Clausius–Clapeyron relation in phase equilibrium:\nWhich mathematical form correctly represents the Clausius–Clapeyron equation connecting saturation pressure and temperature?

Introduction / Context:
The Clausius–Clapeyron equation is a cornerstone of phase equilibrium thermodynamics. It relates how the saturation pressure of a pure substance changes with temperature along a phase boundary (for example, liquid–vapor). Recognizing its correct mathematical form helps in estimating heats of phase change and in understanding why boiling points shift with pressure in distillation, evaporation, and refrigeration.

Given Data / Assumptions:

• One-component system undergoing a reversible phase change at equilibrium.
• ΔH denotes molar enthalpy change for the phase transition (e.g., ΔHvap).
• ΔV is the molar volume difference between phases (V2 − V1).
• Temperature T and saturation pressure P lie on the coexistence curve.

Concept / Approach:
The Clausius–Clapeyron relation is derived from equality of Gibbs free energies of coexisting phases and from the Maxwell relations. It yields a differential connection between P and T along the phase boundary: dP/dT = ΔH / (T * ΔV). For vaporization, ΔV ≈ Vvapor and ΔH = ΔHvap, leading to familiar integrated forms under simplifying assumptions (e.g., ln P vs 1/T lines when vapor behaves ideally and ΔHvap is weakly temperature dependent).

Step-by-Step Solution:
Identify the target: the equation that gives slope of the coexistence curve.Recall the exact differential: dP/dT = ΔH / (T * ΔV).Compare options: only option (a) matches the fundamental relation.Select option (a) as correct.

Verification / Alternative check:
If one assumes ideal-gas vapor and negligible liquid volume, integration produces ln P = −ΔHvap/R * (1/T) + constant, a widely used empirical fit, which is consistent with the differential form in option (a).

Why Other Options Are Wrong:
(b) is the van der Waals equation of state, not a phase boundary slope.(c) is the virial equation expansion; it models non-ideal gases but not the P–T slope of coexistence.(d) is an oversimplified linear law; vapor pressure is not linear in T across wide ranges.

Common Pitfalls:
Confusing equations of state with phase-boundary relations; forgetting ΔV in condensed–condensed transitions is tiny, making slopes very steep; misapplying integrated forms outside their validity.

Final Answer:
dP/dT = ΔH / (T * ΔV)