(a) Draw and discuss the polarographic wave obtained in the case of a dropping mercury electrode in polarography. (10 marks) (b)…

(a) Polarographic wave at a dropping mercury electrode (10 marks)

In polarography the potential applied to a dropping mercury electrode (DME, the microelectrode, connected as cathode) is varied slowly against a large non-polarisable reference (saturated calomel electrode), in an unstirred solution of the analyte containing an excess of inert supporting electrolyte, and the current is recorded. The current-potential curve is the polarogram.

Discussion of the wave:

• Residual current: at potentials before reduction begins, only a small current flows. It arises from the charging of the electrical double layer of the growing drops (the main part) and from trace impurities.
• Rising portion: when the potential reaches the reduction potential of the analyte, Mn+ + ne⁻ → M(Hg), the current rises steeply as the surface concentration of the analyte falls, and the rate is controlled by the electrode reaction and by diffusion.
• Plateau: at more negative potentials the surface concentration falls to zero, so the current is limited by the diffusion of analyte to the electrode. The limiting current less the residual current is the diffusion current i_d, and it is proportional to concentration: i_d = 607 n D^½ m^(2/3) t^(1/6) C (Ilkovic equation, average current in μA, D in cm² s⁻¹, m the mass flow of mercury in mg s⁻¹, t the drop time in s and C in mmol dm⁻³). This is the basis of quantitative analysis, by calibration or standard addition.
• Half-wave potential E½: the potential at which i = i_d/2. It is characteristic of the species and nearly independent of concentration, so it is used for qualitative analysis. For a reversible wave, E = E½ + (0.0592/n) log[(i_d − i)/i] at 25 °C, so a plot of E against log[(i_d − i)/i] is a straight line with slope 0.0592/n, giving n.
• The oscillations in the current follow the growth and fall of each drop, which renews the electrode surface every few seconds.
• Advantages of mercury: a wide cathodic range because of the high hydrogen overpotential, a reproducible fresh surface, and amalgam formation for metals. The excess supporting electrolyte removes the migration current, dissolved oxygen (which gives two waves of its own) is removed with nitrogen, and a maximum-suppressor such as gelatin removes current maxima.

(b)(i) Relaxation time for A ⇌ B + C (15 marks)

The forward step A → B + C has rate constant k_f and the reverse step B + C → A has rate constant k_r, so K = k_f/k_r.

After the temperature jump the system relaxes to its new equilibrium. Let x be the deviation of [A] from its new equilibrium value; then [B] and [C] deviate by −x. Then

dx/dt = −k_f([A]_eq + x) + k_r([B]_eq − x)([C]_eq − x).

At equilibrium, k_f[A]_eq = k_r[B]_eq[C]_eq, and dropping the term in x² (small deviation):

dx/dt = −[k_f + k_r([B]_eq + [C]_eq)] x,

so x decays exponentially with 1/τ = k_f + k_r([B]_eq + [C]_eq).

Here [B]_eq = [C]_eq = 2.0 × 10⁻⁴ mol dm⁻³, so [B]_eq + [C]_eq = 4.0 × 10⁻⁴ mol dm⁻³. Using k_f = K k_r:

1/τ = k_r (K + [B]_eq + [C]_eq) = k_r (2.0 × 10⁻¹⁶ + 4.0 × 10⁻⁴) ≈ 4.0 × 10⁻⁴ k_r,

because K is negligible beside the sum of concentrations. With τ = 3.0 μs, 1/τ = 3.33 × 10⁵ s⁻¹.

k_r = 3.33 × 10⁵/(4.0 × 10⁻⁴) = 8.3 × 10⁸ dm³ mol⁻¹ s⁻¹.

k_f = K k_r = 2.0 × 10⁻¹⁶ × 8.3 × 10⁸ = 1.7 × 10⁻⁷ s⁻¹.

(b)(ii) Parallel first-order reactions (10 marks)

B → D (k₁) and B → D′ (k₂) are parallel first-order reactions, so −d[B]/dt = (k₁ + k₂)[B], and [B] = [B]₀ e^(−(k₁+k₂)t).

With [B]₀ = 0.5 and [B] = 0.05 kmol m⁻³ at t = 40 min:

k₁ + k₂ = (1/t) ln([B]₀/[B]) = ln(10)/40 = 2.303/40 = 0.0576 min⁻¹ = 9.59 × 10⁻⁴ s⁻¹.

The products form in the ratio of the rate constants: [D′]/[D] = k₂/k₁. B reacted = 0.5 − 0.05 = 0.45, of which [D′] = 0.1, so [D] = 0.45 − 0.10 = 0.35 kmol m⁻³. Then k₂/k₁ = 0.10/0.35 = 2/7.

k₁ = 0.0576 × (0.35/0.45) = 0.0448 min⁻¹ = 7.5 × 10⁻⁴ s⁻¹.

k₂ = 0.0576 × (0.10/0.45) = 0.0128 min⁻¹ = 2.1 × 10⁻⁴ s⁻¹.

Check: k₁ + k₂ = 0.0448 + 0.0128 = 0.0576 min⁻¹.

(c) Variable order of a surface-catalysed gas-phase decomposition (15 marks)

Consider the decomposition of a gas A on a solid catalyst. A plausible (Langmuir-Hinshelwood) mechanism has three steps: (1) rapid reversible adsorption of A on a free site, A(g) + ⇌ A, with adsorption coefficient b; (2) unimolecular decomposition of the adsorbed molecule, A* → products, slow and rate-determining, with constant k; (3) desorption of the products.

The rate is proportional to the fraction of surface covered. By the Langmuir isotherm, θ = bP/(1 + bP), so

rate = kθ = k bP/(1 + bP).

• At low pressure or weak adsorption (bP ≪ 1), the rate = k b P: first order in the gas pressure, because the surface is sparsely covered and more gas means more adsorbed molecules.
• At high pressure or strong adsorption (bP ≫ 1), θ → 1 and the rate = k: zero order, since the surface is saturated and more gas cannot increase the rate. Examples are the decomposition of ammonia on tungsten and of hydrogen iodide on gold.
• At intermediate pressures the rate is fractional order (between 0 and 1), which is why the observed order changes with pressure.

If a product is also strongly adsorbed and blocks sites (for example H₂ in the decomposition of ammonia on platinum), the rate = k b_A P_A/(1 + b_A P_A + b_B P_B), and the order in the product is negative (retardation). The order is therefore not a fixed property of the reaction but depends on the surface coverage.