# Thermal energy

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Thermal radiation in visible light can be seen on this hot metalwork. Thermal energy would ideally be the amount of heat required to warm the metal to its temperature, but this quantity is not well-defined, as there are many ways to obtain a given body at a given temperature, and each of them may require a different amount of total heat input. Thermal energy, unlike internal energy, is therefore not a state function.

Thermal energy is a term sometimes used to refer to the internal energy present in a system in a state of thermodynamic equilibrium by virtue of its temperature.[1] The average translational kinetic energy possessed by free particles in a system of free particles in thermodynamic equilibrium (as measured in the frame of reference of the center of mass of that system) may also be referred to as the thermal energy per particle.[2]

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. It originates from the individually random, or disordered, motion of particles in a large ensemble. 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 or by increased separation of particles having mutual forces of attraction, the thermal energy is equally partitioned between potential energy and kinetic energy. Thermal energy is thus equally partitioned between all available degrees of freedom of the particles. As noted, these degrees of freedom may include pure translational motion in gases, rotational motion, vibrational motion and associated potential energies. 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).{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }} Macroscopically, the thermal energy of a system at a given temperature is proportional to its heat capacity. However, since the heat capacity differs according to whether or not constant volume or constant pressure is specified, or phase changes permitted, the heat capacity cannot be used to define thermal energy unless it is done in such a way as to insure that only heat gain or loss (not work) makes any changes in the internal energy of the system. Usually, this means specifying the "constant volume heat capacity" of the system so that no work is done. Also the heat capacity of a system for such purposes must not include heat absorbed by any chemical reaction or process. {{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}

## 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.[3] In engineering, the terms "heat" and "heat transfer" are thus used nearly interchangeably, 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.[4] 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. The heat transfer rate or heating rate is the amount of energy per unit time being transferred as heat, or the heat power.

When two thermodynamic systems with different temperatures are brought into diathermic contact, heat flow occurs from the hotter to the colder system, causing a decrease in the thermal energy of the hotter system, and an increase in the thermal energy of the colder system. Heat flow may cause work to be performed on a system by compressing a system's volume, for example. A heat engine uses the movement of thermal energy (heat flow)to do mechanical work. When two systems have reached a thermodynamic equilibrium, they have attained the same temperature and the net exchange of thermal energy vanishes and heat flow ceases.

## Definitions

Thermal energy is the portion of the thermodynamic or internal energy of a system that is responsible for the temperature of the system.[1][3] 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.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }} 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): ${\displaystyle U_{thermal}=C(T)\cdot T.}$ 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.[5] 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.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}

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.[6]

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

${\displaystyle E_{thermal}=f\cdot {\tfrac {1}{2}}kT\,\!}$

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[7]

${\displaystyle U_{thermal}=N\cdot f\cdot {\tfrac {1}{2}}kT.}$

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.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }} Uthermal is not the total energy of a system. Physical systems also contains static potential energy (such as chemical energy) that arises from interactions between particles, nuclear energy associated with atomic nuclei of particles, and even the rest mass energy due to the equivalence of energy and mass.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}

## 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

${\displaystyle E_{kinetic}={\tfrac {1}{2}}mv^{2}\,\!}$

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:

${\displaystyle U_{thermal}={\tfrac {1}{2}}Nm{\overline {v^{2}}}={\tfrac {3}{2}}NkT,}$

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:

${\displaystyle U=U_{thermal}.\;}$

## 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.[8] 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.[9] 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.[3]

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.

## 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.[10] 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.