Internal Energy

Internal energy is the sum of kinetic and potential energies of all molecules in a system. Kinetic energy includes translational, rotational, and vibrational contributions, while potential energy arises from interactions between molecules. For an ideal monatomic gas, only translational kinetic energy contributes since atoms have no rotational or vibrational energy and no interatomic interactions exist.

For an ideal monatomic gas, internal energy equals three-halves NkBT, where N is the number of atoms, kB is Boltzmann's constant, and T is temperature. Since there are no interatomic interactions, potential energy is zero. When heated at constant volume, only kinetic energy increases, making temperature the sole determinant of internal energy.

Internal energy is path-independent because it is a state function. Boiling water cooled to room temperature has the same internal energy as water from melted ice at the same temperature. The change in internal energy depends only on initial and final states, not on the thermodynamic path taken between them.

When heat Q is added to a system and no work is done, internal energy increases by an amount equal to Q. Conversely, when a system does work W by expanding against its surroundings with no heat added, energy leaves the system and internal energy decreases by W. These relationships form the basis of the first law of thermodynamics problem solving.

Monatomic ideal gases have internal energy from translational kinetic energy only, expressed as three-halves NkBT. Polyatomic molecules include rotational and vibrational kinetic energy contributions. Additionally, polyatomic systems may have potential energy from intermolecular interactions, making their internal energy more complex and temperature-dependent in different ways.

No, neither the system's location nor its bulk motion affects internal energy. Internal energy depends only on the microscopic kinetic and potential energies of molecules within the system. A stationary container and a moving container at the same temperature have identical internal energies regardless of their position or velocity.

For an ideal monatomic gas, internal energy depends only on temperature and is independent of pressure and volume. Changes in pressure and volume do not directly alter internal energy unless they cause temperature changes. This independence distinguishes ideal gases from real gases and is fundamental to understanding adiabatic processes for an ideal gas.