At the boiling point, a liquid is in equilibrium with its vapor. On average, which energy measure is equal for molecules in the two phases at that temperature?

Introduction / Context:
Phase equilibrium at the boiling point means liquid and vapor coexist at the same temperature and pressure with no net mass transfer on average. Molecular interpretations help explain macroscopic equilibrium: temperature reflects the average translational kinetic energy of molecules, while potential energy reflects interactions (which differ across phases).

Given Data / Assumptions:

• Liquid–vapor equilibrium at the boiling temperature.
• System is at steady state with no net heat or mass accumulation.
• Equipartition of translational energy linking temperature and average kinetic energy.

Concept / Approach:
Temperature is directly proportional to the average translational kinetic energy of molecules. At equilibrium, both phases have the same temperature; therefore, the molecules in liquid and in vapor have the same average kinetic energy. However, potential energy differs: vapor molecules are farther apart with higher potential energy on average due to weaker cohesive forces, which is why latent heat is required to vaporize the liquid without changing temperature.

Step-by-Step Solution:

Verification / Alternative check:
Thermodynamically, enthalpy differs by the latent heat h_fg at the same T, which is largely due to a change in potential energy and configuration rather than kinetic energy differences; this supports the conclusion that kinetic energies are equal while total energies are not.

Why Other Options Are Wrong:

• Potential energy and intermolecular forces are not equal across phases; they differ and account for latent heat.
• Total energy includes potential contributions and thus is not equal.
• Vibrational energy only is not the defining equal measure across phases at the same T; the key equality concerns average translational kinetic energy.

Common Pitfalls:
Assuming “same temperature” implies all forms of energy are identical; only the average translational kinetic energy correlates uniquely with temperature in this context.

Final Answer:
kinetic energy