Microscopic interpretation of internal energy (U): which components make up the internal energy of a substance?

Introduction / Context:
Internal energy is the sum of all microscopic forms of energy in a system and is fundamental in the first law of thermodynamics. Understanding its components clarifies heat capacities, phase behavior, and energy storage in molecules.

Given Data / Assumptions:

• Consider molecules in any phase (gas, liquid, solid).
• Assume classical contributions plus quantum restrictions at low temperature.

Concept / Approach:
For molecules, internal energy includes translational energy (center-of-mass motion), rotational energy (angular motion), and vibrational energy (bond stretching/bending). In condensed phases, additional contributions arise from intermolecular potentials; in solids, lattice vibrations dominate. At low temperatures, some modes may be “frozen out,” but in general all three listed categories contribute to U.

Step-by-Step Solution:

Verification / Alternative check:
Heat capacity models (equipartition, statistical mechanics) partition energy into these modes, supporting the comprehensive view.

Why Other Options Are Wrong:

• Selecting only one or two modes ignores complete molecular energy storage.
• None of these: contradicts basic molecular theory.

Common Pitfalls:
For monatomic ideal gases, only translational modes contribute significantly—do not generalize that case to all substances.

Final Answer:
all (a), (b) & (c)