Internal Energy

Every physical system — however large or small — is made up of a huge number of molecules, and those molecules are always in some form of random, disordered motion (and, in liquids and solids, are also held together by intermolecular forces). Internal energy, denoted $U$, is defined as the total energy associated with this random molecular motion and molecular interaction — it is fundamentally different from the macroscopic, ordered kinetic or potential energy of the system as a whole.

A simple example makes the distinction clear: a glass of water resting on a table has zero macroscopic kinetic energy (it isn't moving) and, taking the table as the reference level, essentially zero macroscopic potential energy. Yet the water is far from 'energy-less' — at the molecular level, its water molecules are all moving randomly at speeds set by the water's temperature, exactly as described by the kinetic theory of gases (and, by extension, liquids) covered in Class XI. This random molecular kinetic energy, summed over every molecule in the water, is its internal energy.

The exact composition of a system's internal energy depends on what kind of substance it is:

• For an ideal monatomic gas (such as argon), the molecules are point-like and the internal energy is purely translational kinetic energy — the energy of the atoms' linear, straight-line motion.
• For a polyatomic gas (such as carbon dioxide), the molecules also rotate and vibrate, so their rotational and vibrational kinetic energies must be added to the translational kinetic energy already counted for a monatomic gas. …