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Thermal energy is a term sometimes used loosely as a synonym for more rigorous thermodynamic quantities such as the (entire) internal energy of a system; or for heat or sensible heat which are defined as types of transfer of energy (just as work is another type of transfer of energy), or for the characteristic energy of a degree of freedom in a thermal system ${\displaystyle kT}$, where ${\displaystyle T}$ is temperature and ${\displaystyle k}$ is the Boltzmann constant. Heat and work depend on the way in which an energy transfer occurred, whereas internal energy is a property of the state of a system and can thus be understood even without knowing how the energy got there.

## Relation to heat and internal energy

Heat is energy transferred spontaneously from a hotter to a colder system or body.[1][2] Heat is energy in transfer, not a property of the system; it is not 'contained' within the boundary of the system.[1] On the other hand, internal energy is a property of a system. In an ideal gas, the internal energy is the statistical mean of the kinetic energy of the gas particles, and it is this kinetic motion that is the source and the effect of the transfer of heat across a system boundary. For this reason, the term "thermal energy" is sometimes used synonymously with internal energy. The term "thermal energy" is also applied to the energy carried by a heat flow.[3]

In many statistical physics texts, "thermal energy" refers to the product of Boltzmann's constant and the absolute temperature, typically written ${\displaystyle kT}$ or ${\displaystyle k_{B}T}$.[4] See the article kT (energy) for more details.

## Historical context

In an 1847 lecture entitled "On Matter, Living Force, and Heat," James Prescott Joule characterised various terms that are closely related to thermal energy and heat. He identified the terms latent heat and sensible heat as forms of heat each affecting distinct physical phenomena, namely the potential and kinetic energy of particles, respectively.[5] He described latent energy as the energy of interaction in a given configuration of particles, i.e. a form of potential energy, and the sensible heat as an energy affecting temperature measured by the thermometer due to the thermal energy, which he called the living force.

## Useless thermal energy

If the minimum temperature of a system's environment is ${\displaystyle T_{e}}$ and the system's entropy is ${\displaystyle S}$, then a part of the system's internal energy amounting to ${\displaystyle S\cdot T_{e}}$ cannot be converted into useful work. This is the difference between the internal energy and the Helmholtz free energy.

## References

1. ^ a b Robert F. Speyer (2012). Thermal Analysis of Materials. Materials Engineering. Marcel Dekker, Inc. p. 2. ISBN 0-8247-8963-6.
2. ^ Thomas W. Leland, Jr., G. A. Mansoori, ed., Basic Principles of Classical and Statistical Thermodynamics (PDF)
3. ^ Ashcroft, Neil; Mermin, N. David (1976). Solid State Physics. Harcourt. p. 20. ISBN 0-03-083993-9. We define the thermal current density ${\displaystyle {\bf {j}}^{q}}$ to be a vector parallel to the direction of heat flow, whose magnitude gives the thermal energy per unit time crossing a unit area perpendicular to the flow.
4. ^ Reichl, Linda E. (2016). A Modern Course in Statistical Physics. John Wiley and Sons. p. 154. ISBN 9783527690466.
Kardar, Mehran (2007). Statistical Physics of Particles. Cambridge University Press. p. 243. ISBN 9781139464871.
Feynman, Richard P. (2000). "The Computing Machines in the Future". Selected Papers of Richard Feynman: With Commentary. World Scientific. ISBN 9789810241315.
Feynman, Richard P. (2018). Statistical Mechanics: A Set of Lectures. CRC Press. p. 265. ISBN 9780429972669.
Kittel, Charles (2012). Elementary Statistical Physics. Courier Corporation. p. 60. ISBN 9780486138909.
5. ^ J. P. Joule (1884), "Matter, Living Force, and Heat", The Scientific Papers of James Prescott Joule, The Physical Society of London, p. 274, retrieved 2 January 2013, I am inclined to believe that both of these hypotheses will be found to hold good,—that in some instances, particularly in the case of sensible heat, or such as is indicated by the thermometer, heat will be found to consist in the living force of the particles of the bodies in which it is induced; whilst in others, particularly in the case of latent heat, the phenomena are produced by the separation of particle from particle, so as to cause them to attract one another through a greater space.