# Thermal energy **Thermal energy** is the name given to the part of a system's energy that is associated with its temperature: the disordered [[Kinetic_energy|kinetic]] and [[Potential_energy|potential]] energy of its molecules, as distinct from the organised motion of the system as a whole. The term is used loosely and in at least three incompatible ways — for the whole [[Internal_energy|internal energy]], for the part of it that changes with temperature, and for [[Heat|heat]] in transit — and careful thermodynamics avoids it for exactly that reason, preferring internal energy for the state and heat for the transfer. In the microsim below the reader runs the inelastic-collision preset from the [[Kinetic_energy|kinetic energy]] track: two bodies collide, [[Momentum|momentum]] survives intact, and the ½·m·v² that momentum cannot keep reappears as m·c·ΔT in the two bodies. The energy bar that vanishes on one side of the impact is drawn back on the other as a temperature rise, and the reader can slide the restitution coefficient from a perfectly elastic collision, where nothing is converted, to a perfectly inelastic one, where the conversion is maximal. The microscopic picture of what "m·c·ΔT" means at the molecular scale is carried by the dense [[Kinetic_theory_of_gases|kinetic theory of gases]] child linked beside it. On the [[Energy]] flagship this article is the child of Part I — Forms, section *Thermal energy* (row E9), the destination of the losses that the kinetic and [[Mechanical_energy|mechanical energy]] sections account for and the source term for the heat-engine sections later on the spine. ## Relation between heat and internal energy The cleanest way through the ambiguity is to keep two ideas apart. Internal energy U is a property of a system in a given state: it has a definite value that does not depend on how the system got there. Heat is not a property at all but a quantity of energy in transit, driven across a boundary by a temperature difference, and it exists only during the transfer. Saying that a hot body "contains heat" is the same category error as saying that a full reservoir contains rainfall. The [[First_law_of_thermodynamics|first law]] links them: ΔU = q + w, the heat absorbed plus the work done on the system. Because there are two ways in and only one accumulator, the same rise in U can be produced by heating or by stirring, which is precisely what makes "how much thermal energy does this body have" an unanswerable question — the body does not record which channel the energy arrived through. Engineering thermodynamics therefore writes energy balances in terms of U, enthalpy and work, and treats heat as the boundary term.[^yan-ch1] "Thermal energy" survives as a working phrase because there is a real quantity behind it in the commonest case. For a substance well away from any [[Phase_transition|phase change]], the internal energy that responds to temperature is the part measured by q = m·c·ΔT, where c is the [[Specific_heat_capacity|specific heat capacity]]. That is the quantity the microsim tracks, and the quantity [[Calorimetry|calorimetry]] measures. The historical unit reflects the same practical focus: one calorie is the energy that raises a gram of water by one degree Celsius, 4.184 J, and one kilocalorie is 4,184 J.[^murphy-cal] The [[Second_law_of_thermodynamics|second law]] adds the constraint that makes thermal energy different in kind from the other forms: heat flows spontaneously only from hot to cold, and converting it back into work costs a fraction of it that no design can avoid.[^yan-ch6] ## Macroscopic thermal energy At the macroscopic level thermal energy is bookkeeping with one coefficient. Raising the temperature of a mass m by ΔT takes m·c·ΔT, and worked examples make the scale concrete: 30 g of water raised 5 °C takes 150 cal, about 628 J; 40 kcal raises 2 kg by 20 °C; 250 g raised 35 °C takes 8.75 kcal, about 36 kJ; and taking half a kilogram from 20 °C to 100 °C takes 40 kcal, or 167 kJ, which a 1,000 W element delivers in 167 s.[^murphy-cal] The microsim's collision preset drives exactly this arithmetic from the other direction. In the textbook crash a 1,200 kg car travelling at 27.6 m/s strikes a 3,000 kg truck and pushes it 10 m; about 60 % of the original kinetic energy, some 273 kJ, does not survive the impact.[^os-crash] That is enough to take 0.8 kg of water from 20 °C to boiling (derived from the 167 kJ example above), and in the real collision it goes into deformation, sound and a temperature rise spread through the structure of both vehicles. The sim draws the missing kinetic-energy bar as a temperature bar with a stated heat capacity, which is ILLUSTRATIVE: it assumes all of the lost energy ends as sensible heat in a stated mass, when a real crash spends much of it on permanent plastic deformation that never comes back as warmth. The ranking is right even where the partition is not — a perfectly inelastic collision loses the most, an elastic one loses none, and momentum is untouched throughout.[^os-collisions] ### Chemical internal energy Part of a system's internal energy is held in its chemical bonds rather than in molecular motion, and it does not respond to temperature in the m·c·ΔT way at all. It appears instead as the [[Enthalpy|enthalpy]] change of a reaction, and it can be very much larger: burning carbon releases 394 kJ per 12 g, about 4 [[Electronvolt|electronvolts]] per [[Atom|atom]], against a thermal share of roughly 0.04 eV per particle at room temperature (derived).[^murphy-carbon][^os-kinetic-theory] That factor of a hundred is why [[Chemical_energy|chemical energy]] can raise temperatures by hundreds of degrees while ordinary heating cannot cause chemistry until the thermal share reaches the activation barrier. Whether chemical internal energy is counted inside "thermal energy" is a matter of where the accounting boundary is drawn. For a non-reacting system it is a constant and drops out of every difference; for a reacting one it must be carried explicitly, which is why thermochemical tables give enthalpies of formation relative to the elements rather than absolute internal energies. ### Potential energy of internal interactions The second non-kinetic contribution is the potential energy of the forces between molecules. In a dilute gas these are negligible, which is what makes the [[Ideal_gas_law|ideal gas]] a good model and what makes its internal energy a function of temperature alone. In a liquid or a solid they are not negligible, and they are what a [[Phase_transition|phase change]] pays for: melting ice or boiling water adds energy at constant temperature, so the [[Latent_heat|latent heat]] goes entirely into pulling molecules apart against their mutual attraction rather than into speeding them up. This is why a temperature reading is an incomplete description of how much energy a body holds. Two kilograms of water at 100 °C and two kilograms of steam at 100 °C are at the same temperature and differ by the [[Enthalpy_of_vaporization|enthalpy of vaporization]], a quantity larger than everything spent heating the water from freezing in the first place. The microsim's temperature bar is therefore drawn with a cap at the boiling point of its working substance and a note that further energy goes to the phase change, not to ΔT. ## Microscopic thermal energy Microscopically, thermal energy is the total of the disordered kinetic and potential energy of the particles, and temperature is the variable that measures the average. For an ideal monatomic gas the connection is exact: the mean translational kinetic energy per molecule is (3/2)·k_B·T, with k_B = 1.38×10⁻²³ J/K, which gives the root-mean-square speed v_rms = √(3·k_B·T/m).[^os-kinetic-theory] For argon at 273 K that is 413 m/s (derived), so the particles of a gas at room temperature move at roughly the speed of sound while the gas as a whole stands still — the distinction between thermal energy and bulk kinetic energy in one number.[^os-kinetic-theory] The distribution around that average is the [[Maxwell–Boltzmann_distribution|Maxwell–Boltzmann]] law, and its tail is what makes temperature chemically and astronomically effective: reaction rates and atmospheric escape are governed by the small fraction of molecules far above the mean, not by the mean itself. The [[Equipartition_theorem|equipartition theorem]] generalises the result, giving each quadratic degree of freedom an average of ½·k_B·T, so a gas with d active degrees of freedom has a molar heat capacity C_V = (d/2)·R. Measurement confirms it and also shows its limits: helium, neon and argon give C_V/R = 1.50, as three translational degrees of freedom require, while carbon monoxide gives 2.50 because two rotational modes have joined in.[^os-kinetic-theory-cv] Hydrogen is the clearest case of all, behaving as d = 3 below about 60 K, as d = 5 from just under 300 K to about 600 K, and as d = 7 only above about 3,000 K, because rotation and then vibration need a minimum quantum of energy before they can absorb any at all.[^os-kinetic-theory-cv] In a solid the carriers are lattice vibrations rather than free molecules, and the quantised vibrational modes are [[Phonon|phonons]]; the classical equipartition result, the [[Dulong–Petit_law|Dulong–Petit law]], holds at high temperature and fails below it, which is what the [[Debye_model|Debye model]] was built to describe. In a metal the conduction electrons carry a further share. In every case the pattern is the same: a degree of freedom contributes its ½·k_B·T only once the temperature is high enough to excite it. ## Thermal current density Thermal energy in motion is described by a flux, the thermal current density, measured in watts per square metre. For [[Thermal_conduction|conduction]] the constitutive law is Fourier's, q = −k·∇T: the flux is proportional to the temperature gradient and points down it, with the [[Thermal_conductivity_and_resistivity|thermal conductivity]] k as the material coefficient. The structure is identical to [[Darcy's_law|Darcy's law]] for flow in porous media and to Ohm's law for current, and the analogy is exploited throughout building physics, where a wall is treated as a stack of thermal resistances in series. Combining Fourier's law with energy conservation gives the heat equation, whose rate constant is the [[Thermal_diffusivity|thermal diffusivity]] — the quantity that decides how fast a temperature change travels rather than how much power flows. Conduction is only one of three channels, and which dominates depends on the medium. [[Convection|Convection]] moves internal energy with the fluid that carries it and outruns conduction in any gas or liquid free to circulate; in soil, where circulation is suppressed, conduction dominates.[^ochsner-soil] [[Thermal_radiation|Thermal radiation]] needs no medium at all and rises very steeply with temperature: the emitted flux is J = ε·σ·T⁴ with σ = 5.67×10⁻⁸ W·m⁻²·K⁻⁴, and the peak wavelength falls as λ_max ≈ 2,900/T in micrometres and kelvin, so the Sun at about 5,780 K peaks near 0.5 µm while the Earth at about 287 K peaks near 10 µm.[^ochsner-radiation] Because the two spectra barely overlap, the planetary energy budget can be kept as "shortwave in, longwave out." At a real surface all three run at once and must sum. The surface energy balance writes net radiation as the sum of the fluxes leaving: R_n = LE + H + G, latent heat plus sensible heat plus conduction into the ground. A measured diurnal cycle over Iowa crop residue ran from about −50 W/m² at night to about +300 W/m² near noon, with the sensible-heat term peaking near +200 W/m² and the latent and ground terms staying at or below +100 W/m².[^ochsner-balance] Every term in that budget is thermal current density, and every one changes sign between day and night, which is the practical reason the phrase "thermal energy" has to be handled with care: what is being tracked is a flow, not a stock. ## See also - [[Kinetic_theory_of_gases]] - [[Internal_energy]] - [[Heat]] - [[Specific_heat_capacity]] - [[Latent_heat]] - [[Thermal_conduction]] - [[Equipartition_theorem]] ## References [^yan-ch1]: Yan, Claire Yu (2022). *Introduction to Engineering Thermodynamics*. Chapter 1 "Basic Concepts and Definitions", pp. 31–58 (system, boundary and state; heat as energy crossing a boundary because of a temperature difference; the first-law energy balance; page to pin — most equation displays were lost in extraction). Portal Book 115, https://open.umn.edu/opentextbooks/textbooks/introduction-to-engineering-thermodynamics [^yan-ch6]: Yan (2022), *Introduction to Engineering Thermodynamics*, Chapter 6 "Entropy and the Second Law of Thermodynamics", pp. 239–348 (the direction of spontaneous heat flow; the Carnot bound η = 1 − T_L/T_H on converting heat to work; an actual engine is always below it, and one above it is impossible; page to pin). Portal Book 115. [^murphy-cal]: Murphy, Thomas (2021). *Energy and Human Ambitions on a Finite Planet*. Chapter 5 "Energy and Fossil Fuels", pp. 93–94 (1 cal = 4.184 J and 1 kcal = 4,184 J; 30 g raised 5 °C takes 150 cal ≈ 628 J; 40 kcal raises 2 kg by 20 °C; 250 g raised 35 °C takes 8.75 kcal ≈ 36 kJ; 0.5 kg from 20 to 100 °C takes 40 kcal = 167 kJ, or 167 s at 1,000 W). Portal Book 097, https://open.umn.edu/opentextbooks/textbooks/energy-and-human-ambitions-on-a-finite-planet [^murphy-carbon]: Murphy (2021), *Energy and Human Ambitions on a Finite Planet*, Chapter 5, p. 98 (394 kJ per 12 g of carbon, divided by 6×10²³, is 6.5×10⁻¹⁹ J, about 4 eV per atom and about 1 eV per C–O bond). Portal Book 097. [^os-collisions]: Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 1*. OpenStax. Chapter 9 "Linear Momentum and Collisions", pp. 417–420 (collisions classified by what happens to kinetic energy — explosion, inelastic, perfectly inelastic, elastic; the quadratic kinetic-energy equation and its unphysical root). Portal Book 077, https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-1 [^os-crash]: Sanny and Ling (2016), *University Physics Volume 1*, Chapter 9, Example 9.13, pp. 421–423 (a 1,200 kg car into a 3,000 kg truck that then slides 10 m at μ = 0.62; the truck's 11.0 m/s, the car's 27.6 m/s and the 60 % kinetic-energy loss, about 273 kJ, are derived from the book's inputs; when friction acts, momentum is conserved only across the impact instant). The equivalent 0.8 kg of water taken to boiling is derived against the 167 kJ example in Murphy p. 94. Portal Book 077. [^os-kinetic-theory]: Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 2*. OpenStax. Chapter 2 "The Kinetic Theory of Gases", pp. 85–89 (p·V = (1/3)·N·m·⟨v²⟩ from elastic wall impulses; combined with p·V = N·k_B·T it gives KE_avg = (3/2)·k_B·T; v_rms = √(3·k_B·T/m) = √(3·R·T/M); k_B = 1.38×10⁻²³ J/K). Argon's 413 m/s at 273 K and the ≈0.04 eV thermal share at room temperature are derived. Portal Book 078, https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-2 [^os-kinetic-theory-cv]: Sanny and Ling (2016), *University Physics Volume 2*, Chapter 2, pp. 97–98 (C_V = (d/2)·R; measured C_V/R of 1.50 for He, Ne and Ar and 2.50 for CO; H₂ behaving as d = 3 below about 60 K, d = 5 from just under 300 K to about 600 K, and d = 7 above about 3,000 K). Portal Book 078. [^ochsner-soil]: Ochsner, Tyson (2019). *Rain or Shine*. Chapter 12 "Surface Energy Balance and Evapotranspiration", p. 294 (conduction dominates heat transfer in soil). Portal Book 119, https://open.umn.edu/opentextbooks/textbooks/rain-or-shine [^ochsner-radiation]: Ochsner (2019), *Rain or Shine*, Chapter 12, pp. 278–280 (Wien's law λ_max = b/T with b = 2,900 µm·K; the Stefan–Boltzmann law J = ε·σ·T⁴ with σ = 5.67×10⁻⁸ W·m⁻²·K⁻⁴; the Sun near 5,780 K peaking at about 0.5 µm and the Earth near 287 K at about 10 µm; emissivities of snow, water, vegetation and soil). Both equation displays were lost in extraction and are quoted here in standard form. Portal Book 119. [^ochsner-balance]: Ochsner (2019), *Rain or Shine*, Chapter 12, pp. 292–295 (R_n = (1 − α)·R_s + R_li − R_lo, positive toward the surface, and R_n = LE + H + G, positive away from it, with both reversing at night; the measured Iowa diurnal cycle over corn residue running from about −50 W/m² at night to about +300 W/m² near noon, H peaking near +200 W/m², LE and G at or below +100 W/m²). Portal Book 119. <!-- MATTERSIM:BEGIN g24 — Matter & Energy Cluster microsim (framework build, specs/sims/Thermal_energy.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework), pending deploy:** *Thermal energy* will play here once `https://wikitube-3d-microsims.netlify.app/matter/Thermal_energy.html` is live. <!-- pending: <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Thermal_energy.html" data-title="Thermal energy"></div> --> <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Thermal_energy) : [Wikitube](https://en.wikitube.io/wiki/Thermal_energy) · pinned revision [1371306137](https://en.wikipedia.org/w/index.php?oldid=1371306137) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Energy]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Energy row E9 · sim pending (matter/Thermal_energy).*