Thermal energy is the part of the total potential energy and kinetic energy of an object or sample of matter that results in the system temperature. This quantity may be difficult to determine or even meaningless unless the system has attained its temperature only through warming, and not been subjected to work input or output, or any other energy-changing processes. The internal energy of a system, also often called the thermodynamic energy, includes other forms of energy in a thermodynamic system in addition to thermal energy, namely forms of potential energy that do not influence temperature and do not absorb heat, such as the chemical energy stored in its molecular structure and electronic configuration, and the nuclear binding energy that binds the sub-atomic particles of matter.
Microscopically, the thermal energy may include both the kinetic energy and potential energy of a system's constituent particles, which may be atoms, molecules, electrons, or particles in plasmas. It originates from the individually random, or disordered, motion of particles in a large ensemble, as consequence of absorbing heat. In ideal monatomic gases, thermal energy is entirely kinetic energy. In other substances, in cases where some of thermal energy is stored in atomic vibration, this vibrational part of the thermal energy is stored equally partitioned between potential energy of atomic vibration, and kinetic energy of atomic vibration. Thermal energy is thus equally partitioned between all available quadratic degrees of freedom of the particles. As noted, these degrees of freedom may include pure translational motion in gases, in rotational states, and as potential and kinetic energy in normal modes of vibrations in intermolecular or crystal lattice vibrations. In general, due to quantum mechanical reasons, the availability of any such degrees of freedom is a function of the energy in the system, and therefore depends on the temperature (see heat capacity for discussion of this phenomenon).
Macroscopically, the thermal energy of a system at a given temperature is related proportionally to its heat capacity. However, since the heat capacity differs according to whether or not constant volume or constant pressure is specified, or phase changes permitted, the heat capacity cannot be used define thermal energy unless it is done in such a way as to insure that only heat gain or loss (not work) makes any changes in the internal energy of the system. Usually, this means specifying the "constant volume heat capacity" of the system so that no work is done. Also the heat capacity of a system for such purposes must not include heat absorbed by any chemical reaction or process.
Thermal energy is not a state function, or a property of a system, since the total thermal energy needed to warm a system to a given temperature depends on the path taken to attain the temperature, unless all forms of work and chemical potential change in the system are zero or negligible (in which case thermal energy is a subset of the internal energy). Thus, thermal energy is process-dependent except in systems in which processes to change internal energy other than heating, can be neglected. Nevertheless, when this is true, thermal energy and heat capacity may be a useful concept in the study of heat transfer in solids and liquids, in engineering and other disciplines.
Differentiation from heat 
Heat, in the strict use in physics, is characteristic only of a process, i.e. it is absorbed or produced as an energy exchange, always as a result of a temperature difference. Heat is thermal energy in the process of transfer or conversion across a boundary of one region of matter to another, as a result of a temperature difference. In engineering, the terms "heat" and "heat transfer" are thus used nearly interchangeably (heat transfer is the rate of heat flow in time, or the heat power), since heat is always understood to be in the process of transfer. The energy transferred by heat is called by other terms (such as thermal energy or latent energy) when this energy is no longer in net transfer, and has become static. Thus, heat is not a static property of matter. Matter does not contain heat, but rather thermal energy, and even the thermal energy is subject to transformations into and out of other types of energy, and so can be considered to be "conserved" only when these processes are small.
When two thermodynamic systems with different temperatures are brought into diathermic contact, they spontaneously exchange energy as heat, which is a transfer of thermal energy from the system of higher temperature to the colder system. Heat may cause work to be performed on a system, for example, in form of volume or pressure changes. This work may be used in heat engines to convert thermal energy into other forms of energy. In geothermal power plants it is used for the generation of electricity. When two systems have reached a thermodynamic equilibrium, they have attained the same exact temperature and the net exchange of thermal energy vanishes, and heat flow ceases.
Thermal energy is the portion of the thermodynamic or internal energy of a system that is responsible for the temperature of the system. The thermal energy of a system scales with its size and is therefore an extensive property. It is not a state function of the system unless the system has been constructed so that all changes in internal energy are due to changes in thermal energy, as a result of heat transfer (not work). Otherwise thermal energy is dependent on the way or method by which the system attained its temperature.
From a macroscopic thermodynamic description, the thermal energy of a system is given by its constant volume specific heat capacity C(T), a temperature coefficient also called thermal capacity, at any given absolute temperature (T):
The heat capacity is a function of temperature itself, and is typically measured and specified for certain standard conditions and a specific amount of substance (molar heat capacity) or mass units (specific heat capacity). At constant volume (V), CV it is the temperature coefficient of energy. In practice, given a narrow temperature range, for example the operational range of a heat engine, the heat capacity of a system is often constant, and thus thermal energy changes are conveniently measured as temperature fluctuations in the system.
In the microscopical description of statistical physics, the thermal energy is identified with the mechanical kinetic energy of the constituent particles or other forms of kinetic energy associated with quantum-mechanical microstates.
The distinguishing difference between the terms kinetic energy and thermal energy is that thermal energy is the mean energy of disordered, i.e. random, motion of the particles or the oscillations in the system. The conversion of energy of ordered motion to thermal energy results from collisions.
All kinetic energy is partitioned into the degrees of freedom of the system. The average energy of a single particle with f quadratic degrees of freedom in a thermal bath of temperature T is a statistical mean energy given by the equipartition theorem as
where k is the Boltzmann constant. The total thermal energy of a sample of matter or a thermodynamic system is consequently the average sum of the kinetic energies of all particles in the system. Thus, for a system of N particles its thermal energy is
For gaseous systems, the factor f, the number of degrees of freedom, commonly has the value 3 in the case of the monatomic gas, 5 for many diatomic gases, and 7 for larger molecules at ambient temperatures. In general however, it is a function of the temperature of the system as internal modes of motion, vibration, or rotation become available in higher energy regimes.
Uthermal is not the total energy of a system. Physical systems also contains static potential energy (such as chemical energy) that arises from interactions between particles, nuclear energy associated with atomic nuclei of particles, and even the rest mass energy due to the equivalence of energy and mass.
Thermal energy of the ideal gas 
Thermal energy is most easily defined in the context of the ideal gas, which is well approximated by a monatomic gas at low pressure. The ideal gas is a gas of particles considered as point objects of perfect spherical symmetry that interact only by elastic collisions and fill a volume such that their mean free path between collisions is much larger than their diameter.
The mechanical kinetic energy of a single particle is
where m is the particle's mass and v is its velocity. The thermal energy of the gas sample consisting of N atoms is given by the sum of these energies, assuming no losses to the container or the environment:
where the line over the velocity term indicates that the average value is calculated over the entire ensemble. The total thermal energy of the sample is proportional to the macroscopic temperature by a constant factor accounting for the three translational degrees of freedom of each particle and the Boltzmann constant. The Boltzmann constant converts units between the microscopic model and the macroscopic temperature. This formalism is the basic assumption that directly yields the ideal gas law and it shows that for the ideal gas, the internal energy U consists only of its thermal energy:
Historical context 
In an 1847 lecture entitled On Matter, Living Force, and Heat, James Prescott Joule characterized 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 effecting distinct physical phenomena, namely the potential and kinetic energy of particles, respectively. He describes 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.
Distinction of thermal energy and heat 
In thermodynamics, heat must always be defined as energy in exchange between two systems, or a single system and its surroundings. According to the zeroth law of thermodynamics, heat is exchanged between thermodynamic systems in thermal contact only if their temperatures are different, as this is the condition when the net exchange of thermal energy is non-zero. For the purpose of distinction, a system is defined to be enclosed by a well-characterized boundary. If heat traverses the boundary in direction into the system, the internal energy change is considered to be a positive quantity, while exiting the system, it is negative. As a process variable, heat is never a property of the system, nor is it contained within the boundary of the system.
In contrast to heat, thermal energy exists on both sides of a boundary. It is the statistical mean of the microscopic fluctuations of the kinetic energy of the systems' particles, and it is the source and the effect of the transfer of heat across a system boundary. Statistically, thermal energy is always exchanged between systems, even when the temperatures on both sides is the same, i.e. the systems are in thermal equilibrium. However, at equilibrium, the net exchange of thermal energy is zero, and therefore there is no heat.
Thermal energy may be increased in a system by other means than heat, for example when mechanical or electrical work is performed on the system. No qualitative difference exists between the thermal energy added by other means. Thermal energy is not a state function, although it may be closely related to the internal energy of some systems, which is a state function. There is also no need in classical thermodynamics to characterize the thermal energy in terms of atomic or molecular behavior. A change in thermal energy induced in a system is the product of the change in entropy and the temperature of the system.
Heat exchanged across a boundary may cause changes other than a change in temperature. For example, it may cause phase transitions, such as melting or evaporation, which are changes in the configuration of a material. Since such an energy exchange is not observable by a change in temperature, it is called a latent heat and represents a change in the potential energy of the system.
Rather than being itself the thermal energy involved in a transfer, heat is sometimes also understood as the process of that transfer, i.e. heat functions as a verb.
Today's narrow definition of heat in physics contrasts with its use in common language, in some engineering disciplines, and in the historical scientific development of thermodynamics in the caloric theory of heat. The phenomenon of heat in these instances is today properly identified as the entropy.
The origin of heat energy on Earth 
Earth's proximity to the Sun is the reason that almost everything near Earth's surface is warm with a temperature substantially above absolute zero. Solar radiation constantly replenishes heat energy that Earth loses into space and a relatively stable state of near equilibrium is achieved. Because of the wide variety of heat diffusion mechanisms (one of which is black-body radiation which occurs at the speed of light), objects on Earth rarely vary too far from the global mean surface and air temperature of 287 to 288 K (14 to 15 °C). The more an object's or system's temperature varies from this average, the more rapidly it tends to come back into equilibrium with the ambient environment.
Thermal energy of individual particles 
The term thermal energy is also often used as a property of single particles to designate the kinetic energy of the particles. An example is the description of thermal neutrons having a certain thermal energy, which means that the kinetic energy of the particle is equivalent to the temperature of its surroundings.
See also 
- Thermal energy entry in Britannica Online
- Robert F. Speyer (2012). Thermal Analysis of Materials. Materials Engineering. Marcel Dekker, Inc. p. 2. ISBN 0-8247-8963-6.
- Frank P. Incropera; David P. De Witt and D. P. Dewitt (1990). Fundamentals of Heat and Mass Transfer (3rd ed.). John Wiley & Sons. p. 2. ISBN 0-471-51729-1. See box definition: "Heat transfer (or heat) is energy in transit due to a temperature difference." See page 14 for the definition of the thermal component of the thermodynamic internal energy.
- Hans U. Fuchs (2010). The Dynamics of Heat (2 ed.). Springer. p. 211. ISBN 978-1-4419-7603-1. Unknown parameter
- S. Blundell, K. Blundell (2006). Concepts in Thermal Physics. Oxford University Press. p. 366. ISBN 0-19-856769-3.
- D.V. Schroeder (1999). An Introduction to Thermal Physics. Addison-Wesley. p. 15. ISBN 0-201-38027-7.
- J. P. Joule (1884), The Scientific Paper of James Prescott Joule, The Physical Society of London, p. 274, "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.", Lecture on Matter, Living Force, and Heat. May 5 and 12, 1847
- Thomas W. Leland, Jr., G. A. Mansoori, ed., Basic Principles of Classical and Statistical Thermodynamics
- Hans U. Fuchs (2010). The Dynamics of Heat (2 ed.). Springer. p. 3. ISBN 978-1-4419-7603-1. Unknown parameter
- Friedrich Herrmann, Entropy from the Beginning, Plenary Lecture
- The deepest ocean depths (3 to 10 km) are no colder than about 274.7–275.7 K (1.5–2.5 °C). Even the world-record cold surface temperature established on July 21, 1983 at Vostok Station, Antarctica is 184 K (a reported value of −89.2 °C). The residual heat of gravitational contraction left over from earth's formation, tidal friction, and the decay of radioisotopes in earth's core provide insufficient heat to maintain earth's surface, oceans, and atmosphere "substantially above" absolute zero in this context. Also, the qualification of "most-everything" provides for the exclusion of lava flows, which derive their temperature from these deep-earth sources of heat.