MChem Master of Chemistry Master of Chemistry: Physical Chemistry 5 — Questions and Answers
Question 1: The Clausius-Clapeyron equation dp/dT = ΔH_vap/(TΔV) describes the liquid-vapor coexistence curve. Assuming ideal gas and negligible liquid volume, the equation integrates to give ln(p₂/p₁) proportional to:
- 1/T₂ − 1/T₁
- T₂ − T₁
- (1/T₁ − 1/T₂) (Correct answer)
- ln(T₂/T₁)
Correct answer: (1/T₁ − 1/T₂)
Integration yields ln(p₂/p₁) = −ΔH_vap/R · (1/T₂ − 1/T₁) = ΔH_vap/R · (1/T₁ − 1/T₂).
Question 2: Henry's law states that the partial pressure of a volatile solute above a dilute solution is p_i = K_H · x_i. In contrast, Raoult's law applies to:
- Solutes at very low mole fractions
- The solvent (or any component in the limit x_i → 1) (Correct answer)
- Non-volatile solutes only
- Electrolyte solutions exclusively
Correct answer: The solvent (or any component in the limit x_i → 1)
Raoult's law (p_i = x_i · p*_i) is obeyed by the solvent and by any component as its mole fraction approaches unity.
Question 3: The diffusion coefficient D of a spherical particle in solution is given by the Stokes-Einstein equation D = k_BT/(6πηr). If the solvent viscosity η doubles at lower temperature while T decreases by 10%, D:
- Increases slightly
- Decreases significantly (Correct answer)
- Remains essentially unchanged
- Increases proportionally to T
Correct answer: Decreases significantly
D ∝ T/η; if η doubles and T falls ~10%, D decreases by roughly half (D ≈ 0.9T / 2η ≈ 0.45× original), a significant decrease.
Question 4: In Density Functional Theory (DFT), the Hohenberg-Kohn theorem proves that the ground-state energy of a many-electron system is a unique functional of:
- The many-body wavefunction Ψ
- The electron density ρ(r) (Correct answer)
- The external potential V_ext alone
- The kinetic energy operator
Correct answer: The electron density ρ(r)
The first Hohenberg-Kohn theorem establishes a one-to-one mapping between the ground-state electron density ρ(r) and the external potential, making all ground-state properties functionals of ρ(r).
Question 5: In NMR spectroscopy, the chemical shift δ (in ppm) is defined relative to a reference compound (TMS). A nucleus with greater electron density around it will resonate at:
- Higher δ (downfield)
- Lower δ (upfield) (Correct answer)
- The same δ regardless of shielding
- A δ that depends only on the gyromagnetic ratio
Correct answer: Lower δ (upfield)
Greater electron density increases diamagnetic shielding, reducing the effective field at the nucleus and shifting the resonance upfield (lower δ).
Question 6: The Debye-Hückel limiting law for activity coefficients in dilute electrolyte solutions gives log γ± = −A|z₊z₋|√I, where I is the ionic strength. For a 1:1 electrolyte at concentration c, the ionic strength I equals:
- 2c
- c/2
- c (Correct answer)
- c²
Correct answer: c
For a 1:1 electrolyte, I = ½Σcᵢzᵢ² = ½(c·1² + c·1²) = c.
Question 7: The surface tension γ of a liquid arises from imbalanced intermolecular forces at the interface. The Young-Laplace equation ΔP = γ(1/R₁ + 1/R₂) gives the pressure difference across a curved interface. For a spherical bubble of radius r inside a liquid, ΔP equals:
- γ/r
- 2γ/r (Correct answer)
- 4γ/r
- γ/(2r)
Correct answer: 2γ/r
A bubble has one liquid-gas interface; with R₁ = R₂ = r, ΔP = 2γ/r (compared to 4γ/r for a soap film with two interfaces).
MChem Master of Chemistry Master of Chemistry: Physical Chemistry 5
The Clausius-Clapeyron equation dp/dT = ΔH_vap/(TΔV) describes the liquid-vapor coexistence curve.
Assuming ideal gas and negligible liquid volume, the equation integrates to give ln(p₂/p₁) proportional to: