Thermal energy

In thermal physics, thermal energy is the energy portion of a system that increases with its temperature. In a loose sense, "thermal energy" is a term used to describe the energy content of a system related to heating effects, e.g. temperature increase or decrease. In thermodynamics, thermal energy is the internal energy present in a system in a state of thermodynamic equilibrium by virtue of its temperature. The term is not widely used, however, in a rigorous sense, owing to the result that the phrase "thermal (heat) energy" is counter-intuitive. That is, "thermal energy" can only be defined as any spontaneous flow of energy (energy in transit) from one object to another, caused by a difference in temperature between two objects; thus, an object cannot possess "heat". This is explained by the second law of thermodynamics. Hence, by extrapolation, it is difficult to define quantities of heat energy (thermal energy). In isolated cases, however, a few definitions do exist.

Internal energy
Internal energy – the sum of all microscopic forms of energy of a system. It is related to the molecular structure and the degree of molecular activity and may be viewed as the sum of kinetic and potential energies of the molecules; it consists of the following types of energies:

System of N particles
According to the equipartition theorem, it is possible to define thermal energy. In a system of N molecules, each with f degrees of freedom, and if there are no other (non-quadratic) temperature-dependent forms of energy, then the total thermal energy of the system is:


 * $$U_{thermal} = N \cdot f \cdot \frac{1}{2}kT.$$

To note, Uthermal is almost never the total energy of a system; for instance, there can be static energy that doesn't change with temperature, such as bond energy or rest energy (E=mc2).

Other definitions
Thermal energy per particle is also called the average translational kinetic energy possessed by free particles given by equipartition of energy.

Thermal energy is the difference between the internal energy of an object and the amount that it would have at absolute zero. It includes the quantity of kinetic energy due to the motion of the internal particles of an object, and is increased by heating and reduced by cooling.

In a monatomic ideal gas, the thermal energy is exactly given by the kinetic energy of the constituent particles.