(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 tension of water) = 72 dynes/cm ] (10 marks)

(b) What are ion-selective electrodes? How is glass electrode used in the dete

Question

(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 tension of water) = 72 dynes/cm ] (10 marks)

(b) What are ion-selective electrodes? How is glass electrode used in the determination of pH of a given solution? (20 marks)

(c) (i) Draw and explain the graph of enthalpy of vapourization from the triple point (Tp) to the critical point (Tc). (10 marks)
(ii) A thermally insulated box is separated into two compartments (volumes V₁ and V₂) by a membrane. One of the compartments contains an ideal gas at temperature T, the other is empty (vacuum). The membrane is suddenly removed, and the gas fills up the compartments and reaches the equilibrium. What is the final temperature of the gas? Show that the gas expansion process is irreversible. (10 marks)

UPSC Answer Check · Accepted Answer

Surface phenomena, electrode potentials and phase thermodynamics all express how molecular interactions create macroscopic energy and potential differences.

Part (a) Let the large drop radius be R = 0.4 cm and the number of small drops n = 125. Volume is conserved: (4/3)πR³ = 125(4/3)πr³, so r = R/5 = 0.08 cm. Initial area A_i = 4πR². Final area A_f = 125 × 4πr² = 125 × 4π(R/5)² = 5 × 4πR². Hence ΔA = A_f − A_i = 16πR². Surface energy increase ΔE = γΔA = 72 dyn cm⁻¹ × 16π(0.4 cm)² = 184.32π erg = 5.79 × 10² erg. Since 1 erg = 10⁻⁷ J, ΔE ≈ 5.8 × 10⁻⁵ J, about 58 μJ. This is the work needed to create new liquid-vapour surface; smaller drops have more area per unit volume, so surface energy rises.

Part (b) Ion-selective electrodes are membrane electrodes whose potential responds selectively to the activity of one ion. The glass electrode is the standard H⁺ electrode. It has a thin ion-exchange glass membrane, an internal buffer of fixed H⁺ activity, and an Ag/AgCl internal reference wire. On contact with the test solution, a hydrated gel layer forms; H⁺ exchanges at the outer surface while the inner surface remains fixed, producing a membrane potential that depends on external H⁺ activity. For a reversible response, E = E° + (RT/F) ln a_H⁺ = E° − (2.303RT/F) pH; at 25°C, E = E° − 0.0591 pH. In a pH cell it is paired with a reference electrode, such as calomel, and calibrated with standard buffers, commonly pH 4, 7 and 10; temperature compensation is applied because the Nernst slope changes with T. The measured emf is therefore converted to pH, linking electrochemical potential to the chemical potential of H⁺.

Part (c)(i) A graph of ΔH_vap against temperature has T on the x-axis and ΔH_vap on the y-axis. The x-axis should show 0, then T_p, then T_c; the curve starts at T_p, not at zero temperature, with a finite, near-maximum positive value. Along the liquid-vapour coexistence line, as T rises, intermolecular attractions become less important relative to thermal motion, so the energy required to vaporise one mole falls. The curve declines smoothly and reaches zero at T_c, where liquid and vapour densities become identical and no latent heat is needed. Its shape is gently concave downward. Trouton’s rule is relevant as context: at the normal boiling point many liquids have roughly constant entropy of vaporisation, ΔS_vap = ΔH_vap/T_b, but it does not assume constant latent heat; here ΔH_vap itself falls to zero at T_c.

Part (c)(ii) The gas expands into a vacuum in an insulated box, so q = 0. The opposing external pressure is zero, so w = 0 and ΔU = q + w = 0. For an ideal gas, internal energy depends only on temperature; therefore T_final = T_initial. The final pressure is lower, but the temperature is unchanged. The process is irreversible because the entropy of the gas increases: ΔS = nR ln(V_final/V_initial) = nR ln[(V₁ + V₂)/V₁] > 0. Since the box is isolated, this is the total entropy production, so the process cannot be reversed without changing the surroundings. This is free expansion, not throttling; real gases may show a small temperature change in free expansion, while Joule-Thomson cooling belongs to a separate constant-enthalpy throttling process.

Conclusion: The surface-energy calculation, the Nernstian glass-electrode response, and the thermodynamic limits of vaporisation and free expansion together show how molecular-scale interactions determine measurable energy, potential and phase behaviour.

(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…

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