Clausius–Clapeyron applicability: For deriving/using the Clausius–Clapeyron equation in its common integrated form for liquid–vapour systems, which assumptions are made?

Introduction / Context:The Clausius–Clapeyron equation relates vapour pressure to temperature via the latent heat of phase change. Its commonly used integrated form assumes certain simplifications that make the mathematics tractable and the result accurate over moderate temperature ranges.

Given Data / Assumptions:

• Liquid–vapour equilibrium for a pure substance.
• Moderate pressures, away from the critical point.
• Latent heat treated as weakly temperature dependent over the interval of interest.

Concept / Approach:Starting from the Clapeyron equation, dP/dT = ΔHvap / (T ΔV), the integrated form commonly used is ln P = −ΔHvap/R * (1/T) + constant. To reach this, two main assumptions are made: (1) the vapour behaves ideally so that Vvap ≈ RT/P, and (2) the liquid molar volume is negligible compared to the vapour volume so that ΔV ≈ Vvap. These lead directly to the integrated logarithmic relation.

Step-by-Step Solution:Begin with dP/dT = ΔHvap /(T (Vvap − Vliq)).Assume Vliq ≪ Vvap ⇒ ΔV ≈ Vvap.Assume ideal vapour ⇒ Vvap = RT/P.Substitute and integrate to obtain ln P vs 1/T.

Verification / Alternative check:Comparing with Antoine correlations shows good agreement over limited temperature ranges, validating the assumptions for many substances away from extremes.

Why Other Options Are Wrong:Choosing only one assumption yields a less accurate or non-integrable simple form; “neither” is incorrect; the alternative in option e does not produce the standard integrated expression.

Common Pitfalls:Applying the integrated form near the critical region or at high pressures where the assumptions break down.

Final Answer:both (a) & (b)