(a) A drop of water, 0·4 cm in radius, is split up into 125 tiny drops. Find the increase in surface energy. [γwater (surface…

(a) Surface energy increase on splitting a drop Volume is conserved: (4/3)πR³ = 125 × (4/3)πr³, so r = R/5 = 0.4/5 = 0.08 cm. Initial area = 4πR² = 4π(0.16) = 2.011 cm². Final area = 125 × 4πr² = 125 × 4π(0.0064) = 10.053 cm². Increase in area ΔA = 8.042 cm². Increase in surface energy = γΔA = 72 × 8.042 = 579 ergs = 5.79 × 10⁻⁵ J (work done against surface tension in creating new surface).

(b) Ion-selective electrodes and the glass electrode An ion-selective electrode (ISE) is a membrane electrode whose potential responds selectively to the activity of one ion in solution (examples: glass for H⁺, LaF₃ crystal for F⁻, valinomycin membrane for K⁺, AgCl for Cl⁻). The membrane potential obeys the Nernst equation, E = const + (RT/zF) ln a_ion. Glass electrode: a thin bulb of special glass (about 72% SiO₂, 22% Na₂O, 6% CaO) encloses 0.1 M HCl with an internal Ag/AgCl wire. On soaking, both surfaces form hydrated gel layers in which H⁺ exchanges with Na⁺. Different H⁺ activities inside and outside produce a boundary potential across the membrane. E_G = E°_G + (2.303RT/F) log a_H⁺ = E°_G − 0.0591 pH at 25 °C. For measurement the glass electrode is paired with a reference electrode (saturated calomel or Ag/AgCl) in the test solution, giving a cell whose emf is measured on a high-impedance pH meter: E_cell = K − 0.0591 pH (K includes the reference and liquid junction potentials). It is calibrated with standard buffers (pH 4, 7, 9.2), and works over pH 1–10 with alkaline error at high pH.

(c)(i) Enthalpy of vaporisation from triple point to critical point Plotted against temperature, ΔHvap is highest at the triple point (Tp), decreases continuously as T rises, falls steeply near the critical point, and reaches zero at the critical temperature (Tc), where the curve ends vertically. Reason: ΔHvap = T(Vg − Vl)(dP/dT). As T rises the molar volumes of liquid and vapour converge, since the liquid expands and the vapour becomes denser, and the two phases become identical at Tc, so Vg − Vl → 0, and the latent heat vanishes. The Watson relation ΔH₂/ΔH₁ = [(1 − T_r2)/(1 − T_r1)]^0.38 describes the curve; Clausius–Clapeyron holds only for small ranges, where ΔHvap is nearly constant.

(c)(ii) Free expansion of an ideal gas The box is insulated and the gas expands into vacuum, so q = 0 and w = 0 (no external pressure). By the First Law, ΔU = 0. For an ideal gas U depends only on T, so the final temperature is T, the same as the initial one. Irreversibility: the system entropy change is ΔS = nR ln[(V₁ + V₂)/V₁] > 0, and as the surroundings exchange no heat, ΔS_surr = 0. Hence ΔS_universe > 0, so the process is spontaneous and irreversible. The gas never returns spontaneously to V₁; restoring it requires external work.