Determining latent heat from vapor-pressure data: If the vapor pressures at two temperatures are known for a solid in equilibrium with its liquid, which relation allows you to calculate the latent heat of fusion?

Introduction:Phase-equilibrium thermodynamics connects temperature, pressure, and latent heats. When experimental vapor-pressure data are available at two temperatures for a solid–liquid system, the appropriate relation can be used to back-calculate the latent heat of fusion, essential for solidification and melting process design.

Given Data / Assumptions:

• Two equilibrium points (P1, T1) and (P2, T2) for the same substance.
• Solid–liquid equilibrium considered via the vapor-pressure curve (near melting).
• Latent heat is temperature-insensitive over the interval (engineering approximation).

Concept / Approach:The Clapeyron–Clausius equation in integrated form relates slopes of coexistence curves to latent heats: d ln P / d(1/T) = − ΔH / R for phase changes involving vapor. With two data points, ΔH (here, fusion or sublimation-related depending on the path) can be estimated. For melting, one often uses the Clapeyron form involving ΔV and ΔH; with vapor-pressure data, the Clausius–Clapeyron approximation is common.

Step-by-Step Solution:Write Clausius–Clapeyron: ln(P2/P1) = −ΔH/R * (1/T2 − 1/T1).Solve for ΔH using known P1, P2, T1, T2.Interpret ΔH as the latent heat relevant to the phase transition considered.

Verification / Alternative check:Consistency can be checked by computing ΔH from multiple temperature intervals; results should agree within experimental uncertainty.

Why Other Options Are Wrong:

• Maxwell, Van Laar, Gibbs–Duhem, Nernst: Important relations, but not used directly to extract latent heat from vapor-pressure vs temperature data.

Common Pitfalls:Using Celsius instead of Kelvin; failing to ensure pressure units cancel in the logarithm.

Final Answer:Clapeyron–Clausius equation