_vap ≈ 20 kJ mol⁻¹.
• Locating the state: 20 °C is well below the critical temperature (112 °C), so the gas can be liquefied by pressure alone. The point (20 °C, 18 atm) lies well above the vapour-pressure curve, in the liquid region.
• Isothermal compression at 20 °C: condensation begins at about 5.5 atm. The pressure then stays constant until all the gas has liquefied, after which the liquid is compressed to 18 atm.

So yes, the gas condenses (it will be liquid at 18 atm). This is how CCl₂F₂ (the refrigerant "Freon-12", R-12) worked in refrigerators, before it was phased out under the Montreal Protocol for depleting ozone. Above 112 °C, by contrast, no pressure could liquefy it (the supercritical region).

(d)(ii) Overvoltage and its applications

Overvoltage (overpotential, η) is the extra potential, beyond the reversible (equilibrium) electrode potential, that must be applied to make an electrode reaction proceed at a measurable rate: η = E(applied, at current i) − E(reversible).

Causes

• Activation overvoltage: a slow electron-transfer step, or slow formation of H or O atoms and their recombination into gas molecules. It follows the Tafel equation η = a + b log i.
• Concentration overvoltage: depletion of the reacting species at the electrode.
• Resistance (ohmic) overvoltage: films on the electrode.

What it depends on: the electrode material and surface, the current density, temperature and the gas evolved. Hydrogen overvoltage is almost zero on platinised platinum but about 0.8–1 V on mercury, and it is high on lead and zinc. Cathodic overvoltage applies to reduction, anodic overvoltage to oxidation.

Applications

1. Electrodeposition of active metals: zinc, cadmium, nickel and chromium can be electroplated or electrowon from aqueous acid solution, although their E° is more negative than that of H⁺/H₂. The high hydrogen overvoltage on these metals suppresses H₂ evolution, so the metal is deposited instead.
2. Chlor-alkali (Castner–Kellner) mercury cell: because of hydrogen's high overvoltage on mercury, Na⁺ is discharged to form sodium amalgam instead of H₂. The amalgam later reacts with water to give NaOH. (This process is now being phased out because of mercury pollution.)
3. Lead–acid battery: the high overvoltage for H₂ on lead and for O₂ on PbO₂ allows the cell to be charged to about 2 V without decomposing the water. This limits gassing and self-discharge.
4. Polarography and voltammetry: the dropping mercury electrode can reduce many metal ions and organic compounds that would otherwise be masked by hydrogen evolution.
5. Electro-organic synthesis and corrosion control:

• the choice of electrode material (high or low overvoltage) steers which product forms;
• low-overvoltage metals in contact with a metal speed up its corrosion, whereas high-overvoltage coatings slow it.

6. Water electrolysers and fuel cells: in contrast, low-overvoltage catalysts (Pt, Ni, IrO₂) are chosen to save energy.

(e) Temperature at which the uncatalysed reaction matches the catalysed rate at 20 °C

The Arrhenius equation is k = A e^(−Ea/RT). With the same A, equal rate constants require equal exponents:

Ea(uncat)/(R T) = Ea(cat)/(R × 293.15 K)

T = 293.15 × Ea(uncat)/Ea(cat) = 293.15 × (42/7.0) = 293.15 × 6 = 1759 K ≈ 1486 °C

(The same answer is obtained by taking natural logarithms of k_uncat(T) = k_cat(293 K); the units kJ mol⁻¹ cancel in the ratio.)

Comment:

• Catalase lowers Ea by 35 kJ mol⁻¹. At 20 °C this speeds the reaction by a factor of e^(35000/(8.314 × 293.15)) ≈ e^14.4, about 1.7 × 10⁶.
• Heating alone would have to reach about 1760 K to match it, a temperature at which water and the enzyme could not exist. This shows how powerfully enzymes work at body temperature.
• Assumptions: A and Ea are independent of temperature, and the mechanism does not change.